Parametric Curves Ref: 1, 2

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

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

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

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

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

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)

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

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

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

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

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

C1 Composite Bezier Curves

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

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

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

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 frames Rotation minimizing frame

Generalized Cylinder

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

Frenet Frame (arc-length parameterization)

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

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

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

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

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 BACK