www.KMA.zcu.cz
Reparameterization of rational hypersurfaces with
respect to their convolutions
Miroslav Lávička1
Email: lavicka@kma.zcu.cz
1 Department of Mathematics, Faculty of Applied Sciences
University of West Bohemia in Plzeň, Czech Republic
Centre of Mathematics for Applications, University of Oslo
Geometry Seminar – May 12, 2009
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
1 / 46
www.KMA.zcu.cz
Outline
1 Introduction
Convolution hypersurfaces
Coherent parameterizations
Statement of the problem
2 Reparameterizing one of hypersurfaces
Computing parametric convolutions
Convolution degree of parameterization
Method 1 – Example(s)
3 Finding a new parameterization of one of hypersurfaces
Extended convolution ideal
Method 2 – Example(s)
4 Reparameterizing both hypersurfaces
Motivation: Why the previous stuff is not enough?
Explicit and implicit SF representation
Convolution theory via ISF representation
Method 3 – Examples
5 Finding new parameterizations of both hypersurfaces
Method 4 - Examples
6 Conclusion
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
2 / 46
www.KMA.zcu.cz
Outline
1 Introduction
Convolution hypersurfaces
Coherent parameterizations
Statement of the problem
2 Reparameterizing one of hypersurfaces
Computing parametric convolutions
Convolution degree of parameterization
Method 1 – Example(s)
3 Finding a new parameterization of one of hypersurfaces
Extended convolution ideal
Method 2 – Example(s)
4 Reparameterizing both hypersurfaces
Motivation: Why the previous stuff is not enough?
Explicit and implicit SF representation
Convolution theory via ISF representation
Method 3 – Examples
5 Finding new parameterizations of both hypersurfaces
Method 4 - Examples
6 Conclusion
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
2 / 46
www.KMA.zcu.cz
Outline
1 Introduction
Convolution hypersurfaces
Coherent parameterizations
Statement of the problem
2 Reparameterizing one of hypersurfaces
Computing parametric convolutions
Convolution degree of parameterization
Method 1 – Example(s)
3 Finding a new parameterization of one of hypersurfaces
Extended convolution ideal
Method 2 – Example(s)
4 Reparameterizing both hypersurfaces
Motivation: Why the previous stuff is not enough?
Explicit and implicit SF representation
Convolution theory via ISF representation
Method 3 – Examples
5 Finding new parameterizations of both hypersurfaces
Method 4 - Examples
6 Conclusion
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
2 / 46
www.KMA.zcu.cz
Outline
1 Introduction
Convolution hypersurfaces
Coherent parameterizations
Statement of the problem
2 Reparameterizing one of hypersurfaces
Computing parametric convolutions
Convolution degree of parameterization
Method 1 – Example(s)
3 Finding a new parameterization of one of hypersurfaces
Extended convolution ideal
Method 2 – Example(s)
4 Reparameterizing both hypersurfaces
Motivation: Why the previous stuff is not enough?
Explicit and implicit SF representation
Convolution theory via ISF representation
Method 3 – Examples
5 Finding new parameterizations of both hypersurfaces
Method 4 - Examples
6 Conclusion
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
2 / 46
www.KMA.zcu.cz
Outline
1 Introduction
Convolution hypersurfaces
Coherent parameterizations
Statement of the problem
2 Reparameterizing one of hypersurfaces
Computing parametric convolutions
Convolution degree of parameterization
Method 1 – Example(s)
3 Finding a new parameterization of one of hypersurfaces
Extended convolution ideal
Method 2 – Example(s)
4 Reparameterizing both hypersurfaces
Motivation: Why the previous stuff is not enough?
Explicit and implicit SF representation
Convolution theory via ISF representation
Method 3 – Examples
5 Finding new parameterizations of both hypersurfaces
Method 4 - Examples
6 Conclusion
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
2 / 46
www.KMA.zcu.cz
Outline
1 Introduction
Convolution hypersurfaces
Coherent parameterizations
Statement of the problem
2 Reparameterizing one of hypersurfaces
Computing parametric convolutions
Convolution degree of parameterization
Method 1 – Example(s)
3 Finding a new parameterization of one of hypersurfaces
Extended convolution ideal
Method 2 – Example(s)
4 Reparameterizing both hypersurfaces
Motivation: Why the previous stuff is not enough?
Explicit and implicit SF representation
Convolution theory via ISF representation
Method 3 – Examples
5 Finding new parameterizations of both hypersurfaces
Method 4 - Examples
6 Conclusion
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
2 / 46
Introduction
www.KMA.zcu.cz
Outline
1 Introduction
Convolution hypersurfaces
Coherent parameterizations
Statement of the problem
2 Reparameterizing one of hypersurfaces
Computing parametric convolutions
Convolution degree of parameterization
Method 1 – Example(s)
3 Finding a new parameterization of one of hypersurfaces
Extended convolution ideal
Method 2 – Example(s)
4 Reparameterizing both hypersurfaces
Motivation: Why the previous stuff is not enough?
Explicit and implicit SF representation
Convolution theory via ISF representation
Method 3 – Examples
5 Finding new parameterizations of both hypersurfaces
Method 4 - Examples
6 Conclusion
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
3 / 46
Introduction
www.KMA.zcu.cz
Convolution hypersurface – definition
Definition
Let A and B be smooth hypersurfaces in the affine space Rn+1 . The convolution
hypersurface C = A ? B is defined as
C = {a + b | a ∈ A, b ∈ B and α(a) k β(b)},
(1)
where α(a) and β(b) are the tangent hyperplanes of A and B at points a ∈ A
and b ∈ B. The points a, b are called coherent points.
Pointwise construction
c=a+b
b
a
B
A
Remark
There is a close relation to the offset
theory
I classical offsets – convolutions
with spheres
I general offsets – convolutions
with arbitrary surfaces
0
Reparameterization of hypersurfaces with respect to convolutions
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Introduction
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Convolution hypersurface – definition
Definition
Let A and B be smooth hypersurfaces in the affine space Rn+1 . The convolution
hypersurface C = A ? B is defined as
C = {a + b | a ∈ A, b ∈ B and α(a) k β(b)},
(1)
where α(a) and β(b) are the tangent hyperplanes of A and B at points a ∈ A
and b ∈ B. The points a, b are called coherent points.
Pointwise construction
c=a+b
b
a
B
A
Remark
There is a close relation to the offset
theory
I classical offsets – convolutions
with spheres
I general offsets – convolutions
with arbitrary surfaces
0
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
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Introduction
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Properties of convolutions
Fundamental properties of convolutions
1
Relation for points being coherent is a relation of equivalence
2
If a ∈ A and b ∈ B are coherent points then a ∈ A (or b ∈ B) and
c = a + b ∈ C = A ? B are coherent points
3
A?B = B?A
4
(A ∪ B) ? C = (A ? C) ∪ (B ? C),
5
Despite A, B being irreducible and rational, C = A ? B does not have to be
rational and can be reducible or irreducible
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
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Introduction
www.KMA.zcu.cz
Properties of convolutions
Fundamental properties of convolutions
1
Relation for points being coherent is a relation of equivalence
2
If a ∈ A and b ∈ B are coherent points then a ∈ A (or b ∈ B) and
c = a + b ∈ C = A ? B are coherent points
3
A?B = B?A
4
(A ∪ B) ? C = (A ? C) ∪ (B ? C),
5
Despite A, B being irreducible and rational, C = A ? B does not have to be
rational and can be reducible or irreducible
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
5 / 46
Introduction
www.KMA.zcu.cz
Properties of convolutions
Fundamental properties of convolutions
1
Relation for points being coherent is a relation of equivalence
2
If a ∈ A and b ∈ B are coherent points then a ∈ A (or b ∈ B) and
c = a + b ∈ C = A ? B are coherent points
3
A?B = B?A
4
(A ∪ B) ? C = (A ? C) ∪ (B ? C),
5
Despite A, B being irreducible and rational, C = A ? B does not have to be
rational and can be reducible or irreducible
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
5 / 46
Introduction
www.KMA.zcu.cz
Properties of convolutions
Fundamental properties of convolutions
1
Relation for points being coherent is a relation of equivalence
2
If a ∈ A and b ∈ B are coherent points then a ∈ A (or b ∈ B) and
c = a + b ∈ C = A ? B are coherent points
3
A?B = B?A
4
(A ∪ B) ? C = (A ? C) ∪ (B ? C),
5
Despite A, B being irreducible and rational, C = A ? B does not have to be
rational and can be reducible or irreducible
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
5 / 46
Introduction
www.KMA.zcu.cz
Properties of convolutions
Fundamental properties of convolutions
1
Relation for points being coherent is a relation of equivalence
2
If a ∈ A and b ∈ B are coherent points then a ∈ A (or b ∈ B) and
c = a + b ∈ C = A ? B are coherent points
3
A?B = B?A
4
(A ∪ B) ? C = (A ? C) ∪ (B ? C),
5
Despite A, B being irreducible and rational, C = A ? B does not have to be
rational and can be reducible or irreducible
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
5 / 46
Introduction
www.KMA.zcu.cz
Components of convolution hypersurface
Definition
An irreducible component C0 of the convolution hypersurface C = A ? B is called
simple, special, or degenerate if there exists a dense set S ⊂ C0 such that every
c ∈ S is generated by exactly one, more than one but finitely many, or infinitely
many pair(s) of coherent points a ∈ A, b ∈ B and c = a + b.
Remark
I
We exclude from our further considerations components which are
degenerated (i.e., with dimensions less than n).
I
Every C = A ? B has at least one non-special (i.e., simple) component.
I
If C = A ? B is irreducible then it is simple.
I
Special components of convolution hypersurfaces are typically those ones
when C = (A ? B) ? B − , where B − is centrally symmetric with B – e.g. in
case when offsets to offsets are constructed.
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
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Introduction
www.KMA.zcu.cz
Components of convolution hypersurface
Definition
An irreducible component C0 of the convolution hypersurface C = A ? B is called
simple, special, or degenerate if there exists a dense set S ⊂ C0 such that every
c ∈ S is generated by exactly one, more than one but finitely many, or infinitely
many pair(s) of coherent points a ∈ A, b ∈ B and c = a + b.
Remark
I
We exclude from our further considerations components which are
degenerated (i.e., with dimensions less than n).
I
Every C = A ? B has at least one non-special (i.e., simple) component.
I
If C = A ? B is irreducible then it is simple.
I
Special components of convolution hypersurfaces are typically those ones
when C = (A ? B) ? B − , where B − is centrally symmetric with B – e.g. in
case when offsets to offsets are constructed.
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
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Introduction
www.KMA.zcu.cz
Convolutions of rational hypersurfaces
I
Let be given two regular rational parametric hypersurfaces a(u1 , . . . , un ),
b(s1 , . . . , sn ).
I
In what follows, we denote ū = (u1 , . . . , un ), s̄ = (s1 , . . . , sn ), etc.
I
The normal direction is given simply as the unique direction perpendicular to
all partial derivative vectors, and the convolution formula becomes
∂a
∂b
∂b
∂a
A?B = a(ū) + b(s̄) :
(ū), . . . ,
(ū) =
(s̄), . . . ,
(s̄)
∂u1
∂un
∂s1
∂sn
I
The problem of finding mutually corresponding parametric values ū, s̄ is
highly nonlinear and thus the computation of exact convolutions is too
complicated – in practice various approximation algorithms are used instead.
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
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Introduction
www.KMA.zcu.cz
Convolutions of rational hypersurfaces
I
Let be given two regular rational parametric hypersurfaces a(u1 , . . . , un ),
b(s1 , . . . , sn ).
I
In what follows, we denote ū = (u1 , . . . , un ), s̄ = (s1 , . . . , sn ), etc.
I
The normal direction is given simply as the unique direction perpendicular to
all partial derivative vectors, and the convolution formula becomes
∂a
∂b
∂b
∂a
A?B = a(ū) + b(s̄) :
(ū), . . . ,
(ū) =
(s̄), . . . ,
(s̄)
∂u1
∂un
∂s1
∂sn
I
The problem of finding mutually corresponding parametric values ū, s̄ is
highly nonlinear and thus the computation of exact convolutions is too
complicated – in practice various approximation algorithms are used instead.
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
7 / 46
Introduction
www.KMA.zcu.cz
Convolutions of rational hypersurfaces
I
Let be given two regular rational parametric hypersurfaces a(u1 , . . . , un ),
b(s1 , . . . , sn ).
I
In what follows, we denote ū = (u1 , . . . , un ), s̄ = (s1 , . . . , sn ), etc.
I
The normal direction is given simply as the unique direction perpendicular to
all partial derivative vectors, and the convolution formula becomes
∂a
∂b
∂b
∂a
A?B = a(ū) + b(s̄) :
(ū), . . . ,
(ū) =
(s̄), . . . ,
(s̄)
∂u1
∂un
∂s1
∂sn
I
The problem of finding mutually corresponding parametric values ū, s̄ is
highly nonlinear and thus the computation of exact convolutions is too
complicated – in practice various approximation algorithms are used instead.
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
7 / 46
Introduction
www.KMA.zcu.cz
Convolutions of rational hypersurfaces
I
Let be given two regular rational parametric hypersurfaces a(u1 , . . . , un ),
b(s1 , . . . , sn ).
I
In what follows, we denote ū = (u1 , . . . , un ), s̄ = (s1 , . . . , sn ), etc.
I
The normal direction is given simply as the unique direction perpendicular to
all partial derivative vectors, and the convolution formula becomes
∂a
∂b
∂b
∂a
A?B = a(ū) + b(s̄) :
(ū), . . . ,
(ū) =
(s̄), . . . ,
(s̄)
∂u1
∂un
∂s1
∂sn
I
The problem of finding mutually corresponding parametric values ū, s̄ is
highly nonlinear and thus the computation of exact convolutions is too
complicated – in practice various approximation algorithms are used instead.
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
7 / 46
Introduction
www.KMA.zcu.cz
Coherent parameterizations – definition
Definition
The rational parameterizations a(t̄), b(t̄) of the given hypersurfaces
A, B ⊂ Rn+1 over the same parameter domain Ω ⊂ Rn are called coherent iff the
convolution condition (parallel normals) is automatically satisfied for every
parameter value, i.e.,
∂a
∂a
∂b
∂b
(t̄), . . . ,
(t̄) =
(t̄), . . . ,
(t̄) .
∂t1
∂tn
∂t1
∂tn
Remark
Having coherent parameterizations of hypersurfaces then a parameterization of
(a component of) the convolution is easily computed by
c(t̄) = a(t̄) + b(t̄).
