Interpolating/approximating pp gcurves

Transcription

1 Chap 3 Interpolating Values 1

2 Outline Interpolating/approximating pp gcurves Controlling the motion of a point along a curve Interpolation of orientation Working with paths

3 Interpolation between key frames Parameters to be interpolated Position of an object Normal Joint angle between two joints Transparency attribute of an object 3

4 Interpolation between key frames Interpolation between frames is not trivial Appropriate interpolating function Parameterization ti of fthe function Maintain the desired control of the interpolated values over time Example (-5,0,0) at frame, (5,0,0) art frame 67 Stop at frame and accelerate to reach a max speed by frame 34 Start to decelerate at frame 50 and come to a stop at frame 67 4

5 Interpolation between key frames desired result undesired result Need a smooth interpolation with user control 5

6 Interpolation between key frames Solution Generate a space curve Distribute points evenly along curve Speed control: vary points temporally B A Time = 10 Time = 0 C Time = 35 6 D Time = 60

7 Interpolation functions Interpolation vs. approximation Interpolation Hermite, Catmull-Rom approximation Bezier, B-spline 7

8 Interpolation functions Complexity => computational efficiency Polynomials Lower than cubic No inflection point, may not fit smoothly to some data points Higher than cubic Doesn t provide any significant advantages, costly to evaluate Piecewise cubic Provides sufficient smoothness Allows enough flexibility to satisfy constraints such as end- point position and tangents 8

9 Interpolation functions Continuity within a curve Zero-order First order (tangential) Suffices for most animation applications Second order Important when dealing with time-distance curve Continuity between curve segments Hermite, Ct Catmull-Rom, llr cubic Bezier provide first order continuity between segments 9

10 Interpolation functions 10

11 Continuity position tangent curvature none (0th order) (1st order) (nd order) At junction of two circular arcs 11

12 Interpolation functions Global vs. local control 1

13 Types of Curve Representation Explicit y = f(x) Good for generate points For each input, there is a unique output Implicit f(x,y) = 0 Good for testing ti if a point is on a curve Bad for generating a sequence of points Parametric x = f(u), y = g(u) Good for generating a sequence of points Can be used for multi-valued function of x 13

14 Example: Representing Unit Circle Cannot be represented explicitly as a function of x f>0 f=0 Implicit form: f<0 f(x,y)=x +y =1 Parametric form: x=cos(u), y=sin(u), 0<u<π u 14

15 More on 3-D Parametric Curves Parametric form: P(u) = (P x (u), P y (u), P z (u)) x = P x x( (u), y = P y y( (u), z = P z z( (u) u=0.00 u=/3 u=1/3 u=1.0 Space-curve P = P(u) 0.0 <=u<=1.0 15

16 Polynomial Interpolation An n-th degree polynomial fits a curve to n+1 points Example: fit a second degree curve to three points y(x)= a x + b x + c points to be interpolated (x 1,y 1 ), (x,y ), (x 3,y 3 ) solve for coefficients (a, b, c): 3 linear equations, 3 unknowns called Lagrange Interpolation 16

17 Polynomial Interpolation (cont.) Result is a curve that is too wiggly, change to any control point affects entire curve (nonlocal) this method is poor We usually want the curve to be as smooth as possible minimize the wiggles high-degree g polynomials are bad Higher degree, higher the wiggles 17

18 Composite Segments Divide a curve into multiple segments Represent each in a parametric form Maintain continuity between segments position Tangent (C1-continuity vs. G1-continuity) curvature P 1 (u) P (u) P 3 (u) P n (u) 18

19 Splines: Piecewise Polynomials A spline is a piecewise polynomial - many low degree polynomials are used to interpolate (pass through) the control points Cubic polynomials are the most common lowest order polynomials that interpolate two points and allow the gradient at each point to be defined - C1 continuity is possible Higher or lower degrees are possible, of course 19

20 A Linear Piecewise Polynomial The simple form for interpolating two points. p1 Each segment is of the form: (this is a vector equation) p p ( u) = up1 + (1 u) p 1 u Two basis (blending) functions 0 1 u 0

21 Hermite Interpolation Hermite Curves cubic polynomial for two points 3 P ( u) = au + bu + cu + P ( u ) = ( Px ( u ), Py ( u ), Pz ( u )) Hermite interpolation requires Endpoint positions derivatives at endpoints P(0) d control points/knots P(1) P '(0) P' (1) To create a composite curve, use the end of one as the beginning gof the other and share the tangent vector 1

