1. Parametric Curves
Ref: 1, 2

2. Outline Hermite curves Bezier curves Catmull-Rom splines
Frames along the curve

3. Hermite Curves 3D curve of polynomial bases
Geometrically defined by position and tangents of end points
Able to construct C1 composite curve
In CG, often used as the trace for camera with Frenet frame, or rotation-minimizing frame

4. Math … P(0)= P1, P(1)=P2; P’(0)=T1, P’(1)=T2 h1(s) = 2s3 - 3s2 + 1
h3(s) = s3 - 2s2 + s
h4(s) = s3 - s2
P(s) = P1h1(s) + P2h2(s) + T1h3(s) + T2h4(s)
P’(s)=P1h1’(s) + P2h2’(s) + T1h3’(s) + T2h4’(s)
h1’ = 6s2-6s h2’ = -6s2+6s h3’= 3s2-4s+1 h4’= 3s2 – 2s
P(0)= P1, P(1)=P2;
P’(0)=T1, P’(1)=T2

5. Blending Functions At s = 0: At s = 1: h2(s) h4(s) h3(s) h1(s)
h1 = 1, h2 = h3 = h4 = 0
h1’ = h2’ = h4’ = 0, h3’ = 1
At s = 1:
h1 = h3 = h4 = 0, h2 = 1
h1’ = h2’ = h3’ = 0, h4’ = 1
P(0) = P1
P’(0) = T1
P(1) = P2
P’(1) = T2

6. C1 Composite Curve
P(t)
Q(t)
R(t)
More on Continuity

7. Composite Curve t1 t3 P(t) Q(t) R(t) t0 t2
Each subcurve is defined in [0,1].
The whole curve (PQR) can be defined from [0,3]
To evaluate the position (and tangent)

8. Bezier curves Catmull-Rom splines
Close Relatives
Bezier curves
Catmull-Rom splines

9. Bezier Curve (cubic, ref)
Defined by four control points
de Casteljau algorithm (engineer at Citroën)

10. Bezier Curve (cont)
Also invented by Pierre Bézier (engineer of Renault)
Blending function: Bernstein polynomial
Can be of any degree
Degree n has (n+1) control points

11. First Derivative of Bezier Curves (ref)
Degree-n Bezier curve
Bernstein polynomial
Derivative of Bernstein polynomial
First derivative of Bezier curve
Hodograph

12. Ex: cubic Bezier curve
Hence, to convert to/from Hermite curve:

13. C1 Composite Bezier Curves

14. Bezier Curve Fitting From GraphicsGems
Input: digitized data points in R2
Output: composite Bezier curves in specified error

15. Bezier Marching A path made of composite Bezier curves
Generate a sequence of points along the path with nearly constant step size
Adjust the parametric increment according to (approximated) arc length

16. Catmull-Rom spline (1974, ref)
Given n+1 control points {P0,…,Pn}, we wish to find a curve that interpolates these control points (i.e. passes through them all), and is local in nature (i.e. if one of the control points is moved, it only affects the curve locally).
We define the curve on each segment [Pi,Pi+1] by using the two control points, and specifying the tangent to the curve at each control point to be (Pi+1–Pi-1)/2 and (Pi+2–Pi)/2
Tangents in first and last segments are defined differently

17. PowerPoint Line Tool …
Gives you a Catmull-Rom spline, open or close.

18. Ex: Catmull-Rom Curves

19. Reference Frames Along the Curve
Applications
generalized cylinder
Cinematography
Frenet frames
Rotation minimizing frame

20. Generalized Cylinder

21. Frenet Frame (Farin) tangent vector binormal vector main normal vector
Unit vectors
: cross product

22. Frenet Frame (arc-length parameterization)

23. Frenet-Serret Formula
In this notation, the curve is r(s)
Frenet-Serret Formula
Express T’N’B’
(change rate of TNB)
in terms of TNB   
Orthonormal
expansion

24. Frenet-Serret Formula (cont)
In general parameterization r(t)
Curvature and torsion
r(t)=(x(t),y(t))

25. Geometric Meaning of k and t
curvature
torsion
(s)
x(s+Ds)
Da: angle between t(s) and t(s+Ds)
Db: angle between b(s) and b(s+Ds)
More result on this reference

26. Frenet Frame Problem
Problem: vanishing second derivative at inflection points
(vanishing normal)

27. Rotation Minimizing Frame (ref)
Use the second derivative to define the first frame (if zero, set N0 to any vector T0)
Compute all subsequent Ti; find a rotation from Ti-1 to Ti; rotate Ni and Bi accordingly
If no rotation, use the same frame

28. Continuity BACK Geometric Continuity
A curve can be described as having Gn continuity, n being the increasing measure of smoothness.
G0: The curves touch at the join point.
G1: The curves also share a common tangent direction at the join point.
G2: The curves also share a common center of curvature at the join point.
Parametric Continuity
C0: curves are joined
C1: first derivatives are equal
C2: first and second derivatives are equal
Cn: first through nth derivatives are equal
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