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
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Introduction
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Non-coherent parameterizations
4
1.0
2
0.5
2
-2.0
-1.5
-1.0
4
6
8
-0.5
-0.5
-2
-1.0
-4
Cardioid
a(u) =
2u2 −2u4
4u3
, − u4 +2u
2 +1
u4 +2u2 +1
with the normal vectors for
u = −2, −1, 1, 2
>
Reparameterization of hypersurfaces with respect to convolutions
Tschirnhausen cubic
>
3
b(s) = s2 , s −
s
3
with the normal vectors for
s = −2, −1, 1, 2
May 12, 2009
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Introduction
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Coherent parameterizations
4
1.0
2
0.5
2
-2.0
-1.5
-1.0
4
6
8
-0.5
-0.5
-2
-1.0
-4
Cardioid and Tschirnhausen cubic
The cardioid and the Tschirnhausen cubic parameterized by their coherent
parameterizations ã(t), b̃(t) with the normal vectors for t = −2/3, −2/5, 2/5, 2/3
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
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Introduction
www.KMA.zcu.cz
Computing coherent parameterizations of a(ū) and b(s̄)
1
Rationally reparameterizing one of hypersurfaces. There exists a
reparameterization ū = φ(s̄) giving coherent parameterizations
a(φ(s̄)), b(s̄).
2
Finding a new parameterization of one of hypersurfaces. There exists a new
parameterization ã(s̄) so that
ã(s̄), b(s̄)
3
are coherent. However, Method 1 cannot be used
Reparameterizing both hypersurfaces. There exist reparameterizations
ū 7→ φ(t̄), s̄ 7→ ψ(t̄) giving coherent parameterizations
a(φ(t̄)), b(ψ(t̄)).
4
Finding new parameterizations of both hypersurfaces. There exist new
parameterizations
ã(t̄), b̃(t̄)
which are coherent. However, Method 3 cannot be used.
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
11 / 46
Introduction
www.KMA.zcu.cz
Computing coherent parameterizations of a(ū) and b(s̄)
1
Rationally reparameterizing one of hypersurfaces. There exists a
reparameterization ū = φ(s̄) giving coherent parameterizations
a(φ(s̄)), b(s̄).
2
Finding a new parameterization of one of hypersurfaces. There exists a new
parameterization ã(s̄) so that
ã(s̄), b(s̄)
3
are coherent. However, Method 1 cannot be used
Reparameterizing both hypersurfaces. There exist reparameterizations
ū 7→ φ(t̄), s̄ 7→ ψ(t̄) giving coherent parameterizations
a(φ(t̄)), b(ψ(t̄)).
4
Finding new parameterizations of both hypersurfaces. There exist new
parameterizations
ã(t̄), b̃(t̄)
which are coherent. However, Method 3 cannot be used.
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
11 / 46
Introduction
www.KMA.zcu.cz
Computing coherent parameterizations of a(ū) and b(s̄)
1
Rationally reparameterizing one of hypersurfaces. There exists a
reparameterization ū = φ(s̄) giving coherent parameterizations
a(φ(s̄)), b(s̄).
2
Finding a new parameterization of one of hypersurfaces. There exists a new
parameterization ã(s̄) so that
ã(s̄), b(s̄)
3
are coherent. However, Method 1 cannot be used
Reparameterizing both hypersurfaces. There exist reparameterizations
ū 7→ φ(t̄), s̄ 7→ ψ(t̄) giving coherent parameterizations
a(φ(t̄)), b(ψ(t̄)).
4
Finding new parameterizations of both hypersurfaces. There exist new
parameterizations
ã(t̄), b̃(t̄)
which are coherent. However, Method 3 cannot be used.
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
11 / 46
Introduction
www.KMA.zcu.cz
Computing coherent parameterizations of a(ū) and b(s̄)
1
Rationally reparameterizing one of hypersurfaces. There exists a
reparameterization ū = φ(s̄) giving coherent parameterizations
a(φ(s̄)), b(s̄).
2
Finding a new parameterization of one of hypersurfaces. There exists a new
parameterization ã(s̄) so that
ã(s̄), b(s̄)
3
are coherent. However, Method 1 cannot be used
Reparameterizing both hypersurfaces. There exist reparameterizations
ū 7→ φ(t̄), s̄ 7→ ψ(t̄) giving coherent parameterizations
a(φ(t̄)), b(ψ(t̄)).
4
Finding new parameterizations of both hypersurfaces. There exist new
parameterizations
ã(t̄), b̃(t̄)
which are coherent. However, Method 3 cannot be used.
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
11 / 46
Introduction
www.KMA.zcu.cz
For Further Reading
Bloomenthal, J. and Shoemake, K.:
Convolution Surfaces.
Computer Graphics, Vol. 25, No. 4, pp. 251–256. 1991.
Arrondo, E., Sendra, J., Sendra, J.R.:
Parametric Generalized Offsets to Hypersurfaces.
Journal of Symbolic Computation, Vol. 23, pp.267–285. 1997
Sherstyuk, A.:
Convolution Surfaces in Computer Graphics.
PhD thesis, Monash Univ., Australia. 1999.
Sendra, J.R. and Sendra, J.:
Algebraic analysis of offsets to hypersurfaces.
Mathematische Zeitschrift. Vol. 234, pp. 697-719. Springer Berlin/Heidelberg, 2000.
Lávička, M., Bastl, B., Šír, Z.:
Reparameterization of Curves and Surfaces with Respect to their Convolution.
Proc. of Mathematical Methods of Curves and Surfaces 2008, Tønsberg. [To appear]
Vršek, J., Lávička, M.::
Algebraic analysis of convolutions of hypersurfaces.
[In preparation]
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
12 / 46
Reparameterizing one of hypersurfaces
www.KMA.zcu.cz
Outline
1 Introduction
Convolution hypersurfaces
Coherent parameterizations
Statement of the problem
2 Reparameterizing one of hypersurfaces
Computing parametric convolutions
Convolution degree of parameterization
Method 1 – Example(s)
3 Finding a new parameterization of one of hypersurfaces
Extended convolution ideal
Method 2 – Example(s)
4 Reparameterizing both hypersurfaces
Motivation: Why the previous stuff is not enough?
Explicit and implicit SF representation
Convolution theory via ISF representation
Method 3 – Examples
5 Finding new parameterizations of both hypersurfaces
Method 4 - Examples
6 Conclusion
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
13 / 46
Reparameterizing one of hypersurfaces
www.KMA.zcu.cz
Parametric convolution hypersurface
How to compute convolutions using parameterizations
I
Given rational parameterizations a(ū) and b(s̄) of hypersurfaces A and B
and their tangent hyperplanes α(a) = (α0 , . . . , αn+1 )> (ū) and
β(b) = (β0 , . . . , βn+1 )> (s̄)
I
Find a reparameterization
φ : (s1 , . . . , sn ) 7→ (u1 (s1 , . . . , sn ), . . . , un (s1 , . . . , sn ))
in a way that α(a) k β(b), i.e.,
αj (ui ) = λ · βj (si ),
λ 6= 0, j = 1, . . . , n + 1.
Remark
The parametric convolution hypersurface of two parametric hypersurfaces a(ū)
and b(s̄) is defined as
(a ~ b)(s̄) = a (φ(s̄)) + b(s̄)
Reparameterization of hypersurfaces with respect to convolutions
(a ~ b)(ū) = a(u) + b (φ(ū))
May 12, 2009
14 / 46
Reparameterizing one of hypersurfaces
www.KMA.zcu.cz
Parametric convolution hypersurface
How to compute convolutions using parameterizations
I
Given rational parameterizations a(ū) and b(s̄) of hypersurfaces A and B
and their tangent hyperplanes α(a) = (α0 , . . . , αn+1 )> (ū) and
β(b) = (β0 , . . . , βn+1 )> (s̄)
I
Find a reparameterization
φ : (s1 , . . . , sn ) 7→ (u1 (s1 , . . . , sn ), . . . , un (s1 , . . . , sn ))
in a way that α(a) k β(b), i.e.,
αj (ui ) = λ · βj (si ),
λ 6= 0, j = 1, . . . , n + 1.
Remark
The parametric convolution hypersurface of two parametric hypersurfaces a(ū)
and b(s̄) is defined as
(a ~ b)(s̄) = a (φ(s̄)) + b(s̄)
Reparameterization of hypersurfaces with respect to convolutions
(a ~ b)(ū) = a(u) + b (φ(ū))
May 12, 2009
14 / 46
Reparameterizing one of hypersurfaces
www.KMA.zcu.cz
Parametric convolution hypersurface
How to compute convolutions using parameterizations
I
Given rational parameterizations a(ū) and b(s̄) of hypersurfaces A and B
and their tangent hyperplanes α(a) = (α0 , . . . , αn+1 )> (ū) and
β(b) = (β0 , . . . , βn+1 )> (s̄)
I
Find a reparameterization
φ : (s1 , . . . , sn ) 7→ (u1 (s1 , . . . , sn ), . . . , un (s1 , . . . , sn ))
in a way that α(a) k β(b), i.e.,
αj (ui ) = λ · βj (si ),
λ 6= 0, j = 1, . . . , n + 1.
Remark
The parametric convolution hypersurface of two parametric hypersurfaces a(ū)
and b(s̄) is defined as
(a ~ b)(s̄) = a (φ(s̄)) + b(s̄)
Reparameterization of hypersurfaces with respect to convolutions
(a ~ b)(ū) = a(u) + b (φ(ū))
May 12, 2009
14 / 46
Reparameterizing one of hypersurfaces
www.KMA.zcu.cz
Parametric convolution hypersurface
How to compute convolutions using parameterizations
I
Given rational parameterizations a(ū) and b(s̄) of hypersurfaces A and B
and their tangent hyperplanes α(a) = (α0 , . . . , αn+1 )> (ū) and
β(b) = (β0 , . . . , βn+1 )> (s̄)
I
Find a reparameterization
φ : (s1 , . . . , sn ) 7→ (u1 (s1 , . . . , sn ), . . . , un (s1 , . . . , sn ))
Solving the system of polynomial equations . . .
in a way
that α(a) kφ,β(b),
i.e., Gröbner basis theory is very
For computing
e.g. the
suitable. αj (ui ) = λ · βj (si ), λ 6= 0, j = 1, . . . , n + 1.
Remark
The parametric convolution hypersurface of two parametric hypersurfaces a(ū)
and b(s̄) is defined as
(a ~ b)(s̄) = a (φ(s̄)) + b(s̄)
Reparameterization of hypersurfaces with respect to convolutions
(a ~ b)(ū) = a(u) + b (φ(ū))
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Reparameterizing one of hypersurfaces
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Convolution hypersurfaces – computation
I
To compute the reparameterization φ via the Gröbner basis theory, we
consider the ideal
Ia = hαj (ui ) − λβj (si ), 1 − wλi ⊂ k(si )[w, ui , λ],
the so called convolution ideal of a parameterization a(ui ).
I
Further, we compute the reduced Gröbner basis GIa of Ia with respect to
the lexicographic order for w > u1 > . . . > un > λ
I
Finally, using the Elimination theorem we obtain polynomials
g0 (w, u1 , . . . , un , λ), g1 (u1 , . . . , un , λ), . . . , gn (un , λ), gn+1 (λ)
with
LT(g0 ) = w, LT(gi ) = uri i , 1 ≤ i ≤ n, LT(gn+1 ) = λrn+1
as the generators of the elimination ideals ⇒ reparameterization φ
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
15 / 46
Reparameterizing one of hypersurfaces
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Convolution hypersurfaces – computation
I
To compute the reparameterization φ via the Gröbner basis theory, we
consider the ideal
Ia = hαj (ui ) − λβj (si ), 1 − wλi ⊂ k(si )[w, ui , λ],
the so called convolution ideal of a parameterization a(ui ).
I
Further, we compute the reduced Gröbner basis GIa of Ia with respect to
the lexicographic order for w > u1 > . . . > un > λ
I
Finally, using the Elimination theorem we obtain polynomials
g0 (w, u1 , . . . , un , λ), g1 (u1 , . . . , un , λ), . . . , gn (un , λ), gn+1 (λ)
with
LT(g0 ) = w, LT(gi ) = uri i , 1 ≤ i ≤ n, LT(gn+1 ) = λrn+1
as the generators of the elimination ideals ⇒ reparameterization φ
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
15 / 46
Reparameterizing one of hypersurfaces
www.KMA.zcu.cz
Convolution hypersurfaces – computation
I
To compute the reparameterization φ via the Gröbner basis theory, we
consider the ideal
Ia = hαj (ui ) − λβj (si ), 1 − wλi ⊂ k(si )[w, ui , λ],
the so called convolution ideal of a parameterization a(ui ).
I
Further, we compute the reduced Gröbner basis GIa of Ia with respect to
the lexicographic order for w > u1 > . . . > un > λ
I
Finally, using the Elimination theorem we obtain polynomials
g0 (w, u1 , . . . , un , λ), g1 (u1 , . . . , un , λ), . . . , gn (un , λ), gn+1 (λ)
with
LT(g0 ) = w, LT(gi ) = uri i , 1 ≤ i ≤ n, LT(gn+1 ) = λrn+1
as the generators of the elimination ideals ⇒ reparameterization φ
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
15 / 46
Reparameterizing one of hypersurfaces
www.KMA.zcu.cz
Convolution hypersurfaces – computation
I
To compute the reparameterization φ via the Gröbner basis theory, we
consider the ideal
Ia = hαj (ui ) − λβj (si ), 1 − wλi ⊂ k(si )[w, ui , λ],
the so called convolution ideal of a parameterization a(ui ).
I
I
Further, we compute the reduced Gröbner basis GIa of Ia with respect to
the lexicographic order for w > u1 > . . . > un > λ
Remark
Finally,
using the
Elimination
wewhere
obtainδ =
polynomials
Generally,
φ is
a δ-valuedtheorem
mapping,
(r1 ·r2 · · ·rn+1 )
g0 (w, u1 , . . . , un , λ), g1 (u1 , . . . , un , λ), . . . , gn (un , λ), gn+1 (λ)
with
LT(g0 ) = w, LT(gi ) = uri i , 1 ≤ i ≤ n, LT(gn+1 ) = λrn+1
as the generators of the elimination ideals ⇒ reparameterization φ
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
15 / 46
Reparameterizing one of hypersurfaces
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Convolution degree of a parameterization
Definition
Let GIa be a Gröbner basis of the ideal Ia and let ri are degrees of leading terms
of polynomials g1 , . . . , gn+1 ∈ G. Then the number δ = r1 · · · rn+1 is called the
the convolution degree of a parameterization a(ū). Futher, a(ū) is called δ-SRC
parameterization.
Remark
I
The convolution degree indicates the number of points a(ū) ∈ A
corresponding to the chosen point b(s̄) ∈ B (in the complex extension and
including the multiplicity).
I
If δ = 1 then the parameterization a(ū) is called a GRC parameterization
(hypersurfaces possessing these parameterizations admit rational
convolutions with an arbitrary hypersurface).
Reparameterization of hypersurfaces with respect to convolutions
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Reparameterizing one of hypersurfaces
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Convolution degree of a parameterization
Definition
Let GIa be a Gröbner basis of the ideal Ia and let ri are degrees of leading terms
of polynomials g1 , . . . , gn+1 ∈ G. Then the number δ = r1 · · · rn+1 is called the
the convolution degree of a parameterization a(ū). Futher, a(ū) is called δ-SRC
parameterization.
