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Parametric Curves Ref: 1, 212. Outline Hermite curves Bezier curves Catmull-Rom splines Frames along the curve.

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Presentation on theme: "Parametric Curves Ref: 1, 212. Outline Hermite curves Bezier curves Catmull-Rom splines Frames along the curve."— Presentation transcript:

1 Parametric Curves Ref: 1, 212

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 C 1 composite curve In CG, often used as the trace for camera with Frenet frame, or rotation- minimizing frame

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

5 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) = P1 P ’ (0) = T1 P(1) = P2 P ’ (1) = T2

6 C 1 Composite Curve P(t) Q(t) R(t) More on Continuity

7 Composite Curve P(t) Q(t) R(t) t0t0 t1t1 t2t2 t3t3 Each subcurve is defined in [0,1]. The whole curve (PQR) can be defined from [0,3] To evaluate the position (and tangent)

8 Close Relatives Bezier curves Catmull-Rom splines

9 Bezier Curve (cubic, ref)ref Defined by four control points de Casteljau algorithm (engineer at Citro ë n)

10 Bezier Curve (cont) Also invented by Pierre B é zier (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)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 C 1 Composite Bezier Curves

14 Bezier Curve Fitting From GraphicsGems Input: digitized data points in R 2 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)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 [P i,P i+1 ] by using the two control points, and specifying the tangent to the curve at each control point to be (P i+1 – P i-1 )/2 and (P i+2 – P i )/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  : cross product Unit vectors

22 Frenet Frame (arc-length parameterization)

23 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 )

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

25 (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 referencethis reference

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

27 Rotation Minimizing Frame (ref)ref Use the second derivative to define the first frame (if zero, set N 0 to any vector  T 0 ) Compute all subsequent T i ; find a rotation from T i-1 to T i ; rotate N i and B i accordingly If no rotation, use the same frame

28 Continuity Geometric Continuity A curve can be described as having G n continuity, n being the increasing measure of smoothness. G 0 : The curves touch at the join point. G 1 : The curves also share a common tangent direction at the join point. G 2 : The curves also share a common center of curvature at the join point. Parametric Continuity C 0 : curves are joined C 1 : first derivatives are equal C 2 : first and second derivatives are equal C n : first through n th derivatives are equal BACK

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