22 Hermite Curve Formation Cubic polynomial and its derivative 3 P ( u) = a u + b u + c u + x x x ' P ( u ) = 3a u + b u + c x x x x x d x Given P x (0), P x (1), P x (0), P x (1), solve for a, b, c, d 4 equations are given for 4 unknowns P (0) = P + P = P + x d x x (1) = ax + bx + cx d x ' x(0) c x ' x(1) = 3a x + bx cx

23 Hermite Curve Formation (cont.) Problem: solve for a, b, c, d Solution: P (0) = P + P = P + a b c x d x x (1) = ax + bx + cx d x ' x(0) c x ' x(1) = 3a x + bx cx x x x = = = ( P 3( P P ' x x x (0) d x = P x (0) (0) (1) 3 P P x x (1)) (0)) + P ' x P (0) ' x + (0) P ' x P (1) ' x (1)

24 Hermite Curves in Matrix Form 3 P ( u ) = au + bu + cu + d P( u) = U T = [ u M is B is U 3, T u MB, u,1] is the coefficient matrix the parameter the geometric information P x (0) P x(1) B = ' P x(0) 1 ' Px (1) M = ith segment in composite curves Pi P i+ B = ' Pi ' 1 P i + 1

25 Blending Functions of Hermite Splines Blending Functions of Hermite Splines Each cubic Hermite spline is a linear combination of p 4 blending functions geometric information M B U P T ) ( M B U u P T ) ( = ) ( p p u u u u + = ' ' ) ( p p u u u u p 5 3 p u u

26 Composite Hermite Curve Continuity between connected segments is ensured by using ending tangent vector of one segment as the beginning i tangent tvector of fthe next. 6

27 Bezier Curves Similar to Hermite Curve Instead of endpoints and tangents, four control points are given points P1 and P4 are on the curve points P and P3 are used to control the shape p 1 = P1, p = P4, p 1 ' = 3(P-P1), p ' = 3(P4 - P3) p( u) = [ 1] 3 u u u P P P P4 7

28 Bezier Curves Another representation Blend the control point position using Bernstein polynomials p( u) = n k = 0 B n where k, B k B k ( u), n( u) P k, n( u) = C( n, k) u (1 u) C( n, k) = n! k!( n k are Bernstein polynomials k)! n k 8

29 Bezier Curves 9

30 Composite Bezier Curves How to control the continuity between adjacent Bezier segment? 30

31 De Casteljau Construction of Bezier Curves How to derive a point on a Bezier curve? 31

32 Bezier Curves (cont.) Variant of the Hermite spline basis matrix derived from the Hermite basis (or from scratch) Gives more uniform control knobs (series of points) than Hermite Scale factor (3) is chosen to make velocity approximately constant 3

33 Catmull-Rom Splines With Hermite splines, the designer must arrange for consecutive tangents to be collinear, to get C 1 continuity. Similar for Bezier. This gets tedious. Ct Catmull-Rom: llr an interpolating ti cubic spline with built-in C 1 continuity. Compared to Hermite/Bezier: fewer control points required, but less freedom. 33

34 Catmull-Rom Splines (cont.) Given n control points in 3-D: p 1, p,, p n, Tangent at p i given by s(p i+1 p i-1 ) for i=..n-1, for some s. Curve between p i and p i+1 is determined by p i-1, p i, p i+1,p i+. 34

35 Catmull-Rom Splines (cont.) Given n control points in 3-D: p 1, p,, p n, Tangent at p i given by s(p i+1 p i-1 ) for i=..n-1, for some s Curve between p i and p i+1 is determined by p i-1, p i, p i+1, p i+ What about endpoint tangents? (several good answers: extrapolate, t or use extra control points p 0, p n+1 ) Now we have positions and tangents at each knot a Hermite specification. 35

36 Catmull-Rom Spline Matrix Derived similarly to Hermite and Bezier s is the tension parameter; typically s=1/ When s p ( u ) = U = 1/, T P i 11 1 i 1 Pi 0 Pi + 0 P i

37 Catmull-Rom Spline What about endpoint tangents? Provided by users Several good answers: extrapolate, or use extra control points p 0, p n+1 P 1 = (( P1 ( P 1 = (P P ' 0 P1 )) P0 1 P0 ) ) 37