Remark
I
The convolution degree indicates the number of points a(ū) ∈ A
corresponding to the chosen point b(s̄) ∈ B (in the complex extension and
including the multiplicity).
I
If δ = 1 then the parameterization a(ū) is called a GRC parameterization
(hypersurfaces possessing these parameterizations admit rational
convolutions with an arbitrary hypersurface).
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
16 / 46
Reparameterizing one of hypersurfaces
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Convolution degree of a parameterization
Definition
Let GIa be a Gröbner basis of the ideal Ia and let ri are degrees of leading terms
of polynomials g1 , . . . , gn+1 ∈ G. Then the number δ = r1 · · · rn+1 is called the
the convolution degree of a parameterization a(ū). Futher, a(ū) is called δ-SRC
parameterization.
Remark
RemarkClearly, a hypersurface A can be described by different
parameterizations with different convolution degrees.
I The convolution degree indicates the number of points a(ū) ∈ A
corresponding to the chosen point b(s̄) ∈ B (in the complex extension and
including the multiplicity).
I
If δ = 1 then the parameterization a(ū) is called a GRC parameterization
(hypersurfaces possessing these parameterizations admit rational
convolutions with an arbitrary hypersurface).
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
16 / 46
Reparameterizing one of hypersurfaces
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Example 1 – (Parabola) ? (Circle)
3
2
1
-2
1
-1
2
-1
Parabola
Circle
2 >
a(u) = u, u
Reparameterization of hypersurfaces with respect to convolutions
b(s) =
2
2s
, 1−s
1+s2 1+s2
>
May 12, 2009
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Reparameterizing one of hypersurfaces
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Example 1 – (Parabola) ? (Circle)
1
We compute (polynomial) coordinate vectors of tangents α and β ⇒
>
>
>
2 >
(α1 , α2 ) (ū) = (2u, −1)
a(u(s)) + b(s)
LT(g1 )
LT(g2 )
= u
= λ
u(s) is a rational mapping
⇒ coherent a(u(s)), b(s)
Reparameterization of hypersurfaces with respect to convolutions
(β1 , β2 ) (s) = 2s, 1 − s
b(s(u)) + a(u)
LT(ḡ1 ) =
LT(ḡ2 ) =
s
λ2
s(u) is 2-valued mapping
⇒ non-rational reparameterization
May 12, 2009
18 / 46
Reparameterizing one of hypersurfaces
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Example 1 – (Parabola) ? (Circle)
1
We compute (polynomial) coordinate vectors of tangents α and β ⇒
>
>
>
2 >
(α1 , α2 ) (ū) = (2u, −1)
a(u(s)) + b(s)
LT(g1 )
LT(g2 )
= u
= λ
u(s) is a rational mapping
⇒ coherent a(u(s)), b(s)
Reparameterization of hypersurfaces with respect to convolutions
(β1 , β2 ) (s) = 2s, 1 − s
b(s(u)) + a(u)
LT(ḡ1 ) =
LT(ḡ2 ) =
s
λ2
s(u) is 2-valued mapping
⇒ non-rational reparameterization
May 12, 2009
18 / 46
Reparameterizing one of hypersurfaces
www.KMA.zcu.cz
Example 1 – (Parabola) ? (Circle)
1
We compute (polynomial) coordinate vectors of tangents α and β ⇒
>
>
>
2 >
(α1 , α2 ) (ū) = (2u, −1)
2
(β1 , β2 ) (s) = 2s, 1 − s
The Gröbner basis GIa of the ideal Ia = hα1 − λβ1 , α2 − λβ2 , 1 − wλi
n
o
⊂ Q(s)[w, u, λ] is
1
β1
GIa =
λ+
a(u(s)) + b(s)
LT(g1 )
LT(g2 )
= u
= λ
u(s) is a rational mapping
⇒ coherent a(u(s)), b(s)
Reparameterization of hypersurfaces with respect to convolutions
β2
,u +
2β2
, w + β2
.
b(s(u)) + a(u)
LT(ḡ1 ) =
LT(ḡ2 ) =
s
λ2
s(u) is 2-valued mapping
⇒ non-rational reparameterization
May 12, 2009
18 / 46
Reparameterizing one of hypersurfaces
www.KMA.zcu.cz
Example 1 – (Parabola) ? (Circle)
1
We compute (polynomial) coordinate vectors of tangents α and β ⇒
>
>
>
2 >
(α1 , α2 ) (ū) = (2u, −1)
2
(β1 , β2 ) (s) = 2s, 1 − s
The Gröbner basis GIa of the ideal Ia = hα1 − λβ1 , α2 − λβ2 , 1 − wλi
n
o
⊂ Q(s)[w, u, λ] is
1
β1
GIa =
λ+
a(u(s)) + b(s)
LT(g1 )
LT(g2 )
= u
= λ
u(s) is a rational mapping
⇒ coherent a(u(s)), b(s)
Reparameterization of hypersurfaces with respect to convolutions
β2
,u +
2β2
, w + β2
.
b(s(u)) + a(u)
LT(ḡ1 ) =
LT(ḡ2 ) =
s
λ2
s(u) is 2-valued mapping
⇒ non-rational reparameterization
May 12, 2009
18 / 46
Reparameterizing one of hypersurfaces
www.KMA.zcu.cz
Example 1 – (Parabola) ? (Circle)
1
We compute (polynomial) coordinate vectors of tangents α and β ⇒
>
>
>
2 >
(α1 , α2 ) (ū) = (2u, −1)
2
(β1 , β2 ) (s) = 2s, 1 − s
The Gröbner basis GIa of the ideal Ia = hα1 − λβ1 , α2 − λβ2 , 1 − wλi
n
o
⊂ Q(s)[w, u, λ] is
1
β1
GIa =
λ+
a(u(s)) + b(s)
LT(g1 )
LT(g2 )
= u
= λ
u(s) is a rational mapping
⇒ coherent a(u(s)), b(s)
Reparameterization of hypersurfaces with respect to convolutions
β2
,u +
2β2
, w + β2
.
b(s(u)) + a(u)
LT(ḡ1 ) =
LT(ḡ2 ) =
s
λ2
s(u) is 2-valued mapping
⇒ non-rational reparameterization
May 12, 2009
18 / 46
Reparameterizing one of hypersurfaces
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Example 1 – (Parabola) ? (Circle)
1
We compute (polynomial) coordinate vectors of tangents α and β ⇒
>
>
>
2 >
(α1 , α2 ) (ū) = (2u, −1)
2
(β1 , β2 ) (s) = 2s, 1 − s
The Gröbner basis GIa of the ideal Ia = hα1 − λβ1 , α2 − λβ2 , 1 − wλi
n
o
⊂ Q(s)[w, u, λ] is
1
β1
GIa =
λ+
a(u(s)) + b(s)
LT(g1 )
LT(g2 )
= u
= λ
u(s) is a rational mapping
⇒ coherent a(u(s)), b(s)
Reparameterization of hypersurfaces with respect to convolutions
β2
,u +
2β2
, w + β2
.
b(s(u)) + a(u)
LT(ḡ1 ) =
LT(ḡ2 ) =
s
λ2
s(u) is 2-valued mapping
⇒ non-rational reparameterization
May 12, 2009
18 / 46
Reparameterizing one of hypersurfaces
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Example 1 – (Parabola) ? (Circle)
1
We compute (polynomial) coordinate vectors of tangents α and β ⇒
>
>
>
2 >
(α1 , α2 ) (ū) = (2u, −1)
3
(β1 , β2 ) (s) = 2s, 1 − s
The Gröbner basis GIb of the ideal Ib = hβ1 − λα1 , β2 − λα2 , 1 − wλi
⊂ Q(u)[w, s, λ] is
n
o
α2 λ
GI
b
=
2
λ +
4α2
α2
1
a(u(s)) + b(s)
LT(g1 )
LT(g2 )
= u
= λ
u(s) is a rational mapping
⇒ coherent a(u(s)), b(s)
Reparameterization of hypersurfaces with respect to convolutions
−
4
α2
1
,s−
α1 λ
2
,w −
1
4
− α2
.
b(s(u)) + a(u)
LT(ḡ1 ) =
LT(ḡ2 ) =
s
λ2
s(u) is 2-valued mapping
⇒ non-rational reparameterization
May 12, 2009
18 / 46
Reparameterizing one of hypersurfaces
www.KMA.zcu.cz
Example 1 – (Parabola) ? (Circle)
1
We compute (polynomial) coordinate vectors of tangents α and β ⇒
>
>
>
2 >
(α1 , α2 ) (ū) = (2u, −1)
3
(β1 , β2 ) (s) = 2s, 1 − s
The Gröbner basis GIb of the ideal Ib = hβ1 − λα1 , β2 − λα2 , 1 − wλi
⊂ Q(u)[w, s, λ] is
n
o
α2 λ
GI
b
=
2
λ +
4α2
α2
1
a(u(s)) + b(s)
LT(g1 )
LT(g2 )
= u
= λ
u(s) is a rational mapping
⇒ coherent a(u(s)), b(s)
Reparameterization of hypersurfaces with respect to convolutions
−
4
α2
1
,s−
α1 λ
2
,w −
1
4
− α2
.
b(s(u)) + a(u)
LT(ḡ1 ) =
LT(ḡ2 ) =
s
λ2
s(u) is 2-valued mapping
⇒ non-rational reparameterization
May 12, 2009
18 / 46
Reparameterizing one of hypersurfaces
www.KMA.zcu.cz
Example 1 – (Parabola) ? (Circle)
1
We compute (polynomial) coordinate vectors of tangents α and β ⇒
>
>
>
2 >
(α1 , α2 ) (ū) = (2u, −1)
3
(β1 , β2 ) (s) = 2s, 1 − s
The Gröbner basis GIb of the ideal Ib = hβ1 − λα1 , β2 − λα2 , 1 − wλi
⊂ Q(u)[w, s, λ] is
n
o
α2 λ
GI
b
=
2
λ +
4α2
α2
1
a(u(s)) + b(s)
LT(g1 )
LT(g2 )
= u
= λ
u(s) is a rational mapping
⇒ coherent a(u(s)), b(s)
Reparameterization of hypersurfaces with respect to convolutions
−
4
α2
1
,s−
α1 λ
2
,w −
1
4
− α2
.
b(s(u)) + a(u)
LT(ḡ1 ) =
LT(ḡ2 ) =
s
λ2
s(u) is 2-valued mapping
⇒ non-rational reparameterization
May 12, 2009
18 / 46
Reparameterizing one of hypersurfaces
www.KMA.zcu.cz
Example 1 – (Parabola) ? (Circle)
1
We compute (polynomial) coordinate vectors of tangents α and β ⇒
>
>
>
2 >
(α1 , α2 ) (ū) = (2u, −1)
3
(β1 , β2 ) (s) = 2s, 1 − s
The Gröbner basis GIb of the ideal Ib = hβ1 − λα1 , β2 − λα2 , 1 − wλi
⊂ Q(u)[w, s, λ] is
n
o
α2 λ
GI
b
=
2
λ +
4α2
α2
1
a(u(s)) + b(s)
LT(g1 )
LT(g2 )
= u
= λ
u(s) is a rational mapping
⇒ coherent a(u(s)), b(s)
Reparameterization of hypersurfaces with respect to convolutions
−
4
α2
1
,s−
α1 λ
2
,w −
1
4
− α2
.
b(s(u)) + a(u)
LT(ḡ1 ) =
LT(ḡ2 ) =
s
λ2
s(u) is 2-valued mapping
⇒ non-rational reparameterization
May 12, 2009
18 / 46
Reparameterizing one of hypersurfaces
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For Further Reading
Becker, T., Weispfenning, V.:
Gröbner bases – a computational approach to commutative algebra.
Graduate Texts in Mathematics. Springer-Verlag, New York, 1993.
Peternell, M. and Manhart, F.:
The convolution of a paraboloid and a parametrized surface.
Journal for Geometry and Graphics 7, 157-171. 2003.
Sampoli, M.L., Peternell, M., Jüttler, B.
Rational surfaces with linear normals and their convolutions with rational surfaces.
Computer Aided Geometric Design 23, pp. 179-192, Elsevier, 2006.
Lávička, M., Bastl, B.
Rational Hypersurfaces with Rational Convolutions.
Computer Aided Geometric Design 24, pp. 410-426, Elsevier, 2007.
Lávička, M., Bastl, B.
PN surfaces and their convolutions with rational surfaces.
Computer Aided Geometric Design 25, pp. 763-774, Elsevier, 2008.
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
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Finding a new parameterization of one of hypersurfaces
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Outline
1 Introduction
Convolution hypersurfaces
Coherent parameterizations
Statement of the problem
2 Reparameterizing one of hypersurfaces
Computing parametric convolutions
Convolution degree of parameterization
Method 1 – Example(s)
3 Finding a new parameterization of one of hypersurfaces
Extended convolution ideal
Method 2 – Example(s)
4 Reparameterizing both hypersurfaces
Motivation: Why the previous stuff is not enough?
