Computergrafik. Matthias Zwicker Universität Bern Herbst 2016

Transcription

1 Computergrafik Matthias Zwicker Universität Bern Herbst 2016

2 2 Today Curves Introduction Polynomial curves Bézier curves Drawing Bézier curves Piecewise curves

3 Modeling Creating 3D objects How to construct complicated surfaces? Goal Specify objects with few control points Resulting object should be visually pleasing (smooth) Start with curves, then generalize to surfaces 3

4 4 Usefulness of curves Surface of revolution

5 5 Usefulness of curves Extruded/swept surfaces

6 Usefulness of curves Animation Provide a track for objects Use as camera path 6

7 Usefulness of curves Generalize to surface patches using grids of curves, next class 7

8 8 How to represent curves Specify every point along curve? Hard to get precise, smooth results Too much data, too hard to work with Idea: specify curves using small numbers of control points Mathematics: use polynomials to represent curves Control point

9 9 Interpolating polynomial curves Curve goes through all control points Seems most intuitive Surprisingly, not usually the best choice Hard to predict behavior Overshoots, wiggles Hard to get nice-looking curves Control point Interpolating curve

10 Approximating polynomial curves Curve is influenced by control points Control point Various types & techniques based on polynomial functions Bézier curves, B-splines, NURBS Focus on Bézier curves 10

11 11 Mathematical definition A vector valued function of one variable x(t) Given t, compute a 3D point x=(x,y,z) May interpret as three functions x(t), y(t), z(t) Moving a point along the curve x(t) z y x(0.0) x(0.5) x(1.0) x

12 Tangent vector Derivative A vector that points in the direction of movement Length of x (t) corresponds to speed x(t) z y x (0.0) x (0.5) x (1.0) x 12

13 13 Today Curves Introduction Polynomial curves Bézier curves Drawing Bézier curves Piecewise curves

14 14 Polynomial functions Linear: (1 st order) Quadratic: (2 nd order) Cubic: (3 rd order)

15 15 Polynomial curves Linear Evaluated as a + b b

16 16 Polynomial curves Quadratic: (2 nd order) Cubic: (3 rd order) We usually define the curve for 0 t 1

17 Control points Polynomial coefficients a, b, c, d etc. can be interpreted as 3D control points Remember a, b, c, d have x,y,z components each Unfortunately, polynomial coefficients don t intuitively describe shape of curve Main objective of curve representation is to come up with intuitive control points Position of control points predicts shape of curve 17

18 Control points How many control points? Two points define a line (1 st order) Three points define a quadratic curve (2 nd order) Four points define a cubic curve (3 rd order) k+1 points define a k-order curve Let s start with a line 18

19 First order curve Based on linear interpolation (LERP) Weighted average between two values Value could be a number, vector, color, Interpolate between points p 0 and p 1 with parameter t Defines a curve that is straight (first-order curve) t=0 corresponds to p 0 t=1 corresponds to p 1 t=0.5 corresponds to midpoint p 0. 0<t<1 t=0. p 1 t=1 x(t) Lerpt, p 0, p 1 1 tp 0 t p 1 19

20 20 Linear interpolation Three different ways to write it Equivalent, but different properties become apparent Advantages for different operations, see later 1. Weighted sum of control points 2. Polynomial in t t 0 3. Matrix form

21 Weighted sum of control points x(t) (1 t)p 0 (t)p 1 B 0 (t) p 0 B 1 (t)p 1, where B 0 (t) 1 t and B 1 (t) t Weights B 0 (t), B 1 (t) are functions of t Sum is always 1, for any value of t Also known as basis or blending functions 21

22 22 Linear polynomial x(t) (p 1 p 0 ) vector t p 0 point Curve is based at point p 0 Add the vector, scaled by t p 0...5(p 1 -p 0 ). p 1 -p 0

23 23 Matrix form Geometry matrix Geometric basis Polynomial basis In components

24 24 Tangent For a straight line, the tangent is constant Weighted average Polynomial Matrix form

25 25 Lissajou curves

26 26 Today Curves Introduction Polynomial curves Bézier curves Drawing Bézier curves Piecewise curves