38 Catmull-Rom Spline Example 38

39 Drawback Catmull-Rom Spline An internal tangent is not dependent on the position of the internal point relative to its two neighbors 39

40 p(0) = P p (0) = s(p3 P1) p(0) = P, p (0) = s(p3-p1), p(1) = P3, p (1) = s(p4-p) ) ( P P U T = ) ( P P s s s s U u p T 4 1 P P s s s s = P P P s s s s s s U T P

41 Splines and Other Interpolation Forms See Computer Graphics textbooks Review Appendix B.4 in Parent 41

42 Now What? We have key frames or points We have a way to specify the space curve Now we need to specify velocity to traverse the curve Speed Curves 4

43 Speed Control The speed of tracing a curve needs to be under the direct control of the animator Varying u at a constant t rate will not necessarily generate P(u) at a constant speed. 43

44 Non-uniformity in Parametrization Generally, equally spaced samples in parameter space are not equally spaced along the curve u u s( u ) s ( ) 1 ( u1 B s ( u ) = Px ( u ) + Py ( u ) + Pz ( u ) du A Time = 10 Time = 0 C Time = 35 D 44 Time = 60

45 Arc Length Reparameterization To ensure a constant speed for the interpolated value, the curve has to be parameterized a e ed by arc length (for most applications) Computing arc length Analytic method (many curves do not have, e.g., B- splines) Numeric methods Table and differencing Gaussian quadrature 45

46 Arc Length Reparameterization Space curve vs. time-distance function Relates time to the distance traveled along the curve, i.e., Relates time to the arc length along the curve 46

47 Arc Length Reparameterization Given a space curve with arc length parameterization Allow movement along the curve at a constant speed by stepping at equal arc length intervals Allows acceleration and deceleration along the curve by controlling the distance traveled in a given time interval Problems Given a parametric curve and two parameter values u 1 and u, find arclength(u 1,u ) Given an arc length s, and parameter value u 1, find u such that arclength(u 1,u ) = s 47

48 Arc Length Reparameterization Converting a space curve P(u) to a curve with arc-length ac parameterization a e a Find s=s(u), and u=s -1 (s)=u(s) P*(s)=P(U(s)) P(U( )) Analytic arc-length parameterization is difficult or impossible for most curves, e.g., B- spline curve cannot be parameterized with arc length. Approximate arc-length parameterization ti 48

49 Forward Differencing Sample the curve at small intervals of the parameter Compute the distance between samples Build a table of arc length for the curve u Arc Length

50 Arc Length Reparameterization Using Forward Differencing Given a parameter u, find the arc length Find the entry in the table closest to this u Or take the u before and after it and interpolate arc length linearly u Arc Length

51 Arc Length Reparameterization Using Forward Differencing Given an arc length s and a parameter u 1, find the parametric value u such that arclength(u 1, u )=s Find the entry in the table u Arc Length closest to this u using binary search Or take the u before and after it and interpolate t linearly l

52 Arc Length Reparameterization Using Forward Differencing Easy to implement, intuitive, and fast Introduce errors Super-sampling in forming table 1000 equally spaced parameter values ebtries in each interval Better interpolation Adaptive forward differencing 5

53 Arc Length Reparameterization Using Adaptive forward differencing 53

54 Arc Length Reparameterization Using Arc length integral Numerical Integral s( u) = Px ( u) + Py ( u) + Pz ( u) du Simpson s and trapezoidal integration using evenly spaced intervals Gaussian quadrature using unevenly spaced intervals 54

55 Speed Control Given a arc-length parameterized space curve, how to control o the speed at which the curve is traced? By a speed-control functions that relate an equally spaced time interval to arc length Input time t, output t arc length: s=s(t) Linear function: constant speed control Most common: ease-in/ease-out Smooth motion from stopped position, accelerate, reach a max velocity, and then decelerate to a stop position 55

56 Speed Control Function Relates an equally spaced time interval to arc length Input time t, output arc length: s=s(t) Normalized arc length parameter s=ease(t) Start at 0, slowly increase in value and gain speed until the middle value and then decelerate as it approaches to 1. 56

57 Constant Velocity Speed Curve Moving at 1 m/s if meters and seconds are the units Too simple to be what we want 57

58 Distance Time Function and Speed Control We have a space curve p=p(u) and a speed control o function s=s(t). For a given t, s=s(t) Find the corresponding value u=u(s) by looking up an arc length table for a given s A point on the space curve with parameter u p=p(u(s(t))) ( ( 58