Explicit and implicit SF representation
Convolution theory via ISF representation
Method 3 – Examples
5 Finding new parameterizations of both hypersurfaces
Method 4 - Examples
6 Conclusion
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
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Finding a new parameterization of one of hypersurfaces
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Example 2 – Non-proper parameterization of paraboloid
a(u, v) =
(u2 , v 2 , u4 + v 4 )>
Convolution ideal Ia =
−8u3 v − n1 λ, −8uv 3 − n2 λ,
4uv − n3 λ, 1 − λwi
Basis GIa =
n
w−
n4
3λ
,u
4n1 n2
v2 +
a(u, v) is 4-SRC pararameterization
+
λvn2
3
,
2n2
n2
, λ2
2n3
−
4n1 n2
n4
3
o
Reparameterization of hypersurfaces with respect to convolutions
In this case, rational
reparameterization φ can be obtained
only for exceptional surfaces b(s, t)
May 12, 2009
21 / 46
Finding a new parameterization of one of hypersurfaces
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Example 2 – Non-proper parameterization of paraboloid
a(u, v) =
(u2 , v 2 , u4 + v 4 )>
Convolution ideal Ia =
−8u3 v − n1 λ, −8uv 3 − n2 λ,
4uv − n3 λ, 1 − λwi
Basis GIa =
n
w−
n4
3λ
,u
4n1 n2
v2 +
a(u, v) is 4-SRC pararameterization
+
λvn2
3
,
2n2
n2
, λ2
2n3
−
4n1 n2
n4
3
o
Reparameterization of hypersurfaces with respect to convolutions
In this case, rational
reparameterization φ can be obtained
only for exceptional surfaces b(s, t)
May 12, 2009
21 / 46
Finding a new parameterization of one of hypersurfaces
www.KMA.zcu.cz
Example 2 – Non-proper parameterization of paraboloid
a(u, v) =
(u2 , v 2 , u4 + v 4 )>
Convolution ideal Ia =
−8u3 v − n1 λ, −8uv 3 − n2 λ,
4uv − n3 λ, 1 − λwi
Basis GIa =
n
w−
n4
3λ
,u
4n1 n2
v2 +
a(u, v) is 4-SRC pararameterization
+
λvn2
3
,
2n2
n2
, λ2
2n3
−
4n1 n2
n4
3
o
Reparameterization of hypersurfaces with respect to convolutions
In this case, rational
reparameterization φ can be obtained
only for exceptional surfaces b(s, t)
May 12, 2009
21 / 46
Finding a new parameterization of one of hypersurfaces
www.KMA.zcu.cz
Example 2 – Non-proper parameterization of paraboloid
a(u, v) =
(u2 , v 2 , u4 + v 4 )>
Convolution ideal Ia =
−8u3 v − n1 λ, −8uv 3 − n2 λ,
4uv − n3 λ, 1 − λwi
Basis GIa =
n
w−
n4
3λ
,u
4n1 n2
v2 +
a(u, v) is 4-SRC pararameterization
+
λvn2
3
,
2n2
n2
, λ2
2n3
−
4n1 n2
n4
3
o
Reparameterization of hypersurfaces with respect to convolutions
In this case, rational
reparameterization φ can be obtained
only for exceptional surfaces b(s, t)
May 12, 2009
21 / 46
Finding a new parameterization of one of hypersurfaces
www.KMA.zcu.cz
Example 2 – Non-proper parameterization of paraboloid
a(u, v) =
(u2 , v 2 , u4 + v 4 )>
Convolution ideal Ia =
−8u3 v − n1 λ, −8uv 3 − n2 λ,
4uv − n3 λ, 1 − λwi
Basis GIa =
n
w−
n4
3λ
,u
4n1 n2
v2 +
a(u, v) is 4-SRC pararameterization
+
λvn2
3
,
2n2
n2
, λ2
2n3
−
4n1 n2
n4
3
o
Reparameterization of hypersurfaces with respect to convolutions
In this case, rational
reparameterization φ can be obtained
only for exceptional surfaces b(s, t)
May 12, 2009
21 / 46
Finding a new parameterization of one of hypersurfaces
www.KMA.zcu.cz
Convolution ideal – the old idea updated
I
Given rational parameterized hypersurfaces a(ū) and b(s̄) and their tangent
hyperplanes α(a) = (α0 , . . . , αn+1 )> (ū) and β(b) = (β0 , . . . , βn+1 )> (s̄)
I
Next, we consider the ideal
I˜ = hI, numerator(xj − aj (ū))i ⊂ k(nj )[w, ū, λ, x1 , . . . , xn+1 ],
which is called an extended convolution ideal of a parameterization a(ū).
I
I
Further, we compute the reduced Gröbner basis GI˜a of I˜a with respect to
the lexicographic order for w > u1 > . . . > un > λ > x1 > . . . > xn+1
The rational parameterization of A corresponding to (β1 . . . , βn+1 )> (s̄)
exists if (after substituting concrete rational functions βj (s̄)) the last n + 1
polynomials of GI˜ can be solved for x1 , . . . , xn+1 as rational functions in s̄
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
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Finding a new parameterization of one of hypersurfaces
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Convolution ideal – the old idea updated
I
Given rational parameterized hypersurfaces a(ū) and b(s̄) and their tangent
hyperplanes α(a) = (α0 , . . . , αn+1 )> (ū) and β(b) = (β0 , . . . , βn+1 )> (s̄)
I
Next, we consider the ideal
I˜ = hI, numerator(xj − aj (ū))i ⊂ k(nj )[w, ū, λ, x1 , . . . , xn+1 ],
which is called an extended convolution ideal of a parameterization a(ū).
I
I
Further, we compute the reduced Gröbner basis GI˜a of I˜a with respect to
the lexicographic order for w > u1 > . . . > un > λ > x1 > . . . > xn+1
The rational parameterization of A corresponding to (β1 . . . , βn+1 )> (s̄)
exists if (after substituting concrete rational functions βj (s̄)) the last n + 1
polynomials of GI˜ can be solved for x1 , . . . , xn+1 as rational functions in s̄
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
22 / 46
Finding a new parameterization of one of hypersurfaces
www.KMA.zcu.cz
Convolution ideal – the old idea updated
I
Given rational parameterized hypersurfaces a(ū) and b(s̄) and their tangent
hyperplanes α(a) = (α0 , . . . , αn+1 )> (ū) and β(b) = (β0 , . . . , βn+1 )> (s̄)
I
Next, we consider the ideal
I˜ = hI, numerator(xj − aj (ū))i ⊂ k(nj )[w, ū, λ, x1 , . . . , xn+1 ],
which is called an extended convolution ideal of a parameterization a(ū).
I
I
Further, we compute the reduced Gröbner basis GI˜a of I˜a with respect to
the lexicographic order for w > u1 > . . . > un > λ > x1 > . . . > xn+1
The rational parameterization of A corresponding to (β1 . . . , βn+1 )> (s̄)
exists if (after substituting concrete rational functions βj (s̄)) the last n + 1
polynomials of GI˜ can be solved for x1 , . . . , xn+1 as rational functions in s̄
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
22 / 46
Finding a new parameterization of one of hypersurfaces
www.KMA.zcu.cz
Convolution ideal – the old idea updated
I
Given rational parameterized hypersurfaces a(ū) and b(s̄) and their tangent
hyperplanes α(a) = (α0 , . . . , αn+1 )> (ū) and β(b) = (β0 , . . . , βn+1 )> (s̄)
I
Next, we consider the ideal
I˜ = hI, numerator(xj − aj (ū))i ⊂ k(nj )[w, ū, λ, x1 , . . . , xn+1 ],
which is called an extended convolution ideal of a parameterization a(ū).
I
I
Further, we compute the reduced Gröbner basis GI˜a of I˜a with respect to
the lexicographic order for w > u1 > . . . > un > λ > x1 > . . . > xn+1
The rational parameterization of A corresponding to (β1 . . . , βn+1 )> (s̄)
exists if (after substituting concrete rational functions βj (s̄)) the last n + 1
polynomials of GI˜ can be solved for x1 , . . . , xn+1 as rational functions in s̄
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
22 / 46
Finding a new parameterization of one of hypersurfaces
www.KMA.zcu.cz
Convolution ideal – the old idea updated
I
Given rational parameterized hypersurfaces a(ū) and b(s̄) and their tangent
hyperplanes α(a) = (α0 , . . . , αn+1 )> (ū) and β(b) = (β0 , . . . , βn+1 )> (s̄)
I
Next, we consider the ideal
I
I˜ = hI, numerator(xj − aj (ū))i ⊂ k(nj )[w, ū, λ, x1 , . . . , xn+1 ],
Remark
which
is called
an overcomes
extended convolution
ideal of a of
parameterization
This
approach
some disadvantages
the previous a(ū).
method,
mainly
of
non-proper
parameterizations.
Further, we compute the reduced Gröbner basis G ˜ of I˜a with respect to
Ia
the lexicographic order for w > u1 > . . . > un > λ > x1 > . . . > xn+1
I
The rational parameterization of A corresponding to (β1 . . . , βn+1 )> (s̄)
exists if (after substituting concrete rational functions βj (s̄)) the last n + 1
polynomials of GI˜ can be solved for x1 , . . . , xn+1 as rational functions in s̄
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
22 / 46
Finding a new parameterization of one of hypersurfaces
www.KMA.zcu.cz
Example 2 – revised
a(u, v) =
(u2 , v 2 , u4 + v 4 )>
Extended convolution ideal I˜a =
−8u3 v − n1 λ, −8uv 3 − n2 λ,
4uv − n3 λ, 1 − λw, x − u2 ,
y − v 2 , z − u4 − v 4
Basis GI˜a =
n
w−
n4
λvn2
n2
3λ
, u + 2n23 , v 2 + 2n
,
4n1 n2
3
4n1 n2
n1
n2
2
λ − n4 , x + 2n3 , y + 2n
,
3
3
z−
2
n2
1 +n2
4n2
3
o
.
Reparameterization of hypersurfaces with respect to convolutions
Although φ is generally non-rational, we
have found a rational parameterization
n1
n2
ã(s, t) = − 2n
, − 2n
,
3
3
2
n2
1 +n2
4n2
3
>
(s, t)
The coherent parameterization can be
found for any surface b(s, t) and its
associated normal field (n1 , n2 , n3 )> (s, t).
May 12, 2009
23 / 46
Finding a new parameterization of one of hypersurfaces
www.KMA.zcu.cz
Example 2 – revised
a(u, v) =
(u2 , v 2 , u4 + v 4 )>
Extended convolution ideal I˜a =
−8u3 v − n1 λ, −8uv 3 − n2 λ,
4uv − n3 λ, 1 − λw, x − u2 ,
y − v 2 , z − u4 − v 4
Basis GI˜a =
n
w−
n4
λvn2
n2
3λ
, u + 2n23 , v 2 + 2n
,
4n1 n2
3
4n1 n2
n1
n2
2
λ − n4 , x + 2n3 , y + 2n
,
3
3
z−
2
n2
1 +n2
4n2
3
o
.
Reparameterization of hypersurfaces with respect to convolutions
Although φ is generally non-rational, we
have found a rational parameterization
n1
n2
ã(s, t) = − 2n
, − 2n
,
3
3
2
n2
1 +n2
4n2
3
>
(s, t)
The coherent parameterization can be
found for any surface b(s, t) and its
associated normal field (n1 , n2 , n3 )> (s, t).
May 12, 2009
23 / 46
Finding a new parameterization of one of hypersurfaces
www.KMA.zcu.cz
Example 2 – revised
a(u, v) =
(u2 , v 2 , u4 + v 4 )>
Extended convolution ideal I˜a =
−8u3 v − n1 λ, −8uv 3 − n2 λ,
4uv − n3 λ, 1 − λw, x − u2 ,
y − v 2 , z − u4 − v 4
Basis GI˜a =
n
w−
n4
λvn2
n2
3λ
, u + 2n23 , v 2 + 2n
,
4n1 n2
3
4n1 n2
n1
n2
2
λ − n4 , x + 2n3 , y + 2n
,
3
3
z−
2
n2
1 +n2
4n2
3
o
.
Reparameterization of hypersurfaces with respect to convolutions
Although φ is generally non-rational, we
have found a rational parameterization
n1
n2
ã(s, t) = − 2n
, − 2n
,
3
3
2
n2
1 +n2
4n2
3
>
(s, t)
The coherent parameterization can be
found for any surface b(s, t) and its
associated normal field (n1 , n2 , n3 )> (s, t).
May 12, 2009
23 / 46
Finding a new parameterization of one of hypersurfaces
www.KMA.zcu.cz
Example 2 – revised
a(u, v) =
(u2 , v 2 , u4 + v 4 )>
Extended convolution ideal I˜a =
−8u3 v − n1 λ, −8uv 3 − n2 λ,
4uv − n3 λ, 1 − λw, x − u2 ,
y − v 2 , z − u4 − v 4
Basis GI˜a =
n
w−
n4
λvn2
n2
3λ
, u + 2n23 , v 2 + 2n
,
4n1 n2
3
4n1 n2
n1
n2
2
λ − n4 , x + 2n3 , y + 2n
,
3
3
z−
2
n2
1 +n2
4n2
3
o
.
Reparameterization of hypersurfaces with respect to convolutions
Although φ is generally non-rational, we
have found a rational parameterization
n1
n2
ã(s, t) = − 2n
, − 2n
,
3
3
2
n2
1 +n2
4n2
3
>
(s, t)
The coherent parameterization can be
found for any surface b(s, t) and its
associated normal field (n1 , n2 , n3 )> (s, t).
May 12, 2009
23 / 46
Finding a new parameterization of one of hypersurfaces
www.KMA.zcu.cz
Example 2 – revised
a(u, v) =
(u2 , v 2 , u4 + v 4 )>
Extended convolution ideal I˜a =
−8u3 v − n1 λ, −8uv 3 − n2 λ,
4uv − n3 λ, 1 − λw, x − u2 ,
y − v 2 , z − u4 − v 4
Basis GI˜a =
n
w−
n4
λvn2
n2
3λ
, u + 2n23 , v 2 + 2n
,
4n1 n2
3
4n1 n2
n1
n2
2
λ − n4 , x + 2n3 , y + 2n
,
3
3
z−
2
n2
1 +n2
4n2
3
o
.
Reparameterization of hypersurfaces with respect to convolutions
Although φ is generally non-rational, we
have found a rational parameterization
n1
n2
ã(s, t) = − 2n
, − 2n
,
3
3
2
n2
1 +n2
4n2
3
>
(s, t)
The coherent parameterization can be
found for any surface b(s, t) and its
associated normal field (n1 , n2 , n3 )> (s, t).
May 12, 2009
23 / 46
Finding a new parameterization of one of hypersurfaces
www.KMA.zcu.cz
To sum up . . .
Lemma
If the product of the exponents of leading monomials of the last n + 1
polynomials of GI˜ is equal to 1, a coherent parameterization can be found for any
input normal field n(t̄).
Lemma
Let a(ū) be a parameterized hypersurface in Rn+1 . If δ = 1 then a coherent
parameterization can be found for any input normal field n(t̄) by a rational
reparameterization of a(ū).
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
24 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Outline
1 Introduction
Convolution hypersurfaces
Coherent parameterizations
Statement of the problem
2 Reparameterizing one of hypersurfaces
Computing parametric convolutions
Convolution degree of parameterization
Method 1 – Example(s)
3 Finding a new parameterization of one of hypersurfaces
Extended convolution ideal
Method 2 – Example(s)
4 Reparameterizing both hypersurfaces
Motivation: Why the previous stuff is not enough?
Explicit and implicit SF representation
Convolution theory via ISF representation
Method 3 – Examples
5 Finding new parameterizations of both hypersurfaces
Method 4 - Examples
6 Conclusion
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
25 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Example 3 – (Cardioid) ? (Tschirnhausen cubic)
Cardioid
a(u) =
4
2
3
−2u +2u
, −4u
u4 +2u2 +1 u4 +2u2 +1
>
Reparameterization of hypersurfaces with respect to convolutions
Tschirnhausen cubic
2
1 3 >
b(s) = s , s − 3 s
May 12, 2009
26 / 46
Reparameterizing both hypersurfaces
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Example 3 – (Cardioid) ? (Tschirnhausen cubic)
1
We compute (polynomial) coordinate vectors of tangents α and β ⇒
>
>
2
2
>
>
2
(α1 , α2 ) (u) = ((u − 3)u, 3u − 1)
a(u(s)) + b(s)
LT(g1 )
LT(g2 )
= u
= λ3
u(s) is 3-valued mapping
⇒ non-rational reparam.
Reparameterization of hypersurfaces with respect to convolutions
(β1 , β2 ) (s) = 1 − s , 2s
b(s(u)) + a(u)
LT(ḡ1 ) =
LT(ḡ2 ) =
s
λ2
s(u) is 2-valued mapping
⇒ non-rational reparam.