27 Bézier curves A particularly intuitive way to define control points for polynomial curves Developed for CAD (computer aided design) and manufacturing Before games, before movies, CAD was the big application for CG Pierre Bézier (1962), design of auto bodies for Peugeot, Paul de Casteljau (1959), for Citroen 27

28 28 Bézier curves Higher order extension of linear interpolation Control points p 0, p 1,... p 1 p 1 p 2 p 1 p 3 p 0 p 0 p 0 p 2 Linear Quadratic Cubic

29 Bézier curves Intuitive control over curve given control points Endpoints are interpolated, intermediate points are approximated Convex Hull property Variation-diminishing property Many demo applets online Bezier/bezier.html 29

30 Cubic Bézier curve Cubic polynomials, most common case Defined by 4 control points Two interpolated endpoints Two midpoints control the tangent at the endpoints p 1 x(t) p 0 p 2 Control polyline p 3 30

31 31 Bézier Curve formulation Three alternatives, analogous to linear case 1. Weighted average of control points 2. Cubic polynomial function of t 3. Matrix form Algorithmic construction de Casteljau algorithm

32 de Casteljau Algorithm A recursive series of linear interpolations Works for any order, not only cubic Not terribly efficient to evaluate Other forms more commonly used Why study it? Intuition about the geometry Useful for subdivision (later today) 32

33 33 de Casteljau Algorithm Given the control points A value of t Here t 0.25 p 0 p 1 p 2 p 3

34 34 de Casteljau Algorithm p 1 q 1 q 0 (t) Lerpt,p 0,p 1 q 1 (t) Lerpt,p 1,p 2 q 2 (t) Lerpt,p 2,p 3 p 0 q 0 q 2 p 2 p 3

35 35 de Casteljau Algorithm q 0 r 0 q 1 r 0 (t) Lerpt,q 0 (t),q 1 (t) r 1 (t) Lerpt,q 1 (t),q 2 (t) r 1 q 2

36 36 de Casteljau Algorithm r 0 x r 1 x(t) Lerpt,r 0 (t),r 1 (t)

37 de Casteljau algorithm p 1 p 0 x p 2 Applets p 3 37

38 38 de Casteljau Algorithm Linear Quadratic Cubic Quartic

39 39 Recursive linear interpolation x Lerp t,r 0,r 1 r Lerp t,q,q r 1 Lerpt,q 1,q 2 q 0 Lerpt,p 0,p 1 q 1 Lerpt,p 1,p 2 q 2 Lerpt,p 2,p 3 p 0 p 1 p 2 p 3 p 1 q 0 r 0 p 2 x q 1 r 1 p 3 q 2 p 4

40 40 Recursive linear interpolation x Lerp t,r 0,r 1 r Lerp t,q,q r 1 Lerpt,q 1,q 2 q 0 Lerpt,p 0,p 1 q 1 Lerpt,p 1,p 2 q 2 Lerpt,p 2,p 3 p 0 p 1 p 2 p 3 p 1 q 0 r 0 p 2 x q 1 r 1 p 3 q 2 p 4

41 41 Recursive linear interpolation x Lerp t,r 0,r 1 r Lerp t,q,q r 1 Lerpt,q 1,q 2 q 0 Lerpt,p 0,p 1 q 1 Lerpt,p 1,p 2 q 2 Lerpt,p 2,p 3 p 0 p 1 p 2 p 3 p 1 q 0 r 0 p 2 x q 1 r 1 p 3 q 2 p 4

42 42 Recursive linear interpolation x Lerp t,r 0,r 1 r Lerp t,q,q r 1 Lerpt,q 1,q 2 q 0 Lerpt,p 0,p 1 q 1 Lerpt,p 1,p 2 q 2 Lerpt,p 2,p 3 p 0 p 1 p 2 p 3 p 1 q 0 r 0 p 2 x q 1 r 1 p 3 q 2 p 4

