Introduction to
Polynomial Curves
Part II
Peter Schröder
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Polar Forms
Blossom
Q
For every polynomial F(u) of degree
n there exists a unique symmetric
multiaffine map f(u1,...,un) for
which F(u)=f(u,...,u). This map is
called the polar form. F(u) is called
its diagonal and f is called the
blossom of F.
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Page 1
Example
Cubic case
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Polar Form
Correspondence
Differentiation
Q
differencing
well defined, i.e., independent of un
Q linear in τ
Q
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Page 2
Polar Form
Differentiation
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Polar Form
Differentiation
Q
let f be a polar form and 0· p· n,
and u ∈ R (fixed) then the
following are equivalent
Proof
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Continuity
Two polynomials
Q
let F and G be two polynomials of
degree n and let u be a point on
the real line. Then F and G are Cq
continuous at u iff
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Bezier Segments
Continuity conditions
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Bezier Points
Finding control points
Q
the Bezier control points of F with
respect to [s,t] are
Proof
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de Casteljau Alg.
Evaluate Bezier form
very stable
Q convex combinations only
Q
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de Casteljau
Properties
Q
segment splitting, derivatives
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Multiple Segments
How to stick them together?
can match first derivative easily
Q second derivative?
Q
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Modeling
Bezier curves
requires to keep track of continuity
conditions explicitly... Possible, but
very messy
Q better: use basis which has
continuity conditions built in!
Q
B-Splines
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B-Splines
Basic idea
Q
share arguments of control points
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Page 7
Argument “Bags”
Knotvector
Q
definition
non-decreasing sequence, can
have multiple entries (multiplicity)
Q curve:
Q
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Knot Vector
Example
Q
Bezier segments
Q
B-splines segments
Q
2n knots yield n+1 control points
Q
moving window of size n (why?)
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Page 8
Continuity
Simple knots example
Q
n=3 just drawing arguments
(pqr)
(qrs)
(qrv1)
(rst)
(rsv1)
(rv1v2)
(stx)
(stv1)
(rst)
(rsv1)
(sv1v2)
evaluation at s
(stx)
(stv1)
(sv1v2)
(v1v2v3)
Q
(qrs)
(txy)
(txv1)
(tv1v2)
(v1v2v3)
C2 at s: cubic B-splines are C2
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Continuity
Doubling a knot
Q
lose one order of continuity
(pqr)
(qrs)
(qrv1)
(rss)
(rsv1)
(rv1v2)
(rss)
(ssv1)
(sst)
(ssv1)
(sv1v2)
(v1v2v3)
Q
(sst)
(stv1)
(sv1v2)
evaluation at s
(stx)
(txy)
(txv1)
(tv1v2)
(v1v2v3)
now only C1
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B-Splines
Terms
degree: highest exponent in
polynomial
Q order: 1+degree
Q multiplicity of a knot: how many
times repeated
Q continuity: Cn-m with multiplicty m
Q
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B-Splines
Definition
Q
recursive
Q
support
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Page 10
B-Spline Definition
Basis functions
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de Boor Algorithm
Recursion
Q
recursive definition of the basis is
reflected in recursive structure on
control points
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Page 11
Evaluation
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Evaluation
Cubic spline
Q
two successive segments
(pqr)
(qrs)
(qrv1)
(rst)
(rsv1)
(rv1v2)
(stx)
(stv1)
(sv1v2)
(qrs)
(rst)
(rsv1)
(stv1)
(sv1v2)
(v1v2v3)
(stx)
(txy)
(txv1)
(tv1v2)
(v1v2v3)
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Page 12
B-Splines
de Boor Algorithm
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B-Splines
Knot insertion
Q
create new degree of freedom
along curve
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Page 13
Knot Insertion
Example
Q
cubic spline: new knot
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Knot Doubling
Example
Q
increasing multiplicity:
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Page 14
Other Properties
Integrals and derivatives
Q
derivatives as differences
Q
Q
B-spline:
integrals
Q
Bernstein basis:
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Degree Elevation
Re-express as higher order curve
Q
by uniqueness
Q
example: quadratic to cubic
Q
Bezier control points?
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Page 15
Polar Forms
Curve manipulations
splitting
Q differentiation
Q knot insertion
Q increasing knot multiplicity
Q degree elevation
Q
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What Else?
Rational curves
Q
conic sections in homogeneous
coordinates
w
y
x
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