May 12, 2009
27 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Example 3 – (Cardioid) ? (Tschirnhausen cubic)
1
We compute (polynomial) coordinate vectors of tangents α and β ⇒
>
>
2
2
>
>
2
(α1 , α2 ) (u) = ((u − 3)u, 3u − 1)
a(u(s)) + b(s)
LT(g1 )
LT(g2 )
= u
= λ3
u(s) is 3-valued mapping
⇒ non-rational reparam.
Reparameterization of hypersurfaces with respect to convolutions
(β1 , β2 ) (s) = 1 − s , 2s
b(s(u)) + a(u)
LT(ḡ1 ) =
LT(ḡ2 ) =
s
λ2
s(u) is 2-valued mapping
⇒ non-rational reparam.
May 12, 2009
27 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Example 3 – (Cardioid) ? (Tschirnhausen cubic)
1
We compute (polynomial) coordinate vectors of tangents α and β ⇒
>
>
2
2
>
>
2
(α1 , α2 ) (u) = ((u − 3)u, 3u − 1)
2
(β1 , β2 ) (s) = 1 − s , 2s
The Gröbner basis Ga of the ideal Ia = hα1 − λβ1 , α2 − λβ2 , 1 − wλi
⊂ Q(s)[w, u, λ] is
n
Ga =
2
3
2
2
2
3
2
2
64w + λ β2 + (−27β1 − 15β2 )λ + 48β2 , 72β1 u − λ β2 + (7β2 + 27β1 )λ + 8β2 ,
3
3
2
2
o
2
λ β3 + (−27β1 − 15β2 )λ + 48λβ2 + 64
a(u(s)) + b(s)
LT(g1 )
LT(g2 )
= u
= λ3
u(s) is 3-valued mapping
⇒ non-rational reparam.
Reparameterization of hypersurfaces with respect to convolutions
b(s(u)) + a(u)
LT(ḡ1 ) =
LT(ḡ2 ) =
s
λ2
s(u) is 2-valued mapping
⇒ non-rational reparam.
May 12, 2009
27 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Example 3 – (Cardioid) ? (Tschirnhausen cubic)
1
We compute (polynomial) coordinate vectors of tangents α and β ⇒
>
>
2
2
>
>
2
(α1 , α2 ) (u) = ((u − 3)u, 3u − 1)
2
(β1 , β2 ) (s) = 1 − s , 2s
The Gröbner basis Ga of the ideal Ia = hα1 − λβ1 , α2 − λβ2 , 1 − wλi
⊂ Q(s)[w, u, λ] is
n
Ga =
2
3
2
2
2
3
2
2
64w + λ β2 + (−27β1 − 15β2 )λ + 48β2 , 72β1 u − λ β2 + (7β2 + 27β1 )λ + 8β2 ,
3
3
2
2
o
2
λ β3 + (−27β1 − 15β2 )λ + 48λβ2 + 64
a(u(s)) + b(s)
LT(g1 )
LT(g2 )
= u
= λ3
u(s) is 3-valued mapping
⇒ non-rational reparam.
Reparameterization of hypersurfaces with respect to convolutions
b(s(u)) + a(u)
LT(ḡ1 ) =
LT(ḡ2 ) =
s
λ2
s(u) is 2-valued mapping
⇒ non-rational reparam.
May 12, 2009
27 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Example 3 – (Cardioid) ? (Tschirnhausen cubic)
1
We compute (polynomial) coordinate vectors of tangents α and β ⇒
>
>
2
2
>
>
2
(α1 , α2 ) (u) = ((u − 3)u, 3u − 1)
2
(β1 , β2 ) (s) = 1 − s , 2s
The Gröbner basis Ga of the ideal Ia = hα1 − λβ1 , α2 − λβ2 , 1 − wλi
⊂ Q(s)[w, u, λ] is
n
Ga =
2
3
2
2
2
3
2
2
64w + λ β2 + (−27β1 − 15β2 )λ + 48β2 , 72β1 u − λ β2 + (7β2 + 27β1 )λ + 8β2 ,
3
3
2
2
o
2
λ β3 + (−27β1 − 15β2 )λ + 48λβ2 + 64
a(u(s)) + b(s)
LT(g1 )
LT(g2 )
= u
= λ3
u(s) is 3-valued mapping
⇒ non-rational reparam.
Reparameterization of hypersurfaces with respect to convolutions
b(s(u)) + a(u)
LT(ḡ1 ) =
LT(ḡ2 ) =
s
λ2
s(u) is 2-valued mapping
⇒ non-rational reparam.
May 12, 2009
27 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Example 3 – (Cardioid) ? (Tschirnhausen cubic)
1
We compute (polynomial) coordinate vectors of tangents α and β ⇒
>
>
2
2
>
>
2
(α1 , α2 ) (u) = ((u − 3)u, 3u − 1)
2
(β1 , β2 ) (s) = 1 − s , 2s
The Gröbner basis Ga of the ideal Ia = hα1 − λβ1 , α2 − λβ2 , 1 − wλi
⊂ Q(s)[w, u, λ] is
n
Ga =
2
3
2
2
2
3
2
2
64w + λ β2 + (−27β1 − 15β2 )λ + 48β2 , 72β1 u − λ β2 + (7β2 + 27β1 )λ + 8β2 ,
3
3
2
2
o
2
λ β3 + (−27β1 − 15β2 )λ + 48λβ2 + 64
a(u(s)) + b(s)
LT(g1 )
LT(g2 )
= u
= λ3
u(s) is 3-valued mapping
⇒ non-rational reparam.
Reparameterization of hypersurfaces with respect to convolutions
b(s(u)) + a(u)
LT(ḡ1 ) =
LT(ḡ2 ) =
s
λ2
s(u) is 2-valued mapping
⇒ non-rational reparam.
May 12, 2009
27 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Example 3 – (Cardioid) ? (Tschirnhausen cubic)
1
We compute (polynomial) coordinate vectors of tangents α and β ⇒
>
>
2
2
>
>
2
(α1 , α2 ) (u) = ((u − 3)u, 3u − 1)
3
(β1 , β2 ) (s) = 1 − s , 2s
The Gröbner basis Gb of the ideal Ib = hβ1 − λα1 , β2 − λα2 , 1 − wλi
⊂ Q(u)[w, s, λ] is
n
o
Gb =
2
a(u(s)) + b(s)
LT(g1 )
LT(g2 )
= u
= λ3
2
2
4w − λα2 − 4α1 , 2s + λα2 , λ α2 + 4λα1 − 4
u(s) is 3-valued mapping
⇒ non-rational reparam.
Reparameterization of hypersurfaces with respect to convolutions
b(s(u)) + a(u)
LT(ḡ1 ) =
LT(ḡ2 ) =
s
λ2
s(u) is 2-valued mapping
⇒ non-rational reparam.
May 12, 2009
27 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Example 3 – (Cardioid) ? (Tschirnhausen cubic)
1
We compute (polynomial) coordinate vectors of tangents α and β ⇒
>
>
2
2
>
>
2
(α1 , α2 ) (u) = ((u − 3)u, 3u − 1)
3
(β1 , β2 ) (s) = 1 − s , 2s
The Gröbner basis Gb of the ideal Ib = hβ1 − λα1 , β2 − λα2 , 1 − wλi
⊂ Q(u)[w, s, λ] is
n
o
Gb =
2
a(u(s)) + b(s)
LT(g1 )
LT(g2 )
= u
= λ3
2
2
4w − λα2 − 4α1 , 2s + λα2 , λ α2 + 4λα1 − 4
u(s) is 3-valued mapping
⇒ non-rational reparam.
Reparameterization of hypersurfaces with respect to convolutions
b(s(u)) + a(u)
LT(ḡ1 ) =
LT(ḡ2 ) =
s
λ2
s(u) is 2-valued mapping
⇒ non-rational reparam.
May 12, 2009
27 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Example 3 – (Cardioid) ? (Tschirnhausen cubic)
1
We compute (polynomial) coordinate vectors of tangents α and β ⇒
>
>
2
2
>
>
2
(α1 , α2 ) (u) = ((u − 3)u, 3u − 1)
3
(β1 , β2 ) (s) = 1 − s , 2s
The Gröbner basis Gb of the ideal Ib = hβ1 − λα1 , β2 − λα2 , 1 − wλi
⊂ Q(u)[w, s, λ] is
n
o
Gb =
2
a(u(s)) + b(s)
LT(g1 )
LT(g2 )
= u
= λ3
2
2
4w − λα2 − 4α1 , 2s + λα2 , λ α2 + 4λα1 − 4
u(s) is 3-valued mapping
⇒ non-rational reparam.
Reparameterization of hypersurfaces with respect to convolutions
b(s(u)) + a(u)
LT(ḡ1 ) =
LT(ḡ2 ) =
s
λ2
s(u) is 2-valued mapping
⇒ non-rational reparam.
May 12, 2009
27 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Example 3 – (Cardioid) ? (Tschirnhausen cubic)
1
We compute (polynomial) coordinate vectors of tangents α and β ⇒
>
>
2
2
>
>
2
(α1 , α2 ) (u) = ((u − 3)u, 3u − 1)
3
(β1 , β2 ) (s) = 1 − s , 2s
The Gröbner basis Gb of the ideal Ib = hβ1 − λα1 , β2 − λα2 , 1 − wλi
⊂ Q(u)[w, s, λ] is
n
o
Gb =
2
a(u(s)) + b(s)
LT(g1 )
LT(g2 )
= u
= λ3
2
2
4w − λα2 − 4α1 , 2s + λα2 , λ α2 + 4λα1 − 4
u(s) is 3-valued mapping
⇒ non-rational reparam.
Reparameterization of hypersurfaces with respect to convolutions
b(s(u)) + a(u)
LT(ḡ1 ) =
LT(ḡ2 ) =
s
λ2
s(u) is 2-valued mapping
⇒ non-rational reparam.
May 12, 2009
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Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Example 3 – again
1
We use the concept of extended convolution ideals and compute the bases
GI˜a and GI˜b
2
Next, we substitute the normal fields
>
>
nb = 1 − s2 , 2s , na = (u2 − 3)u, 3u2 − 1
into the last two polynomials of GI˜a and GI˜b , respectively, and obtain
(−76s2 − 76s6 − 282s4 + s8 + 1)x + (−12s8 − 48s6 − 72s4 − 48s2 − 12)y 2 +
+(−96s5 − 32s7 + 96s3 + 32s)y + 2s8 + 16s6 + 2 − 36s4 + 16s2 =
0,
(16s8 + 64s6 + 96s4 + 64s2 + 16)y 3 + (−3s8 − 108s6 − 210s4 −
−108s2 − 3)y + 8s7 − 24s5 + 24s3 − 8s =
0,
(−21u4 + 24u2 − 1 + 2u6 )x + (3u + 3u5 − 10u3 )y + 9u4 − 6u2 + 1
6
4
2
2
9
3
5
(27u − 27u + 9u − 1)y + (8u − 216u + 216u − 72u )y +
+36u8 − 336u6 + 504u4 − 144u2 + 4
Reparameterization of hypersurfaces with respect to convolutions
=
0,
=
0.
7
May 12, 2009
28 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Example 3 – again
1
We use the concept of extended convolution ideals and compute the bases
GI˜a and GI˜b
2
Next, we substitute the normal fields
>
>
nb = 1 − s2 , 2s , na = (u2 − 3)u, 3u2 − 1
into the last two polynomials of GI˜a and GI˜b , respectively, and obtain
(−76s2 − 76s6 − 282s4 + s8 + 1)x + (−12s8 − 48s6 − 72s4 − 48s2 − 12)y 2 +
+(−96s5 − 32s7 + 96s3 + 32s)y + 2s8 + 16s6 + 2 − 36s4 + 16s2 =
0,
(16s8 + 64s6 + 96s4 + 64s2 + 16)y 3 + (−3s8 − 108s6 − 210s4 −
−108s2 − 3)y + 8s7 − 24s5 + 24s3 − 8s =
0,
(−21u4 + 24u2 − 1 + 2u6 )x + (3u + 3u5 − 10u3 )y + 9u4 − 6u2 + 1
6
4
2
2
9
3
5
(27u − 27u + 9u − 1)y + (8u − 216u + 216u − 72u )y +
+36u8 − 336u6 + 504u4 − 144u2 + 4
Reparameterization of hypersurfaces with respect to convolutions
=
0,
=
0.
7
May 12, 2009
28 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Example 3 – again
1
We use the concept of extended convolution ideals and compute the bases
GI˜a and GI˜b
2
Next, we substitute the normal fields
>
>
nb = 1 − s2 , 2s , na = (u2 − 3)u, 3u2 − 1
into the last two polynomials of GI˜a and GI˜b , respectively, and obtain
(−76s2 − 76s6 − 282s4 + s8 + 1)x + (−12s8 − 48s6 − 72s4 − 48s2 − 12)y 2 +
+(−96s5 − 32s7 + 96s3 + 32s)y + 2s8 + 16s6 + 2 − 36s4 + 16s2 =
0,
(16s8 + 64s6 + 96s4 + 64s2 + 16)y 3 + (−3s8 − 108s6 − 210s4 −
−108s2 − 3)y + 8s7 − 24s5 + 24s3 − 8s =
0,
(−21u4 + 24u2 − 1 + 2u6 )x + (3u + 3u5 − 10u3 )y + 9u4 − 6u2 + 1
6
4
2
2
9
3
5
(27u − 27u + 9u − 1)y + (8u − 216u + 216u − 72u )y +
+36u8 − 336u6 + 504u4 − 144u2 + 4
Reparameterization of hypersurfaces with respect to convolutions
=
0,
=
0.
7
May 12, 2009
28 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Example 3 – again
1
We use the concept of extended convolution ideals and compute the bases
GI˜a and GI˜b
2
Next, we substitute the normal fields
>
>
nb = 1 − s2 , 2s , na = (u2 − 3)u, 3u2 − 1
into the last two polynomials of GI˜a and GI˜b , respectively, and obtain
(−76s2 − 76s6 − 282s4 + s8 + 1)x + (−12s8 − 48s6 − 72s4 − 48s2 − 12)y 2 +
+(−96s5 − 32s7 + 96s3 + 32s)y + 2s8 + 16s6 + 2 − 36s4 + 16s2 =
0,
(16s8 + 64s6 + 96s4 + 64s2 + 16)y 3 + (−3s8 − 108s6 − 210s4 −
−108s2 − 3)y + 8s7 − 24s5 + 24s3 − 8s =
:-(
0,
In both cases with non-rational solutions.
(−21u4 + 24u2 − 1 + 2u6 )x + (3u + 3u5 − 10u3 )y + 9u4 − 6u2 + 1
6
4
2
2
9
3
5
(27u − 27u + 9u − 1)y + (8u − 216u + 216u − 72u )y +
+36u8 − 336u6 + 504u4 − 144u2 + 4
Reparameterization of hypersurfaces with respect to convolutions
=
0,
=
0.