43 43 Expand the LERPs q 0 (t) Lerp t,p 0,p 1 1 tp 0 tp 1 q 1 (t) Lerpt,p 1,p 2 1 tp 1 tp 2 q 2 (t) Lerpt,p 2,p 3 1 tp 2 tp 3 r 0 (t) Lerpt,q 0 (t),q 1 (t) 1 t 1 tp 0 tp 1 t 1 tp 1 tp 2 r 1 (t) Lerpt,q 1 (t),q 2 (t) 1 t 1 tp 1 tp 2 t 1 tp 2 tp 3 x(t) Lerp t,r 0 (t),r 1 (t) 1 t 1 t 1 tp 0 tp 1 t 1 tp 1 tp 2 t 1 t 1 tp 1 tp 2 t 1 tp 2 tp 3

44 44 Expand the LERPs q 0 (t) Lerp t,p 0,p 1 1 tp 0 tp 1 q 1 (t) Lerpt,p 1,p 2 1 tp 1 tp 2 q 2 (t) Lerpt,p 2,p 3 1 tp 2 tp 3 r 0 (t) Lerpt,q 0 (t),q 1 (t) 1 t 1 tp 0 tp 1 t 1 tp 1 tp 2 r 1 (t) Lerpt,q 1 (t),q 2 (t) 1 t 1 tp 1 tp 2 t 1 tp 2 tp 3 x(t) Lerp t,r 0 (t),r 1 (t) 1 t 1 t 1 tp 0 tp 1 t 1 tp 1 tp 2 t 1 t 1 tp 1 tp 2 t 1 tp 2 tp 3

45 45 Expand the LERPs q 0 (t) Lerp t,p 0,p 1 1 tp 0 tp 1 q 1 (t) Lerpt,p 1,p 2 1 tp 1 tp 2 q 2 (t) Lerpt,p 2,p 3 1 tp 2 tp 3 r 0 (t) Lerpt,q 0 (t),q 1 (t) 1 t 1 tp 0 tp 1 t 1 tp 1 tp 2 r 1 (t) Lerpt,q 1 (t),q 2 (t) 1 t 1 tp 1 tp 2 t 1 tp 2 tp 3 x(t) Lerp t,r 0 (t),r 1 (t) 1 t 1 t 1 tp 0 tp 1 t 1 tp 1 tp 2 t 1 t 1 tp 1 tp 2 t 1 tp 2 tp 3

46 46 Expand the LERPs q 0 (t) Lerp t,p 0,p 1 1 tp 0 tp 1 q 1 (t) Lerpt,p 1,p 2 1 tp 1 tp 2 q 2 (t) Lerpt,p 2,p 3 1 tp 2 tp 3 r 0 (t) Lerpt,q 0 (t),q 1 (t) 1 t 1 tp 0 tp 1 t 1 tp 1 tp 2 r 1 (t) Lerpt,q 1 (t),q 2 (t) 1 t 1 tp 1 tp 2 t 1 tp 2 tp 3 x(t) Lerp t,r 0 (t),r 1 (t) 1 t 1 t 1 tp 0 tp 1 t 1 tp 1 tp 2 t 1 t 1 tp 1 tp 2 t 1 tp 2 tp 3

47 47 Expand the LERPs q 0 (t) Lerp t,p 0,p 1 1 tp 0 tp 1 q 1 (t) Lerpt,p 1,p 2 1 tp 1 tp 2 q 2 (t) Lerpt,p 2,p 3 1 tp 2 tp 3 r 0 (t) Lerpt,q 0 (t),q 1 (t) 1 t 1 tp 0 tp 1 t 1 tp 1 tp 2 r 1 (t) Lerpt,q 1 (t),q 2 (t) 1 t 1 tp 1 tp 2 t 1 tp 2 tp 3 x(t) Lerp t,r 0 (t),r 1 (t) 1 t 1 t 1 tp 0 tp 1 t 1 tp 1 tp 2 t 1 t 1 tp 1 tp 2 t 1 tp 2 tp 3

