Representing Curves
Foley & Van Dam, Chapter 11
Representing Curves
• Motivations
• Techniques for Object Representation
• Curves Representation
• Free Form Representation
• Approximation and Interpolation
• Parametric Polynomials
• Parametric and Geometric Continuity
• Polynomial Splines
• Hermite Interpolation
3D Objects Representation
• Solid Modeling attempts to develop methods
and algorithms to model and represent real
objects by computers
Objects Representation
• Three types of objects in 3D:
• 1D curves
• 2D surfaces
• 3D objects
• We need to represent objects when:
• Modeling of existing objects (3D scan)
- modeling is not precise
• Modeling a new object “from scratch” (CAD)
- modeling is precise
- interactive sculpting capabilities
General Techniques
• Primitive Based:
A composition of “simple” components
• Not precise
• Efficient and simple
• Free Form:
Global representation, curved manifolds
• Precise
• Complicated
• Statistical:
Modeling of objects generated by statistical
phenomena, such as fog, trees, rocks
Curves Representation
20
15
10
5
0
1
1
0 .5
0.5
0
0
-0 .5
-0 .5
-1
-1
Primitive Based Representation
• Line segments: A curve is approximated by
a collection of connected line segments
Free Form Representations
• Explicit form: z = f(x, y)
• f(x,y) must be a function
• Not a rotation invariant representation
• Difficult to represent vertical tangents
• Implicit form: f(x, y, z) = 0
• Difficult to connect two curves in a smooth manner
• Not efficient for drawing
• Useful for testing object inside/outside
• Parametric: x(t), y(t), z(t)
• A mapping from [0,1] → R3
• Very common in modeling
Free Form Representations
Example: A Circle of radius R
• Implicit:
x2 + y2 + z2 - R2 = 0 & z = 0
z
• Parametric:
x(θ) = R cos(θ)
y(θ) = R sin(θ)
z(θ) = 0
θ
x
y
Approximated vs. Interpolated Curves
• Given a set of control points Pi known to be
on the curve, find a parametric curve that
interpolates/approximates the points
Interpolating Curve
P1
P0
P3
P2
P4
Approximating Curve
P1
P3
P0
P2
P4
Parametric Polynomials
• For interpolating n points we need a polynomial of
degree n-1
x (u ) = a
x
+ bxu + c xu
y (u ) = a
y
+ b yu + c yu
z(u ) = a
z
p1
+ b zu + c zu
+ "
2
2
2
+ "
+ "
p3
p0
p2
p4
• Example: Linear polynomial. For interpolating 2
points we need a polynomial of degree 1
x
( u
)
=
a
y
( u
)
=
a
z ( u
)
=
a
x
+
b
+
b
y
z
+
b
u
x
y
z
u
p1
u
p0
Example: Linear Polynomial
• The geometrical constraints for x(u) are:
x (0 ) = a
=
x
P 0x
;
x (1 ) = a
+ b
x
=
x
P1 x
• Solving the coefficients for x(u) we get:
a
⇒
x
= P0x
; b
= P1 x − P 0 x
x
x ( u ) = P 0 x + ( P1 x − P 0 x )
u
• Solving for [x(u) y(u) z(u)] we get:
V
( u )
=
 x ( u ) 
 y ( u ) 


 z ( u ) 
=
P
0
+
( P
1
u=1
u=0
p1
p0
−
P
0
)
u
Example: Linear Polynomial
u=0
p0
u=1
p1
V
1
 x (u ) 
 y (u )  =


 z ( u ) 
(u ) =
p1
p0
V1(u)
P
0
+ ( P1 −
P
0
)
p3
p2
V2(u)
 x (u

V ( u ) =  y (u
 z (u

)
) 
) 
w h ere
V3(u)
=
∑
V4(u)
p4
V i (u )
i
V i ( u )
Vi (u ) = 
0

if
u ∈ [ u i , u i +1 ]
o th erw ise
u
Parametric Polynomials
• Polynomial interpolation has several
disadvantages:
• Polynomial coefficients are geometrically
meaningless
• Polynomials of high degree introduce unwanted
wiggles
• Polynomials of low degree give little flexibility
• Solution: Polynomial Splines
p1
p0
p3
p2
p4
Polynomial Splines
• Piecewise, low degree, polynomial curves, with
continuous joints
 x (u