7
May 12, 2009
28 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Support function representation
I
Recently, the (explicit) support function representation of hypersurfaces has
been introduced to CAGD.
Šír, Z., Gravesen, J., Jüttler, B.:
Computing convolutions and Minkowski sums via support functions.
In: Curve and Surface Design: Avignon 2006, pp. 244-253. Nashboro Press, 2007.
Gravesen, J., Jüttler, B., Šír, Z.:
On rationally supported surfaces.
Computer Aided Geometric Design, Vol. 25, pp. 320-331. Elsevier, 2008.
I
The SF representation of hypersurfaces is a certain kind of dual
representation. A hypersurface is described as the envelope of its tangent
hyperplanes
Tn := {x : n · x = h(n)},
where the support function h(n) is a function defined on the sphere
Sn ⊂ Rn+1 (or its suitable subset).
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
29 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Support function representation
I
Recently, the (explicit) support function representation of hypersurfaces has
been introduced to CAGD.
Šír, Z., Gravesen, J., Jüttler, B.:
Computing convolutions and Minkowski sums via support functions.
In: Curve and Surface Design: Avignon 2006, pp. 244-253. Nashboro Press, 2007.
Gravesen, J., Jüttler, B., Šír, Z.:
On rationally supported surfaces.
Computer Aided Geometric Design, Vol. 25, pp. 320-331. Elsevier, 2008.
I
The SF representation of hypersurfaces is a certain kind of dual
representation. A hypersurface is described as the envelope of its tangent
hyperplanes
Tn := {x : n · x = h(n)},
where the support function h(n) is a function defined on the sphere
Sn ⊂ Rn+1 (or its suitable subset).
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
29 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Support function representation
I
Recently, the (explicit) support function representation of hypersurfaces has
been introduced to CAGD.
Šír, Z., Gravesen, J., Jüttler, B.:
Computing convolutions and Minkowski sums via support functions.
In: Curve and Surface Design: Avignon 2006, pp. 244-253. Nashboro Press, 2007.
Switching
fromB.,
h Šír,
to the
Gravesen,
J., Jüttler,
Z.: parametric representation
On rationally supported surfaces.
Composing the mapping x
Computer Aided Geometric Design, hVol. 25, pp. 320-331. Elsevier, 2008.
I
of dual
xh :of
n hypersurfaces
7→ xh (n) := h(n)n
+ ∇S hkind
The SF representation
is a certain
n
representation. A hypersurface is described as the envelope of its tangent
with any rational parameterization of the sphere we
hyperplanes
obtain a parametric
representation
the hypersurface.
Tn :=
{x : n · x =ofh(n)},
where the support function h(n) is a function defined on the sphere
Sn ⊂ Rn+1 (or its suitable subset).
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
29 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Support function representation
I
Recently, the (explicit) support function representation of hypersurfaces has
been introduced to CAGD.
Šír, Z., Gravesen, J., Jüttler, B.:
Computing convolutions and Minkowski sums via support functions.
In: Curve and Surface Design: Avignon 2006, pp. 244-253. Nashboro Press, 2007.
ConditionJ.,ofJüttler,
pseudoconvexicity
Gravesen,
B., Šír, Z.:
On
rationally
supportedor
surfaces.
Given
a parametric
implicit representation of a hypersurface,
Computer Aided Geometric Design, Vol. 25, pp. 320-331. Elsevier, 2008.
I
it is not always possible to represent it via SF – mainly due to
the representation
fact, that for each
vector n only
value kind
of h of
is possible.
The SF
of hypersurfaces
is one
a certain
dual
representation. A hypersurface is described as the envelope of its tangent
hyperplanes
Tn := {x : n · x = h(n)},
where the support function h(n) is a function defined on the sphere
Sn ⊂ Rn+1 (or its suitable subset).
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
29 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Implicit Support Function (ISF) representation
I
A hypersurface is again represented as an envelope of hyperplanes
Tn,h := {x : n · x = h},
where n ∈ Sn and h satisfy the implicit homogeneous polynomial equation
D(n, h) = 0.
I
From now on, not only one value of h is assumed – multivalued or implicit
support function (ISF) representation (available for all algebraic
hypersurfaces with non-degenerated Gauss image).
I
The dual representation does not generally require the normalized normal
vectors n – on the contrary, SF is a function defined only on the unit
sphere Sn
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
30 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Implicit Support Function (ISF) representation
I
A hypersurface is again represented as an envelope of hyperplanes
Tn,h := {x : n · x = h},
where n ∈ Sn and h satisfy the implicit homogeneous polynomial equation
D(n, h) = 0.
I
From now on, not only one value of h is assumed – multivalued or implicit
support function (ISF) representation (available for all algebraic
hypersurfaces with non-degenerated Gauss image).
I
The dual representation does not generally require the normalized normal
vectors n – on the contrary, SF is a function defined only on the unit
sphere Sn
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
30 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Implicit Support Function (ISF) representation
I
A hypersurface is again represented as an envelope of hyperplanes
Tn,h := {x : n · x = h},
where n ∈ Sn and h satisfy the implicit homogeneous polynomial equation
D(n, h) = 0.
I
From now on, not only one value of h is assumed – multivalued or implicit
support function (ISF) representation (available for all algebraic
hypersurfaces with non-degenerated Gauss image).
I
The dual representation does not generally require the normalized normal
vectors n – on the contrary, SF is a function defined only on the unit
sphere Sn
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
30 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Implicit Support Function (ISF) representation
I
A hypersurface is again represented as an envelope of hyperplanes
Tn,h := {x : n · x = h},
I
I
Rational
implicit homogeneous polynomial equation
where
n ∈ parameterization
Sn and h satisfy of
theISF
Rationality of the dual hypersurface does not guarantee the
D(n, h) =of0.the corresponding ISF –
existence of a rational parameterization
which is a simultaneous rational parameterization of D(n, h) = 0
and of
n on,
∈ Snnot
. only one value of h is assumed – multivalued or implicit
From
now
support function (ISF) representation (available for all algebraic
hypersurfaces with non-degenerated Gauss image).
The dual representation does not generally require the normalized normal
vectors n – on the contrary, SF is a function defined only on the unit
sphere Sn
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
30 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
ISFs – computation
How to compute ISF for a hypersurface given parametrically?
Input: Parameterization x : Rn → Rn+1 : (u1 , . . . , un ) → (x1 , . . . , xn+1 ).
Output: Implicit support function representation D(n, h) = 0.
∂x
∂x
1: I := hn · ∂u
, . . . , n · ∂u
, n · x − h, 1 − whi
1
n
2: ≺:= a term order such that w and each ui is greater than any ni and h
3: G := a Gröbner basis of I w.r.t ≺
4: D := G ∩ k[n1 , . . . , nn+1 , h]
5: return D
How to compute ISF for a hypersurface given implicitly?
Input: Polynomial F (x1 , . . . , xn+1 ) which zero set represents a hypersurface.
Output: Implicit support function representation D(n, h) = 0.
∂F
1: I := hF, ∂x
− λn1 , . . . , ∂x∂F
− λnn+1 , n · x − h, 1 − whi
1
n+1
2: ≺:= a term order such that w, λ and each xi is greater than any ni and h
3: G := a Gröbner basis of I w.r.t ≺
4: D := G ∩ k[n1 , . . . , nn+1 , h]
5: return D
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
31 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
ISFs – computation
How to compute ISF for a hypersurface given parametrically?
Input: Parameterization x : Rn → Rn+1 : (u1 , . . . , un ) → (x1 , . . . , xn+1 ).
Output: Implicit support function representation D(n, h) = 0.
∂x
∂x
1: I := hn · ∂u
, . . . , n · ∂u
, n · x − h, 1 − whi
1
n
2: ≺:= a term order such that w and each ui is greater than any ni and h
3: G := a Gröbner basis of I w.r.t ≺
4: D := G ∩ k[n1 , . . . , nn+1 , h]
5: return D
How to compute ISF for a hypersurface given implicitly?
Input: Polynomial F (x1 , . . . , xn+1 ) which zero set represents a hypersurface.
Output: Implicit support function representation D(n, h) = 0.
∂F
1: I := hF, ∂x
− λn1 , . . . , ∂x∂F
− λnn+1 , n · x − h, 1 − whi
1
n+1
2: ≺:= a term order such that w, λ and each xi is greater than any ni and h
3: G := a Gröbner basis of I w.r.t ≺
4: D := G ∩ k[n1 , . . . , nn+1 , h]
5: return D
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
31 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Example 4 – Enneper surface
a(u, v) =
3
3
(u − u3 + uv 2 , −v − u2 v + v3 , u2 − v 2 )>
8
4
F (x, y, z) =
9
0
7
2 6
2 6
−64z + 1152z + 432x z − 432y z +
3888x2 z 5 + 3888y 2 z 5 − 5184z 5 +
6480x2 z 4 − 6480y 2 z 4 + 1215x4 z 3 +
1215y 4 z 3 − 3888x2 z 3 + 6318x2 y 2 z 3 −
3888y 2 z 3 + 4374x4 z 2 − 4374y 4 z 2 −
729x4 z − 729y 4 z + 1458x2 y 2 z +
729x6 − 729y 6 + 2187x2 y 4 − 2187x4 y 2
Reparameterization of hypersurfaces with respect to convolutions
-4
-20
-8
-20
-10
0
-10
0
10
10
20
20
D(n1 , n2 , n3 , h) =
−4n61 +9h2 n41 +4n22 n41 −3n23 n41 −18hn3 n41 +
4n42 n21 −12hn33 n21 +18h2 n22 n21 +6n22 n23 n21 −
4n62 +9h2 n42 +12hn22 n33 −3n42 n23 +18hn42 n3
May 12, 2009
32 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Example 4 – Enneper surface
a(u, v) =
3
3
(u − u3 + uv 2 , −v − u2 v + v3 , u2 − v 2 )>
8
4
F (x, y, z) =
9
0
7
2 6
2 6
−64z + 1152z + 432x z − 432y z +
3888x2 z 5 + 3888y 2 z 5 − 5184z 5 +
6480x2 z 4 − 6480y 2 z 4 + 1215x4 z 3 +
1215y 4 z 3 − 3888x2 z 3 + 6318x2 y 2 z 3 −
3888y 2 z 3 + 4374x4 z 2 − 4374y 4 z 2 −
729x4 z − 729y 4 z + 1458x2 y 2 z +
729x6 − 729y 6 + 2187x2 y 4 − 2187x4 y 2
Reparameterization of hypersurfaces with respect to convolutions
-4
-20
-8
-20
-10
0
-10
0
10
10
20
20
D(n1 , n2 , n3 , h) =
−4n61 +9h2 n41 +4n22 n41 −3n23 n41 −18hn3 n41 +
4n42 n21 −12hn33 n21 +18h2 n22 n21 +6n22 n23 n21 −
4n62 +9h2 n42 +12hn22 n33 −3n42 n23 +18hn42 n3
May 12, 2009
32 / 46
Reparameterizing both hypersurfaces
Example 4 – Enneper surface
www.KMA.zcu.cz
h(n1 , n2 , n2 ) =
2
2
2
2
n3 (3n2
1 +2n3 +3n2 )±2(n1 −n2 )
√
(n2
+n2
+n2
)3
1
2
3
(n2
+n2
)2
1
2
a(u, v) =
3
3
(u − u3 + uv 2 , −v − u2 v + v3 , u2 − v 2 )>
8
4
F (x, y, z) =
9
0
7
2 6
2 6
−64z + 1152z + 432x z − 432y z +
3888x2 z 5 + 3888y 2 z 5 − 5184z 5 +
6480x2 z 4 − 6480y 2 z 4 + 1215x4 z 3 +
1215y 4 z 3 − 3888x2 z 3 + 6318x2 y 2 z 3 −
3888y 2 z 3 + 4374x4 z 2 − 4374y 4 z 2 −
729x4 z − 729y 4 z + 1458x2 y 2 z +
729x6 − 729y 6 + 2187x2 y 4 − 2187x4 y 2
Reparameterization of hypersurfaces with respect to convolutions
-4
-20
-8
-20
-10
0
-10
0
10
10
20
20
D(n1 , n2 , n3 , h) =
−4n61 +9h2 n41 +4n22 n41 −3n23 n41 −18hn3 n41 +
4n42 n21 −12hn33 n21 +18h2 n22 n21 +6n22 n23 n21 −
4n62 +9h2 n42 +12hn22 n33 −3n42 n23 +18hn42 n3
May 12, 2009
32 / 46
Reparameterizing both hypersurfaces
Example 4 – Enneper surface
www.KMA.zcu.cz
h(n1 , n2 , n2 ) =
2
2
2
2
n3 (3n2
1 +2n3 +3n2 )±2(n1 −n2 )
√
(n2
+n2
+n2
)3
1
2
3
(n2
+n2
)2
1
2
a(u, v) =
3
3
(u − u3 + uv 2 , −v − u2 v + v3 , u2 − v 2 )>
8
4
F (x, y, z) =
9
0
7
2 6
2 6
−64z + 1152z + 432x z − 432y z +
3888x2 z 5 + 3888y 2 z 5 − 5184z 5 +
6480x2 z 4 − 6480y 2 z 4 + 1215x4 z 3 +
1215y 4 z 3 − 3888x2 z 3 + 6318x2 y 2 z 3 −
3888y 2 z 3 + 4374x4 z 2 − 4374y 4 z 2 −
729x4 z − 729y 4 z + 1458x2 y 2 z +
729x6 − 729y 6 + 2187x2 y 4 − 2187x4 y 2
Reparameterization of hypersurfaces with respect to convolutions
-4
-20
-8
-20
-10
0
-10
0
10
10
20
20
D(n1 , n2 , n3 , h) =
−4n61 +9h2 n41 +4n22 n41 −3n23 n41 −18hn3 n41 +
4n42 n21 −12hn33 n21 +18h2 n22 n21 +6n22 n23 n21 −
4n62 +9h2 n42 +12hn22 n33 −3n42 n23 +18hn42 n3
May 12, 2009
32 / 46
Reparameterizing both hypersurfaces
Example 4 – Enneper surface
www.KMA.zcu.cz
h(n1 , n2 , n2 ) =
2
2
2
2
n3 (3n2
1 +2n3 +3n2 )±2(n1 −n2 )
(n2
+n2
)2
1
2
a(u, v) =
3
3
(u − u3 + uv 2 , −v − u2 v + v3 , u2 − v 2 )>
8
4
F (x, y, z) =
9
0
7
2 6
2 6
−64z + 1152z + 432x z − 432y z +
3888x2 z 5 + 3888y 2 z 5 − 5184z 5 +
6480x2 z 4 − 6480y 2 z 4 + 1215x4 z 3 +
1215y 4 z 3 − 3888x2 z 3 + 6318x2 y 2 z 3 −
3888y 2 z 3 + 4374x4 z 2 − 4374y 4 z 2 −
729x4 z − 729y 4 z + 1458x2 y 2 z +
729x6 − 729y 6 + 2187x2 y 4 − 2187x4 y 2
Reparameterization of hypersurfaces with respect to convolutions
-4
-20
-8
-20
-10
0
-10
0
10
10
20
20
D(n1 , n2 , n3 , h) =
−4n61 +9h2 n41 +4n22 n41 −3n23 n41 −18hn3 n41 +
4n42 n21 −12hn33 n21 +18h2 n22 n21 +6n22 n23 n21 −
4n62 +9h2 n42 +12hn22 n33 −3n42 n23 +18hn42 n3
May 12, 2009
32 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Dual and ISF representation
Properties of D(n, h)
1
A is irreducible if and only if D(n, h) is irreducible.