48 48 Weighted average of control points Regroup x(t) 1 t 1 t t 1 t x(t) 1 t 1 tp 0 tp 1 t 1 tp 1 tp 2 t 1 tp 2 tp 3 1 tp 1 tp 2 3 p 0 31 t 2 tp 1 31 tt 2 p 2 t 3 p 3 B 0 (t ) x(t) t 3 3t 2 3t 1 3t 3 3t 2 B 2 (t ) p 2 t 3 B 1 (t ) p 0 3t 3 6t 2 3t B 3 (t ) p 3 p 1

49 49 Weighted average of control points Regroup x(t) 1 t 1 t t 1 t x(t) 1 t 1 tp 0 tp 1 t 1 tp 1 tp 2 t 1 tp 2 tp 3 1 tp 1 tp 2 3 p 0 31 t 2 tp 1 31 tt 2 p 2 t 3 p 3 B 0 (t ) x(t) t 3 3t 2 3t 1 3t 3 3t 2 B 2 (t ) p 2 t 3 B 1 (t ) p 0 3t 3 6t 2 3t B 3 (t ) p 3 p 1

50 50 Weighted average of control points Regroup x(t) 1 t 1 t t 1 t x(t) 1 t 1 tp 0 tp 1 t 1 tp 1 tp 2 t 1 tp 2 tp 3 1 tp 1 tp 2 3 p 0 31 t 2 tp 1 31 tt 2 p 2 t 3 p 3 B 0 (t ) x(t) t 3 3t 2 3t 1 3t 3 3t 2 B 2 (t ) p 2 t 3 B 1 (t ) p 0 3t 3 6t 2 3t B 3 (t ) p 3 p 1 Bernstein polynomials

51 51 Cubic Bernstein polynomials x(t) B 0 t p 0 B 1 t p 1 B 2 t p 2 B 3 t p 3 The cubic Bernstein polynomials : B 0 t t 3 3t 2 3t 1 B 1 t 3t 3 6t 2 3t B 2 t 3t 3 3t 2 t t 3 B 3 B i (t) 1 Partition of unity, at each t always add to 1 Endpoint interpolation, B 0 and B 3 go to 1

52 52 General Bernstein polynomials B 1 0 B 1 1 t t 1 B 2 0 t t 2 2t 1 B 3 0 t t 3 3t 2 3t 1 t t B 2 1 t 2t 2 2t B 3 1 t 3t 3 6t 2 3t B 2 2 t t 2 B 3 2 t 3t 3 3t 2 B 3 3 t t 3 B i n t n i 1 t ni t i B n i t 1 n i n! i! n i!

53 53 General Bernstein polynomials B 1 0 B 1 1 t t 1 B 2 0 t t 2 2t 1 B 3 0 t t 3 3t 2 3t 1 t t B 2 1 t 2t 2 2t B 3 1 t 3t 3 6t 2 3t B 2 2 t t 2 B 3 2 t 3t 3 3t 2 B 3 3 t t 3 B i n t n i 1 t ni t i B n i t 1 n i n! i! n i!

54 54 General Bernstein polynomials B 1 0 B 1 1 t t 1 B 2 0 t t 2 2t 1 B 3 0 t t 3 3t 2 3t 1 t t B 2 1 t 2t 2 2t B 3 1 t 3t 3 6t 2 3t B 2 2 t t 2 B 3 2 t 3t 3 3t 2 B 3 3 t t 3 B i n t n i 1 t ni t i B n i t 1 n i n! i! n i!

55 General Bernstein polynomials B 1 0 B 1 1 t t 1 B 2 0 t t 2 2t 1 B 3 0 t t 3 3t 2 3t 1 t t B 2 1 t 2t 2 2t B 3 1 t 3t 3 6t 2 3t B 2 2 t t 2 B 3 2 t 3t 3 3t 2 B 3 3 t t 3 Order n: B i n t n i 1 t ni t i B n i t 1 n i n! i! n i! Partition of unity, endpoint interpolation 55

56 56 General Bézier curves nth-order Bernstein polynomials form nth-order Bézier curves Bézier curves are weighted sum of control points using nth-order Bernstein polynomials Bernstein polynomials of order n: Bézier curve of order n: B i n t n i 1 tni t n i0 xt B n i t p i i

57 57 Bézier curve properties Convex hull property Variation diminishing property Affine invariance

