Parametric Curves

Ref: 1, 2

Outline

Hermite curvesBezier curvesCatmull-Rom splinesFrames along the curve

Hermite Curves

3D curve of polynomial basesGeometrically defined by position and tangents of end pointsAble to construct C1 composite curveIn CG, often used as the trace for camera with Frenet frame, or rotation-minimizing frame

Math …

h1(s) = 2s3 - 3s2 + 1h2(s) = -2s3 + 3s2

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

Blending Functions

At s = 0: 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

h1(s) h2(s) h3(s) h4(s)

P(0) = P1P’(0) = T1

P(1) = P2P’(1) = T2

C1 Composite Curve

P(t)

Q(t) R(t)

More on Continuity

Composite Curve

P(t)

Q(t) R(t)

t0

t1

t2

t3

Each subcurve is defined in [0,1].The whole curve (PQR) can be defined from [0,3]

To evaluate the position (and tangent)

Close Relatives

Bezier curvesCatmull-Rom splines

Bezier Curve (cubic, ref)

Defined by four control pointsde Casteljau algorithm (engineer at Citroën)

Bezier Curve (cont)

Also invented by Pierre Bézier (engineer of Renault) Blending function: Bernstein polynomialCan be of any degree

Degree n has (n+1) control points

First Derivative of Bezier Curves (ref)

Degree-n Bezier curveBernstein polynomialDerivative of Bernstein polynomialFirst derivative of Bezier curve

Hodograph

iii PPnQ 1

Ex: cubic Bezier curve

22312

201

2

012,

312313

3)()(

tPPttPPtPP

PPtBtPdt

d

iiii

23

01

3)1(

3)0(

PPP

PPP

Hence, to convert to/from Hermite curve:

23

01

3

0

3)1(

3)0(

)1(

)0(

PPP

PPP

PP

PP

3

2

1

0

1

0

1

0

3300

0033

1000

0001

P

P

P

P

p

p

p

p

1

0

1

0

31

31

3

2

1

0

0010

010

001

0001

p

p

p

p

P

P

P

P

C1 Composite Bezier Curves

Bezier Curve Fitting

From GraphicsGemsInput: digitized data points in R2

Output: composite Bezier curves in specified error

Bezier Marching

A path made of composite Bezier curvesGenerate a sequence of points along the path with nearly constant step sizeAdjust the parametric increment according to (approximated) arc length

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)/2Tangents in first and last segments are defined differently

PowerPoint Line Tool …

Gives you a Catmull-Rom spline, open or close.

Ex: Catmull-Rom Curves

Reference Frames Along the Curve

Applications generalized cylinder Cinematography

Frenet framesRotation minimizing frame

Generalized Cylinder

Frenet Frame (Farin)

tangent vector

binormal vector

main normal vector

: cross productUnit vectors

Frenet Frame (arc-length parameterization)

Frenet-Serret Formula

Orthonormal expansion

Express T’N’B’ (change rate of TNB)in terms of TNB

In this notation, the curve is r(s)

Frenet-Serret Formula (cont)

In general parameterization r(t)

Curvature and torsion

r(t)=(x(t),y(t))

(s)

Geometric Meaning of and

x(s+s)

: angle between t(s) and t(s+s): angle between b(s) and b(s+s)

curvature torsion

More result on this reference

Frenet Frame Problem

Problem: vanishing second derivative at inflection points

(vanishing normal)

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

ContinuityGeometric 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 ContinuityC0: curves are joinedC1: first derivatives are equalC2: first and second derivatives are equalCn: first through nth derivatives are equal

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