C ( u ) =  y (u
 z (u

)
) 
) 
w h ere
=
∑
C i (u )
i
 C i (u )
C i (u ) = 
0

if
u ∈ [ u i , u i +1 ]
o th erw ise
• Advantages:
• Rich representation
• Geometrically meaning coefficients
• Local effects
• Interactivep sculpting capabilities
p3
1
p0
c1(u)
c2(u)
p2
c3(u)
c4(u)
p4
Tangent Vector
• Let V(u)=[x(u), y(u), z(u)], u→[0,1] be a continuous
univariate parametric curve in R3
• The tangent vector at u0, T(u0), is:
G
dy dz 
dV ( u )

dx
T ( u 0 ) = V ′( u 0 ) =
= 

du
du
du
du

u=u0
u=u
• V(u) may be thought of as the trajectory of a point
in time
• In this case, T(u0) is the instantaneous velocity
vector at time u0
u0
u=0
T(u0)
u=1
0
Parametric Continuity
• Let V1(u) and V2(u) , u→[0,1], be two parametric
curves
•Level of parametric continuity of the curves at the
joint between V1(1) and V2(0):
• C-1: The joint is discontinuous, V1(1)≠V2(0)
• C0: Positional continuos, V1(1)=V2(0)
• C1: Tangent continuos, C0 & V’1(1)=V’2(0)
• Ck, k>0: Continuous up to the k-th derivative,
V1(j) (1)=V2(j) (0), 0 ≤ j ≤ k
C-1
V1(1)
V2(0)
C0
C1
Geometric Continuity
• In computer aided geometry design, we also consider the
notion of geometric continuity:
• G-1, G0: Same as C-1 and C0
• G1: Same tangent direction: V’1(1)=αV’2(0)
• Gk: All derivatives up to the k-th order are proportional
• Given a set of points {pi}:
• A piecewise constant interpolant is C-1
• A piecewise linear interpolant is C0
Parametric and Geometric Continuity
• S-C0 is C0
• S-C1 is C1
• S-C2 is C2
• In general, Ci implies Gi
(not vice versa)
• Exception when the tangents
are zero
• Q1-Q2 both C1 and G1
• Q1-Q3 is G1 but not C1
Parametric Cubic Curves
• Cubic polynomials defining a curve in R3 have the
form:
3
2
x (u ) = a
x
u
y (u ) = a
y
u
z (u ) = a
z
u
+ b
3
3
+ b
+ b
2
u
y
z
+ c
u
x
u
2
+ c
+ c
u + d
x
y
z
x
u + d
u + d
y
z
Where u is in [0,1]. Defining:
U
T
[
(u ) = u
3
u
2
u
1
]
and
The curve can be rewritten as:
[ x (u )
y (u )
a x
b
x
Q = 
cx

 d x
a
b
c
d
y
y
y
y
z (u ) ] = V T (u ) = U T (u ) Q
az
bz
cz
d z






Parametric Cubic Curves
• The coefficients Q are unknown and should
be determined
• For this purpose we have to supply 4
geometrical constraints
• Different types of constraints define different
types of Splines
Hermite Curves
• Assume we have n control points {pk} with
their tangents {Tk}
• W.L.O.G. V(u) represents a parametric cubic
function for the section between pk and pk+1
• For V(u) we have the following geometric
constraints:
V(0)=pk; V(1)=pk+1
V'(0)=Tk; V'(1)=Tk+1
pk
V(0)
Tk
V(1)
V(u)
Pk+1 Tk+1
Hermite Curves
Since
V
T
( u ) =  u
3
u
2
1 
u
Q
we have that
( V ′ ) T ( u ) =  3 u
2
2u
1
0 
Q
We can write the constraints in a matrix form:
G
And thus
Where
= MQ
⇒
 p k
 p
k +1

 Tk

 T k +1
G
V
T
M
(u ) = U
−1


= 



2
− 3
0
1

 0

 1
 = 

 0


 3

0
1
0
1
0
1
2
1
1 
1 
Q
0 

0 
M
T
(u )Q = U
− 2
3
0
0
1
− 2
1
0
T
1 
− 1 
0 

0 
(u ) M
−1
G
Hermite Curves
V (u ) = [u
3
u
2
u
 2
−3
1] 
 0

 1
−2
1
3
−2
0
0
1
0
1 
− 1 
0 

0 
pk 
p k + 1 
Tk 

T k + 1 
Geometry
matrix
Blending
functions
T
2u 3 − 3u 2 + 1

3
2 
− 2u
+ 3u

V (u ) = 
 u 3 − 2u 2 + u 


3
2
− u
u


T
 H 0(u )
 p k
 H (u ) 
 p
1
k +1


= 
 H 2(u )
 T k



H
(
u
)
3


 T k +1






 p k
 p
k +1

 T k

 T k +1








 =



Hermite Curves
1.2
H1=H(1)
1
H0=H(0)
0.8
0.6
0.4
H2=H’(0)
0.2
0
-0 . 2
H3=H’(1)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Hermite Blending Functions
0.9
1
Hermite Curves
y(u)
Change in Magnitude of T0
T0
P0
T1
P1
x(u)
y(u)
Change in Direction of T0
x(u)
Hermite Curves
Properties:
• The Hermite curve is composed of a linear combinations of
tangents and locations (for each u)
• Alternatively, the curve is a linear combination of Hermite
basis functions (the matrix M)
• It can be used to create geometrically intuitive curves
• The piecewise interpolation scheme is C1 continuous
• The blending functions have local support; changing a
control point or a tangent vector, changes its local
neighborhood while leaving the rest unchanged
Hermite Curves
• Main Drawback:
Requires the specification of the tangents
This information is not always available