58 58 Convex hull, convex combination Convex hull of a set of points Smallest polyhedral volume such that (i) all points are in it (ii) line connecting any two points in the volume lies completely inside it (or on its boundary) Convex combination of the points Weighted average of the points, where weights all between 0 and 1, sum up to 1 Any convex combination always lies within the convex hull p 1 p 3 Convex hull p 0 p 2

59 59 Convex hull property Bézier curve is a convex combination of the control points Bernstein polynomials add to 1 at each value of t Curve is always inside the convex hull of control points Makes curve predictable Allows efficient culling, intersection testing, adaptive tessellation p 1 p 3 p 0 p 2

60 Variation diminishing property If the curve is in a plane, this means no straight line intersects a Bézier curve more times than it intersects the curve's control polyline Curve is not more wiggly than control polyline Yellow line: 7 intersections with control polyline 3 intersections with curve 60

61 Affine invariance Two ways to transform Bézier curves 1. Transform the control points, then compute resulting point on curve 2. Compute point on curve, then transform it Either way, get the same transform point! Curve is defined via affine combination of points (convex combination is special case of an affine combination) Invariant under affine transformations Convex hull property always remains 61

62 62 Cubic polynomial form Start with Bernstein form: x(t) t 3 3t 2 3t 1p 0 3t 3 6t 2 3tp 1 3t 3 3t 2 p 2 t 3 p 3 Regroup into coefficients of t : x(t) p 0 3p 1 3p 2 p 3 t 3 3p 0 6p 1 3p 2 t 2 3p 0 3p 1 t p 0 1 x(t) at 3 bt 2 ct d a p 0 3p 1 3p 2 p 3 b 3p 0 6p 1 3p 2 c 3p 0 3p 1 d p 0 Good for fast evaluation, precompute constant coefficients (a,b,c,d) Not much geometric intuition

63 63 Cubic polynomial form Start with Bernstein form: x(t) t 3 3t 2 3t 1p 0 3t 3 6t 2 3tp 1 3t 3 3t 2 p 2 t 3 p 3 Regroup into coefficients of t : x(t) p 0 3p 1 3p 2 p 3 t 3 3p 0 6p 1 3p 2 t 2 3p 0 3p 1 t p 0 1 x(t) at 3 bt 2 ct d a p 0 3p 1 3p 2 p 3 b 3p 0 6p 1 3p 2 c 3p 0 3p 1 d p 0 Good for fast evaluation, precompute constant coefficients (a,b,c,d) Not much geometric intuition

64 64 Cubic polynomial form Start with Bernstein form: x(t) t 3 3t 2 3t 1p 0 3t 3 6t 2 3tp 1 3t 3 3t 2 p 2 t 3 p 3 Regroup into coefficients of t : x(t) p 0 3p 1 3p 2 p 3 t 3 3p 0 6p 1 3p 2 t 2 3p 0 3p 1 t p 0 1 x(t) at 3 bt 2 ct d a p 0 3p 1 3p 2 p 3 b 3p 0 6p 1 3p 2 c 3p 0 3p 1 d p 0 Good for fast evaluation, precompute constant coefficients (a,b,c,d) Not much geometric intuition

65 65 Cubic polynomial form Start with Bernstein form: x(t) t 3 3t 2 3t 1p 0 3t 3 6t 2 3tp 1 3t 3 3t 2 p 2 t 3 p 3 Regroup into coefficients of t : x(t) p 0 3p 1 3p 2 p 3 t 3 3p 0 6p 1 3p 2 t 2 3p 0 3p 1 t p 0 1 x(t) at 3 bt 2 ct d a p 0 3p 1 3p 2 p 3 b 3p 0 6p 1 3p 2 c 3p 0 3p 1 d p 0 Good for fast evaluation, precompute constant coefficients (a,b,c,d) Not much geometric intuition

66 66 Cubic matrix form x(t) a b c d t 3 t 2 t 1 a p 0 3p 1 3p 2 p 3 b 3p 0 6p 1 3p 2 c 3p 0 3p 1 d p 0 x(t) p 0 p 1 p 2 p 3 G Bez Can construct other cubic curves by just using different basis matrix B Hermite, Catmull-Rom, B-Spline, t t t T B Bez