59 Speed Control P(u( * ) u* Arc Length Table u Arc Length s* Speed Control Curve t* 59

60 Distance Time Function Assumptions on distance time function The entire arc length of the curve is to be traversed during the given total time Additional optional assumptions The function should be monotonic in t The function should be continuous 60

61 Ease-in/Ease-out Most useful and most common ways to control motion o along acurve Arc length (distance) Time Equally spaced samples in time specify arc length required for that t frame 61

62 s Ease-in/Ease-out Sine Interpolation 1 π 1 = ease( t) = sin ( tπ ) +, 0 t 1 1 π/ π/ Arc Length -1 sin( α), π α π Time 6

63 Ease-in/Ease-out Sine Interpolation The speed of s w.r.t. t is never constant over an interval but rather is always accelerating e a or decelerating. Zero derivatives at t=0 and t=1 indicating smooth accelerating and deceleration at end points. 63

64 Ease-in/Ease-out Piecing i Curves Together for Ease In/Out Arc Length Linear segment Time Sinusoidal segments 64

65 Ease-in/Ease-out Piecing i Curves Together for Ease In/Out 65

66 Ease-in/Ease-out Integrating to avoid the sine function acceleration A velocity Find v such that area = 1 v time D t 1 t arc length t 1 t V 1 = V A*t 1 = - D*(1-t ) time time 66

67 Ease-in/Ease-out Integrating to avoid the sine function To avoid the transcendental function evaluation while providing a constant speed interval between ease-in and ease-out intervals Acceleration-time time curve Velocity-time curve Velocity-time curve is defined by integral of the acceleration-time curve Distance-time curve is defined by integral of the velocity-time curve 67

68 Ease-in/Ease-out Integrating to avoid the sine function In ease-in/ease-out function Velocity starts out at 0 and ends at 0 V 1 = V inplies A*t 1 = - D*(1-t ) 68

69 Ease-in/Ease-out Integrating to avoid the sine function Total distance = 1 implies 1 = ½ V 0 t + V 0 (t -t 1 )+1/ V 0 (1-t ) Users specifies any two of three variables t 1, t, and V 0, and the system can solve for the third 69

70 Ease-in/Ease-out Integrating to avoid the sine function The result is Parabolic ease-in/ease-out More flexible than the sinusoidal ease-in/ease-out 70

71 Videos Bunny (Blue Sky Studios, 1998) 71

72 q Review - Quaternions Similar to axis-angleangle representations 4-tuple of real numbers q=(s, x, y, z) or [s, v]s is a scalar; v is a vector The quaternion for rotating an angle about an axis (an axis-angle rotation): = Rot ( θ,( a, a, a )) x y θ θ = cos( ),sin( ) ( ax, ay, az ) If a is unit length, then q will be also z Y Z a=(a x, a y, a z ) θ X Animation(U), Chap, Tech. Background 7

73 Review - Quaternions Review Quaternions If a is unit length, then q will be also = q q q q q sin sin sin cos = θ θ θ θ z y x a a a ( ) sin cos = θ θ y a a a ( ) sin cos = θ θ θ θ z y x a a a sin cos sin cos + = + = θ θ θ θ a Animation(U), Chap, Tech. Background Animation(U), Chap, Tech. Background 73 1 =1 =

74 Review - Quaternions Rotating some angle around an axis is the same as rotating the negative angle around the negated axis θ θ Rot θ, ( x, y, z) = cos( ), sin( )( x, y, z) = q Animation(U), Chap, Tech. Background 74

75 Addition Review - Quaternion Math [ s, v ] + [ s, v ] = [ s + s, v + v ] Multiplication [ s, v ][ s, v ] = [ s s v v, s v + s v + v v ] Multiplication is associative but not commutative q ( = q 1 qq3) ( q1q ) 3 q 1q qq1 Animation(U), Chap, Tech. Background 75

76 Review - Quaternion Math (cont.) Multiplicative identity: [1, 0,0,0] Inverse [ 1, 0, 0, 0 ] q q = [ ] 1 s, v 1 q = qq 1 = [ 1, 0, 0, 0 ] q qq 1 = Normalization for unit quaternion [ 1, 0, 0, 0] Normalization for unit quaternion ) q q = ) qq q = q q = s + x + y + z z q = s + x + y + Animation(U), Chap, Tech. Background 76