2
There exists a rational representation of A if and only if the zero locus of
the corresponding D(n, h) = 0 is rational.
3
This representation is very suitable for describing convolutions of
hypersurfaces as this operation corresponds to the sum of the associated
support functions, i.e., h3 = h1 + h2 .
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
33 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Dual and ISF representation
Properties of D(n, h)
1
A is irreducible if and only if D(n, h) is irreducible.
2
There exists a rational representation of A if and only if the zero locus of
the corresponding D(n, h) = 0 is rational.
3
This representation is very suitable for describing convolutions of
hypersurfaces as this operation corresponds to the sum of the associated
support functions, i.e., h3 = h1 + h2 .
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
33 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Dual and ISF representation
Properties of D(n, h)
1
A is irreducible if and only if D(n, h) is irreducible.
2
There exists a rational representation of A if and only if the zero locus of
the corresponding D(n, h) = 0 is rational.
3
This representation is very suitable for describing convolutions of
hypersurfaces as this operation corresponds to the sum of the associated
support functions, i.e., h3 = h1 + h2 .
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
33 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Support functions and convolutions
c=a+b
h1 + h2 = h3
b
h2
B
a
h1
A
0
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
34 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Convolution hypersurfaces – computation 2
How to compute convolutions using ISFs
Input: Polynomials D1 (n1 , . . . , nn+1 , h1 ) and D2 (n1 , . . . , nn+1 , h2 ) which
represent hypersurfaces A and B.
Output: Implicit support function D3 (n1 , . . . , nn+1 , h3 ) of the convolution
hypersurface C = A ? B.
1: I := hD1 , D2 , h3 − h1 − h2 i
2: ≺:= a term order such that h1 and h2 are greater than h3 and any ni
3: G := a Gröbner basis of I w.r.t ≺
4: D3 := G ∩ k[h3 , n1 , . . . , nn+1 ]
5: return D3
Remark
The operation of convolution A ? B is characterized by the simple condition
h3 = h1 + h2 , as in the (explicit) SF case.
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
35 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Convolution hypersurfaces – computation 2
How to compute convolutions using ISFs
Input: Polynomials D1 (n1 , . . . , nn+1 , h1 ) and D2 (n1 , . . . , nn+1 , h2 ) which
represent hypersurfaces A and B.
Output: Implicit support function D3 (n1 , . . . , nn+1 , h3 ) of the convolution
hypersurface C = A ? B.
1: I := hD1 , D2 , h3 − h1 − h2 i
2: ≺:= a term order such that h1 and h2 are greater than h3 and any ni
3: G := a Gröbner basis of I w.r.t ≺
4: D3 := G ∩ k[h3 , n1 , . . . , nn+1 ]
5: return D3
Remark
The operation of convolution A ? B is characterized by the simple condition
h3 = h1 + h2 , as in the (explicit) SF case.
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Convolution degree of a hypersurface
Definition
Let D(n, h) = 0 be the implicit support function of A. Then the number
∆(A) = degh (D(n, h))
is called the convolution degree of the hypersurface A.
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Convolution degree of a hypersurface
Definition
Let D(n, h) = 0 be the implicit support function of A. Then the number
∆(A) = degh (D(n, h))
is called the convolution degree of the hypersurface A.
Remark
I
The convolution degree indicates the number of points at A corresponding
to the chosen direction.
I
If ∆(A) = 1 then A is a LN hypersurface and we can easily switch to the
standard (explicit) support function description
h(n) =
p(n)
,
q(n)
n ∈ Sn , deg(q) = deg(p) − 1
with the property h(n) = −h(−n).
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Convolution degree of a hypersurface
Definition
Let D(n, h) = 0 be the implicit support function of A. Then the number
∆(A) = degh (D(n, h))
is called the convolution degree of the hypersurface A.
Remark
I
For a hypersurface A parameterized by a(ū)
∆(A) · index(a(ū)) = δ(a(ū))
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SFs and components of convolution hypersurfaces
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SFs and components of convolution hypersurfaces
Simple and special components
Analogously to primal hypersurfaces, we distinguish between simple
and special components of the corresponding dual hypersurface
described by the implicit support function.
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Example 5 – (Cardioid) ? (Tschirnhausen cubic) . . . again
1
We compute ISFs of cardioid and T-cubic
D1 (n1 , n2 , h1 ) = 16h31 + 24n1 h21 − 27n22 h1 − 15n21 h1 + 2n31
D2 (n1 , n2 , h2 ) = 9h22 n22 − 12h2 n31 − 18h2 n1 n22 − 3n21 n22 − 4n42
2
. . . and applying the presented algorithm we find ISF of the convolution curve
6
6
6
3
4
5
8
6
2
D3 (n1 , n2 , h3 ) = (186624n2 )h3 + (−559872n1 n2 − 746496n1 n2 )h3 + (−878688n2 − 676512n2 n1 +
4
4
2
6
4
9
8
6
3
4
5
3
+1119744n2 n1 + 995328n2 n1 )h3 + (−442368n1 + 2072304n2 n1 + 4944240n2 n1 + 3297024n 2 n1 )h3 +
10
8 2
4 6
6 4
2 8
10
2
+(−663552n1 − 3504384n1 n2 − 4316247n1 n2 − 6453216n1 n2 − 63990n1 n2 + 642033n2 )h3 +
11
9 2
1
5 6
7 4
3 8
+(414720n1 + 1938816n1 n2 − 753570n1 n2 0 + 772362n1 n2 + 2832948n1 n2 − 1290204n1 n2 )h3 +
8 4
4 8
2 10
10 2
6 6
12
12
+(289224n1 n2 + 458010n1 n2 − 108387n1 n2 − 82944n1 n2 + 755109n1 n2 − 55296n1 − 128164n2 )
Result
Computing genus, we can determine that the convolution curve of cardioid and
Tschirnhausen cubic is a rational curve.
Reparameterization of hypersurfaces with respect to convolutions
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Reparameterizing both hypersurfaces
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Example 5 – (Cardioid) ? (Tschirnhausen cubic) . . . again
1
We compute ISFs of cardioid and T-cubic
D1 (n1 , n2 , h1 ) = 16h31 + 24n1 h21 − 27n22 h1 − 15n21 h1 + 2n31
D2 (n1 , n2 , h2 ) = 9h22 n22 − 12h2 n31 − 18h2 n1 n22 − 3n21 n22 − 4n42
2
. . . and applying the presented algorithm we find ISF of the convolution curve
6
6
6
3
4
5
8
6
2
D3 (n1 , n2 , h3 ) = (186624n2 )h3 + (−559872n1 n2 − 746496n1 n2 )h3 + (−878688n2 − 676512n2 n1 +
4
4
2
6
4
9
8
6
3
4
5
3
+1119744n2 n1 + 995328n2 n1 )h3 + (−442368n1 + 2072304n2 n1 + 4944240n2 n1 + 3297024n 2 n1 )h3 +
10
8 2
4 6
6 4
2 8
10
2
+(−663552n1 − 3504384n1 n2 − 4316247n1 n2 − 6453216n1 n2 − 63990n1 n2 + 642033n2 )h3 +
11
9 2
1
5 6
7 4
3 8
+(414720n1 + 1938816n1 n2 − 753570n1 n2 0 + 772362n1 n2 + 2832948n1 n2 − 1290204n1 n2 )h3 +
8 4
4 8
2 10
10 2
6 6
12
12
+(289224n1 n2 + 458010n1 n2 − 108387n1 n2 − 82944n1 n2 + 755109n1 n2 − 55296n1 − 128164n2 )
Result
Computing genus, we can determine that the convolution curve of cardioid and
Tschirnhausen cubic is a rational curve.
Reparameterization of hypersurfaces with respect to convolutions
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Reparameterizing both hypersurfaces
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Example 5 – (Cardioid) ? (Tschirnhausen cubic) . . . again
1
We compute ISFs of cardioid and T-cubic
D1 (n1 , n2 , h1 ) = 16h31 + 24n1 h21 − 27n22 h1 − 15n21 h1 + 2n31
D2 (n1 , n2 , h2 ) = 9h22 n22 − 12h2 n31 − 18h2 n1 n22 − 3n21 n22 − 4n42
2
. . . and applying the presented algorithm we find ISF of the convolution curve
6
6
6
3
4
5
8
6
2
D3 (n1 , n2 , h3 ) = (186624n2 )h3 + (−559872n1 n2 − 746496n1 n2 )h3 + (−878688n2 − 676512n2 n1 +
4
4
2
6
4
9
8
6
3
4
5
3
+1119744n2 n1 + 995328n2 n1 )h3 + (−442368n1 + 2072304n2 n1 + 4944240n2 n1 + 3297024n 2 n1 )h3 +
10
8 2
4 6
6 4
2 8
10
2
+(−663552n1 − 3504384n1 n2 − 4316247n1 n2 − 6453216n1 n2 − 63990n1 n2 + 642033n2 )h3 +
11
9 2
1
5 6
7 4
3 8
+(414720n1 + 1938816n1 n2 − 753570n1 n2 0 + 772362n1 n2 + 2832948n1 n2 − 1290204n1 n2 )h3 +
8 4
4 8
2 10
10 2
6 6
12
12
+(289224n1 n2 + 458010n1 n2 − 108387n1 n2 − 82944n1 n2 + 755109n1 n2 − 55296n1 − 128164n2 )
Result
Computing genus, we can determine that the convolution curve of cardioid and
Tschirnhausen cubic is a rational curve.
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
38 / 46
Reparameterizing both hypersurfaces
www.KMA.zcu.cz
Example 5 – (Cardioid) ? (Tschirnhausen cubic) . . . again
1
We compute ISFs of cardioid and T-cubic
D1 (n1 , n2 , h1 ) = 16h31 + 24n1 h21 − 27n22 h1 − 15n21 h1 + 2n31
D2 (n1 , n2 , h2 ) = 9h22 n22 − 12h2 n31 − 18h2 n1 n22 − 3n21 n22 − 4n42
2
. . . and applying the presented algorithm we find ISF of the convolution curve
6
6
6
3
4
5
8
6
2
D3 (n1 , n2 , h3 ) = (186624n2 )h3 + (−559872n1 n2 − 746496n1 n2 )h3 + (−878688n2 − 676512n2 n1 +
4
4
2
6
4
9
8
6
3
4
5
3
+1119744n2 n1 + 995328n2 n1 )h3 + (−442368n1 + 2072304n2 n1 + 4944240n2 n1 + 3297024n 2 n1 )h3 +
10
8 2
4 6
6 4
2 8
10
2
+(−663552n1 − 3504384n1 n2 − 4316247n1 n2 − 6453216n1 n2 − 63990n1 n2 + 642033n2 )h3 +
11
9 2
1
5 6
7 4
3 8
+(414720n1 + 1938816n1 n2 − 753570n1 n2 0 + 772362n1 n2 + 2832948n1 n2 − 1290204n1 n2 )h3 +
8 4
4 8
2 10
10 2
6 6
12
12
+(289224n1 n2 + 458010n1 n2 − 108387n1 n2 − 82944n1 n2 + 755109n1 n2 − 55296n1 − 128164n2 )
Result
Computing genus, we can determine that the convolution curve of cardioid and
Tschirnhausen cubic is a rational curve.
Reparameterization of hypersurfaces with respect to convolutions
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38 / 46
Reparameterizing both hypersurfaces
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Example 5 – (Cardioid) ? (Tschirnhausen cubic) . . . again
1
We compute ISFs of cardioid and T-cubic
D1 (n1 , n2 , h1 ) = 16h31 + 24n1 h21 − 27n22 h1 − 15n21 h1 + 2n31
D2 (n1 , n2 , h2 ) = 9h22 n22 − 12h2 n31 − 18h2 n1 n22 − 3n21 n22 − 4n42
2
. . . and applying the presented algorithm we find ISF of the convolution curve
6
6
6
3
4
5
8
6
2
D3 (n1 , n2 , h3 ) = (186624n2 )h3 + (−559872n1 n2 − 746496n1 n2 )h3 + (−878688n2 − 676512n2 n1 +
4
4
2
6
4
9
8
6
3
4
5
3
+1119744n2 n1 + 995328n2 n1 )h3 + (−442368n1 + 2072304n2 n1 + 4944240n2 n1 + 3297024n 2 n1 )h3 +
10
8 2
4 6
6 4
2 8
10
2
+(−663552n1 − 3504384n1 n2 − 4316247n1 n2 − 6453216n1 n2 − 63990n1 n2 + 642033n2 )h3 +
11
9 2
1
5 6
7 4
3 8
+(414720n1 + 1938816n1 n2 − 753570n1 n2 0 + 772362n1 n2 + 2832948n1 n2 − 1290204n1 n2 )h3 +
8 4
4 8
2 10
10 2
6 6
12
12
+(289224n1 n2 + 458010n1 n2 − 108387n1 n2 − 82944n1 n2 + 755109n1 n2 − 55296n1 − 128164n2 )
Result
Computing genus, we can determine that the convolution curve of cardioid and
Tschirnhausen cubic is a rational curve.
Reparameterization of hypersurfaces with respect to convolutions
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Reparameterizing both hypersurfaces
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Fundamental property
Theorem
Let a(ū) and b(s̄) be parameterizations of the rational hypersurfaces A and B,
respectively. Let DC (n, hC ) be the dual representation of the convolution
hypersurface C = A ? B. If (n(t̄), hC (t̄))> is a rational parameterization of a
simple component of DC (n, hC ), then there exist parameterizations a(t̄) and
b(t̄) of A, B which are coherent with respect to the normal field n(t̄) and
therefore with respect to each other.
Proof.
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Fundamental property
Theorem
Let a(ū) and b(s̄) be parameterizations of the rational hypersurfaces A and B,
respectively. Let DC (n, hC ) be the dual representation of the convolution
hypersurface C = A ? B. If (n(t̄), hC (t̄))> is a rational parameterization of a
simple component of DC (n, hC ), then there exist parameterizations a(t̄) and
b(t̄) of A, B which are coherent with respect to the normal field n(t̄) and
therefore with respect to each other.
Proof.
M
)
DA (n, hA ) = 0
n+1
(n : hC : hA : hB ) ∈ k
× k × k × k DB (n, hB ) = 0
D (n, h ) = 0
C
C
=
(
=
M1 ∪ M2 ∪ . . . ∪ Ml .