77 Review Rotating Vectors Using Quaternion A point in space, v, is represented as [0, v] To rotate a vector v using quaternion q Represent the vector as v = [0, v] Represent the rotation as a quaternion q Using quaternion multiplication v' = Rot ( v) = qvq q The proof isn t that hard Note that tthe result v always has zero scalar value 1 Animation(U), Chap, Tech. Background 77

78 Review - Compose Rotations Rotating a vector v by first quaternion p followed by a quaternion q is like rotation using qp Rot ( Rot ( v )) = q p = = Rot q qpvp p ( pvp 1 1 q ( qp) v( qp) = Rot qp (v) 1 ) 1 Animation(U), Chap, Tech. Background 78

79 Review - Compose Rotations To rotate a vector v by quaternion q followed by its inverse quaternion q -1 Rot 1 ( Rot ( v )) = Rot 1 q q = = q v 1 q qvq qq ( qvq 1 q 1 ) Animation(U), Chap, Tech. Background 79

80 Review - Quaternion Interpolation A quaternion is a point on a 4D unit sphere Unit quaternion: q=(s,x,y,z), q = 1 Form a subspace: a 4D sphere Interpolating quaternion means moving between two points on the 4D unit sphere A unit quaternion at each step another point on the 4D unit sphere Move with constant angular velocity along the greatest circle between the two points on the 4D unit sphere Animation(U), Chap, Tech. Background 80

81 Review - Linear Interpolation Linear interpolation generates unequal spacing of points after projecting to circle Animation(U), Chap, Tech. Background 81

82 Review Spherical Linear Interpolation (slerp) Want equal increment along arc connecting two quaternion on the spherical surface Spherical linear interpolation (slerp) sin((1 u) θ ) slerp q, q, u) = q1 + sin θ sin( uθ ) sin θ ( 1 q 1 where u goes from 0 to1andθ = cos ( q1 q Normalize to regain unit quaternion ) Animation(U), Chap, Tech. Background 8

83 Review Spherical Linear Interpolation (slerp) Let q = α q1 + βq We can solve for given : q = 1, q 1 q = θ, q1 q to give = u θ sin((1 u) θ ) sin( uθ ) slerp( q1, q, u) = q1 + q sin θ sin θ Animation(U), Chap, Tech. Background 83

84 Review Spherical Linear Interpolation (slerp) q 1 q > 0 q 1 q q Animation(U), Chap, Tech. Background 84

85 Review Spherical Linear Interpolation (slerp) Recall that q and q represent same rotation What is the difference between: Slerp(u, q 1, q ) and Slerp(u, q 1, -q )? One of these will travel less than 90 degrees while the other will travel more than 90 degrees across the sphere This corresponds to rotating the short way or the long way Usually, we want to take the short way, so we negate one of them if their dot product is < 0 Animation(U), Chap, Tech. Background 85

86 Review Spherical Linear Interpolation (slerp) If we have an intermediate position q, the interpolation from q 1 -->q -->q 3 will not necessarily follow the same path as the interpolation from q 1 to q 3. Animation(U), Chap, Tech. Background 86

87 Interpolating a Series of Quaternions As linear interpolation in Euclidean space, we can have first order discontinuity problem when interpolating a series of orientations. Need a cubic curve interpolation to maintain first order continuity in Euclidean space Similarly, slerp can have 1st order discontinuity We also need a cubic curve interpolation in 4D spherical space for 1st order continuity 87

88 Interpolating a Series of Quaternions Interpolated points [, p n-1, p n, p n+1,.]in3d Between each pair of points, two control points will be constructed. e.g., For p n a n : the one intermediate after p n b n : the one immediately before p n p n +(p n -p n-1 ) Take average 88

89 Interpolating a Series of Quaternions Interpolated points [, p n-1, p n, p n+1,.] in 3D Between each pair of points, two control points will be constructed. e.g., For p n a n : the one intermediate after p n b n : the one immediately before p n b n= p n +(p n -a n ) 89

90 Interpolating a Series of Quaternions Interpolated points [, p n-1, p n, p n+1,.] in 3D End conditions 90

91 Interpolating a Series of Quaternions Interpolated points [, p n-1, p n, p n+1,.] in 3D Between any interpolated points p n and p n+1, a cubic Bezier curve segment is defined by p n, a n, b n+1, and p n+1 91

92 Interpolating a Series of Quaternions How to evaluate a point on Bezier curve? De Casteljau Construction Constructing Bezier curve by multiple linear interpolation u=1/3 p(1/3) 9

93 Interpolating a Series of Quaternions Animation: See gif 93

94 Interpolating a Series of Quaternions Bezier interpolation on 4D sphere? How are control points generated? How are cubic Beziers defined? Control points are automatically ti generated as it is not intuitive to manually adjust them on a 4D sphere Derive Bezier curve points by iteratively subdivision s on p n, a n, b n+1, and p n+1 94

95 Interpolating a Series of Quaternions Connecting Segments on 4D Sphere Automatically generating interior (spherical) control points b n = double( an, qn) b n q n q n a n. Bisect the span Bi sect( q' n, q n+ 1 ) = q' q' n n + q + q n+ 1 n+ 1 q n+1 q n-1 1. Double the arc q ' n = double q ( qn 1, qn ) = ( qn 1 qn ) qn qn 1 95

96 Interpolating a Series of Quaternions De Casteljau Construction on 4D Sphere p 1 = slerp(q n,a n, 1 3) p = slerp(a n,b n +1, 1 3) p 3 = slerp(b n +1,q n +1, 1 3) p = slerp(p 1 1 1, p, 3) 3 u=1/3 p 1 a n p P(1/3) b n+1 p 3 p 3 = slerp(p, p 3, 1 3) p = slerp(p 1, p 3, 1 3) q n q n+1 96

97 Path Following Issues about path following Arc length parameterization and speed control Changing the orientation Smoothing a path Path along a surface 97

98 Orientation Along a Curve A local coordinate system (u, v, w) is defined for an object to be animated a Position along the path P(s), denoted as POS w-axis: the direction the object tis facing v-axis: up vector u-axis: perpendicular to w and v (u, v, w): a right-handed coordinate system 98

99 Orientation Along a Curve 99

100 Orientation Along a Curve Frenet Frame Frenet frame: a moving coordinate system determined by curve tangent and curvature w = P '( s ) u = P"( s) v = u w P'( s) 100

101 Problems Orientation Along a Curve Frenet Frame No concept of up vector Is undefined in segments that has no curvature; i.e., P (u)=0 Can be dealt with by interpolating a Frenet frame from the Frenet frames at segment s boundary, which differ by only a rotation around w A discontinuity in the curvature vector Frenet frame has a discontinuous jump in orientation Resulting motions are usually too extreme and not natural looking 101

102 Orientation Along a Curve Frenet Frame 10

103 Orientation Along a Curve Frenet Frame 103

104 Orientation Along a Curve Camera Path Following COI: center of interest A fixed point in the scene Center point of an object in the scene Points along the path itself Points on a separate path in the scene Points on the interpolation between points in the scene w = COI POS u = w y_axis w = C ( s ) P ( s ) w = C ( s ) P ( s ) u = w y_axis u = w ( U ( s) P( s)) v = u w v = u w v = u w 104

105 Smoothing a Path with Linear Interpolation Two points on either sides of a point are averaged, and this point is averaged with the original point. Repeated application P + P + P i 1 i+ 1 i P' i = = P i 1 + P i + P i

106 Smoothing a Path with Linear Interpolation 106

107 Smoothing a Path with Cubic Interpolation Four points on either sides of a point are fitted by a cubic curve, and the midpoint of the curve is averaged with the original point. 3 P ( u ) = au + bu + cu + d P i = P (0) = i d Pi 1 = P(1/ 4) = a + b + c + d P i + 1 = P(3/ 4) = a + b + c + d P i = P(1 = a + b + c + d + ) 107

108 Smoothing a Path with Cubic Interpolation 108

109 Smoothing a Path with Cubic Interpolation 109

110 Determine a Path on a Surface If one object is to move across the surface of another object, we need to specify a path across the surface. Given start and destination points Find a shortest path between the points expensive Determine a plane that contain two points and dis generally perpendicular to the surface (averaged normal of two points) Intersection of the plane with the surface (mesh) 110

111 Determine a Path on a Surface 111

112 Determine a Path on a Parametric Surface Given start and destination points on parametric domain Define the straight line connecting two points in the parametric space The curve on the surface corresponding to this line 11