Reparameterization of hypersurfaces with respect to convolutions
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Fundamental property
Theorem
Let a(ū) and b(s̄) be parameterizations of the rational hypersurfaces A and B,
respectively. Let DC (n, hC ) be the dual representation of the convolution
hypersurface C = A ? B. If (n(t̄), hC (t̄))> is a rational parameterization of a
simple component of DC (n, hC ), then there exist parameterizations a(t̄) and
b(t̄) of A, B which are coherent with respect to the normal field n(t̄) and
therefore with respect to each other.
Proof.
M
)
DA (n, hA ) = 0
n+1
(n : hC : hA : hB ) ∈ k
× k × k × k DB (n, hB ) = 0
D (n, h ) = 0
C
C
=
(
=
M1 ∪ M2 ∪ . . . ∪ Ml .
Further, we consider
π2 :
π3 :
π4 :
the natural projections
M → V(DC ) : (n : hC : hA : hB ) 7→ (n : hC ),
M → V(DA ) : (n : hC : hA : hB ) 7→ (n : hA ),
M → V(DB ) : (n : hC : hA : hB ) 7→ (n : hB ),
Reparameterization of hypersurfaces with respect to convolutions
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Fundamental property
Theorem
Let a(ū) and b(s̄) be parameterizations of the rational hypersurfaces A and B,
respectively. Let DC (n, hC ) be the dual representation of the convolution
hypersurface C = A ? B. If (n(t̄), hC (t̄))> is a rational parameterization of a
simple component of DC (n, hC ), then there exist parameterizations a(t̄) and
b(t̄) of A, B which are coherent with respect to the normal field n(t̄) and
therefore with respect to each other.
Proof.
All natural projections π2 , π3 , π4 are rational. Moreover, for simple component C1 ,
π2 : M1 → V(DC1 ) is 1:1, i.e. birational.
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Fundamental property
Theorem
Let a(ū) and b(s̄) be parameterizations of the rational hypersurfaces A and B,
respectively. Let DC (n, hC ) be the dual representation of the convolution
hypersurface C = A ? B. If (n(t̄), hC (t̄))> is a rational parameterization of a
simple component of DC (n, hC ), then there exist parameterizations a(t̄) and
b(t̄) of A, B which are coherent with respect to the normal field n(t̄) and
therefore with respect to each other.
Proof.
All natural projections π2 , π3 , π4 are rational. Moreover, for simple component C1 ,
π2 : M1 → V(DC1 ) is 1:1, i.e. birational.
Hence, we obtain two maps
ξ : π3 ◦ π2−1 ,
ζ : π4 ◦ π2−1 ,
which are rational. This implies that any rational parameterization of DC provides
rational parameterizations of DA and DB .
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Example 5 – continued
I We find a rational parameterization of D3
n1 (t) =
6t−20t3 +6t5
,
t6 +3t4 +16t3 +3t2 +1
5
9
7
11
n2 (t) =
6
1−15t2 +15t4 −t6
t6 +3t4 +16t3 +3t2 +1
3
4
2
8
10
12
1−3t−54t −23t −54t −3t +264t −23t −9t −12t −9t −12t +t
h3 (t) = − 32 1+14t
3 +t12 +14t9 +6t11 +6t−300t6 +6t10 +111t8 +12t7 +111t4 +12t5 +6t2
I Applying Method 1 we obtain the following rational reparameterizations u = φi (t)
and s = ψj (t)
φ1 : u =
and
√
√
√
t4 3 − 8t3 + 2 3t2 + 8t + 3
,
t4 − 14t2 + 1
√
√
√
t4 3 + 8t3 + 2 3t2 − 8t + 3
φ3 : u = −
,
t4 − 14t2 + 1
2t
,
1 − t2
ψ1 : s = −
φ2 : u =
t3 + 3t2 − 3t − 1
,
t3 − 3t2 − 3t + 1
ψ2 : s =
t3 − 3t2 − 3t + 1
.
t3 + 3t2 − 3t − 1
I An arbitrary pair (φi , ψj ), i = 1, 2, 3, j = 1, 2, yields coherent parameterizations
a(φi (t)), b(ψj (t)) of the cardioid and the Tschirnhausen cubic, respectively.
Reparameterization of hypersurfaces with respect to convolutions
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Example 5 – continued
I We find a rational parameterization of D3
n1 (t) =
6t−20t3 +6t5
,
t6 +3t4 +16t3 +3t2 +1
5
9
7
11
n2 (t) =
6
1−15t2 +15t4 −t6
t6 +3t4 +16t3 +3t2 +1
3
4
2
8
10
12
1−3t−54t −23t −54t −3t +264t −23t −9t −12t −9t −12t +t
h3 (t) = − 32 1+14t
3 +t12 +14t9 +6t11 +6t−300t6 +6t10 +111t8 +12t7 +111t4 +12t5 +6t2
I Applying Method 1 we obtain the following rational reparameterizations u = φi (t)
and s = ψj (t)
φ1 : u =
and
√
√
√
t4 3 − 8t3 + 2 3t2 + 8t + 3
,
t4 − 14t2 + 1
√
√
√
t4 3 + 8t3 + 2 3t2 − 8t + 3
φ3 : u = −
,
t4 − 14t2 + 1
2t
,
1 − t2
ψ1 : s = −
φ2 : u =
t3 + 3t2 − 3t − 1
,
t3 − 3t2 − 3t + 1
ψ2 : s =
t3 − 3t2 − 3t + 1
.
t3 + 3t2 − 3t − 1
I An arbitrary pair (φi , ψj ), i = 1, 2, 3, j = 1, 2, yields coherent parameterizations
a(φi (t)), b(ψj (t)) of the cardioid and the Tschirnhausen cubic, respectively.
Reparameterization of hypersurfaces with respect to convolutions
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Reparameterizing both hypersurfaces
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Example 5 – continued
I We find a rational parameterization of D3
n1 (t) =
6t−20t3 +6t5
,
t6 +3t4 +16t3 +3t2 +1
5
9
7
11
n2 (t) =
6
1−15t2 +15t4 −t6
t6 +3t4 +16t3 +3t2 +1
3
4
2
8
10
12
1−3t−54t −23t −54t −3t +264t −23t −9t −12t −9t −12t +t
h3 (t) = − 32 1+14t
3 +t12 +14t9 +6t11 +6t−300t6 +6t10 +111t8 +12t7 +111t4 +12t5 +6t2
I Applying Method 1 we obtain the following rational reparameterizations u = φi (t)
and s = ψj (t)
φ1 : u =
and
√
√
√
t4 3 − 8t3 + 2 3t2 + 8t + 3
,
t4 − 14t2 + 1
√
√
√
t4 3 + 8t3 + 2 3t2 − 8t + 3
φ3 : u = −
,
t4 − 14t2 + 1
2t
,
1 − t2
ψ1 : s = −
φ2 : u =
t3 + 3t2 − 3t − 1
,
t3 − 3t2 − 3t + 1
ψ2 : s =
t3 − 3t2 − 3t + 1
.
t3 + 3t2 − 3t − 1
I An arbitrary pair (φi , ψj ), i = 1, 2, 3, j = 1, 2, yields coherent parameterizations
a(φi (t)), b(ψj (t)) of the cardioid and the Tschirnhausen cubic, respectively.
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For Further Reading
Sendra, J.R., Winkler, F.:
Symbolic Parametrization of Curves.
Journal of Symbolic Computation, Vol. 12, pp. 607-631. Academic Press, 1990.
Schicho, J.:
Rational parametrization of surfaces.
Journal of Symbolic Computation, Vol. 26, pp. 1-29. Academic Press, 1998.
Šír, Z., Gravesen, J., Jüttler, B.:
Computing convolutions and Minkowski sums via support functions.
In: Curve and Surface Design: Avignon 2006, pp. 244-253. Nashboro Press, 2007.
Gravesen, J., Jüttler, B., Šír, Z.:
On rationally supported surfaces.
Computer Aided Geometric Design, Vol. 25, pp. 320-331. Elsevier, 2008.
Aigner, M., Jüttler, B., Gonzales-Vega, L. and Schicho, J.:
Parameterizing surfaces with certain special support functions, including offsets of quadrics
and rationally supported surface.
Journal of Symbolic Computation. Vol. 44, pp. 180-191. Elsevier, 2009.
Reparameterization of hypersurfaces with respect to convolutions
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Outline
1 Introduction
Convolution hypersurfaces
Coherent parameterizations
Statement of the problem
2 Reparameterizing one of hypersurfaces
Computing parametric convolutions
Convolution degree of parameterization
Method 1 – Example(s)
3 Finding a new parameterization of one of hypersurfaces
Extended convolution ideal
Method 2 – Example(s)
4 Reparameterizing both hypersurfaces
Motivation: Why the previous stuff is not enough?
Explicit and implicit SF representation
Convolution theory via ISF representation
Method 3 – Examples
5 Finding new parameterizations of both hypersurfaces
Method 4 - Examples
6 Conclusion
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Example 6
Let us consider two rational hypersurfaces A and B parameterized by a(ū) and
b(s̄), respectively, with rational convolution C = A ? B. However, there exist only
non-rational dependencies ū = φ(s̄), and s̄ = ψ(ū) yielding parallel normal fields
(typically, this is the case when both parameterizations a(ū), b(s̄) are not proper
parameterizations).
I
Firstly, we apply Method 3 and compute from DC (n, hC ) a suitable rational
parameterization n(t̄)
I
Then, we use Method 2 and set the obtained n(t̄) to polynomials in Gröbner
bases of the extended convolution ideals I˜a and I˜b
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Example 6
Let us consider two rational hypersurfaces A and B parameterized by a(ū) and
b(s̄), respectively, with rational convolution C = A ? B. However, there exist only
non-rational dependencies ū = φ(s̄), and s̄ = ψ(ū) yielding parallel normal fields
(typically, this is the case when both parameterizations a(ū), b(s̄) are not proper
parameterizations).
I
Firstly, we apply Method 3 and compute from DC (n, hC ) a suitable rational
parameterization n(t̄)
I
Then, we use Method 2 and set the obtained n(t̄) to polynomials in Gröbner
bases of the extended convolution ideals I˜a and I˜b
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
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Finding new parameterizations of both hypersurfaces
www.KMA.zcu.cz
Example 6
Let us consider two rational hypersurfaces A and B parameterized by a(ū) and
b(s̄), respectively, with rational convolution C = A ? B. However, there exist only
non-rational dependencies ū = φ(s̄), and s̄ = ψ(ū) yielding parallel normal fields
(typically, this is the case when both parameterizations a(ū), b(s̄) are not proper
parameterizations).
I
Firstly, we apply Method 3 and compute from DC (n, hC ) a suitable rational
parameterization n(t̄)
I
Then, we use Method 2 and set the obtained n(t̄) to polynomials in Gröbner
bases of the extended convolution ideals I˜a and I˜b
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
43 / 46
Conclusion
www.KMA.zcu.cz
Outline
1 Introduction
Convolution hypersurfaces
Coherent parameterizations
Statement of the problem
2 Reparameterizing one of hypersurfaces
Computing parametric convolutions
Convolution degree of parameterization
Method 1 – Example(s)
3 Finding a new parameterization of one of hypersurfaces
Extended convolution ideal
Method 2 – Example(s)
4 Reparameterizing both hypersurfaces
Motivation: Why the previous stuff is not enough?
Explicit and implicit SF representation
Convolution theory via ISF representation
Method 3 – Examples
5 Finding new parameterizations of both hypersurfaces
Method 4 - Examples
6 Conclusion
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
44 / 46
Conclusion
www.KMA.zcu.cz
Summary
I
The talk was devoted to the analysis of reparameterizations of given
hypersurfaces with respect to their convolution.
I
Coherent parameterizations of rational hypersurfaces were introduced and
presented.
I
The presented approach is an “economic” one – we try to find the simplest
possible way to coherent parameterizations (if possible, one of the input
parameterizations is kept unchanged).
I
Using concepts of the extended convolution ideal and the dual
representation in connection with implicit support function, we are able to
describe a general algorithm which finds coherent parameterizations (if they
exist) in an optimal way.
I
In special case, the algorithm can be reformulated for computing PH/PN
parameterizations.
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
45 / 46
Conclusion
www.KMA.zcu.cz
Summary
I
The talk was devoted to the analysis of reparameterizations of given
hypersurfaces with respect to their convolution.
I
Coherent parameterizations of rational hypersurfaces were introduced and
presented.
I
The presented approach is an “economic” one – we try to find the simplest
possible way to coherent parameterizations (if possible, one of the input
parameterizations is kept unchanged).
I
Using concepts of the extended convolution ideal and the dual
representation in connection with implicit support function, we are able to
describe a general algorithm which finds coherent parameterizations (if they
exist) in an optimal way.
I
In special case, the algorithm can be reformulated for computing PH/PN
parameterizations.
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
45 / 46
Conclusion
www.KMA.zcu.cz
Summary
I
The talk was devoted to the analysis of reparameterizations of given
hypersurfaces with respect to their convolution.
I
Coherent parameterizations of rational hypersurfaces were introduced and
presented.
I
The presented approach is an “economic” one – we try to find the simplest
possible way to coherent parameterizations (if possible, one of the input
parameterizations is kept unchanged).
I
Using concepts of the extended convolution ideal and the dual
representation in connection with implicit support function, we are able to
describe a general algorithm which finds coherent parameterizations (if they
exist) in an optimal way.
I
In special case, the algorithm can be reformulated for computing PH/PN
parameterizations.
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
45 / 46
Conclusion
www.KMA.zcu.cz
Summary
I
The talk was devoted to the analysis of reparameterizations of given
hypersurfaces with respect to their convolution.
I
Coherent parameterizations of rational hypersurfaces were introduced and
presented.
I
The presented approach is an “economic” one – we try to find the simplest
possible way to coherent parameterizations (if possible, one of the input
parameterizations is kept unchanged).
I
Using concepts of the extended convolution ideal and the dual
representation in connection with implicit support function, we are able to
describe a general algorithm which finds coherent parameterizations (if they
exist) in an optimal way.
I
In special case, the algorithm can be reformulated for computing PH/PN
parameterizations.
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
45 / 46
Conclusion
www.KMA.zcu.cz
Summary
I
The talk was devoted to the analysis of reparameterizations of given
hypersurfaces with respect to their convolution.
I
Coherent parameterizations of rational hypersurfaces were introduced and
presented.
I
The presented approach is an “economic” one – we try to find the simplest
possible way to coherent parameterizations (if possible, one of the input
parameterizations is kept unchanged).
I
Using concepts of the extended convolution ideal and the dual
representation in connection with implicit support function, we are able to
describe a general algorithm which finds coherent parameterizations (if they
exist) in an optimal way.
I
In special case, the algorithm can be reformulated for computing PH/PN
parameterizations.
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
45 / 46
Conclusion
www.KMA.zcu.cz
THANK YOU FOR YOUR ATTENTION!
Reparameterization of hypersurfaces with respect to convolutions
May 12, 2009
46 / 46