67 67 Cubic matrix form 3 parallel equations, in x, y and z: x x (t) p 0 x p 1x p 2 x p 3x x y (t) p 0 y p 1y p 2 y p 3y x z (t) p 0z p 1z p 2z p 3z t t t t t t t t t

68 Matrix form Bundle into a single matrix x(t) t 3 p 0 x p 1x p 2 x p 3x p 0 y p 1y p 2 y p 3y t t p 0z p 1z p 2z p 3z x(t) G Bez B Bez T x(t) C T Efficient evaluation Precompute C Take advantage of existing 4x4 matrix hardware support 68

69 69 Today Curves Introduction Polynomial curves Bézier curves Drawing Bézier curves Piecewise curves

70 70 Drawing Bézier curves Generally no low-level support for drawing smooth curves I.e., GPU draws only straight line segments Need to break curves into line segments or individual pixels Approximating curves as series of line segments called tessellation Tessellation algorithms Uniform sampling Adaptive sampling Recursive subdivision

71 Uniform sampling Approximate curve with N-1 straight segments N chosen in advance Evaluate x i x t i where t i i N for i 0, 1,, N x i a i 3 N b i2 3 N c i 2 N d Connect the points with lines Too few points? x(t) x 2 x 3 x 4 Bad approximation Curve is faceted Too many points? x 1 x 0 Slow to draw too many line segments Segments may draw on top of each other 71

72 72 Adaptive Sampling Use only as many line segments as you need Fewer segments where curve is mostly flat More segments where curve bends Segments never smaller than a pixel Various schemes for sampling, checking results, deciding whether to sample more x(t)

73 73 Recursive Subdivision Any cubic (or k-th order) curve segment can be expressed as a cubic (or k-th order) Bézier curve Any piece of a cubic (or k-th order) curve is itself a cubic (or k-th order) curve Therefore, any Bézier curve can be subdivided into smaller Bézier curves

74 de Casteljau subdivision p 1 p 0 q r 0 0 r x 1 p 2 q 2 de Casteljau construction points are the control points of two Bézier sub-segments (p 0,q 0,r 0,x) and (x,r 1,q 2,p 3 ) p 3 74

75 Adaptive subdivision algorithm 1. Use de Casteljau construction to split Bézier segment in middle (t=0.5) 2. For each half If flat enough : draw line segment Else: recurse from 1. for each half Curve is flat enough if hull is flat enough Test how far away midpoints are from straight segment connecting start and end If about a pixel, then hull is flat enough 75

76 76 Today Curves Introduction Polynomial curves Bézier curves Drawing Bézier curves Piecewise curves

77 77 More control points Cubic Bézier curve limited to 4 control points Cubic curve can only have one inflection Need more control points for more complex curves k-1 order Bézier curve with k control points Hard to control and hard to work with Intermediate points don t have obvious effect on shape Changing any control point changes the whole curve Want local support Each control point only influences nearby portion of curve

78 78 Piecewise curves (splines) Sequence of simple (low-order) curves, end-to-end Piecewise polynomial curve, or splines Sequence of line segments Piecewise linear curve (linear or first-order spline) Sequence of cubic curve segments Piecewise cubic curve, here piecewise Bézier (cubic spline)

79 79 Piecewise cubic Bézier curve Given 3N 1 points p 0,p 1, Define N Bézier segments:,p 3N x 0 (t) B 0 (t)p 0 B 1 (t)p 1 B 2 (t)p 2 B 3 (t)p 3 x 1 (t) B 0 (t)p 3 B 1 (t)p 4 B 2 (t)p 5 B 3 (t)p 6 x N 1 (t) B 0 (t)p 3N 3 B 1 (t)p 3N 2 B 2 (t)p 3N 1 B 3 (t)p 3N p 0 p p 2 1 x 0 (t) p 3 x 1 (t) 7 p 8 p x 2 (t) 6p p 9 x 3 (t) p p p 11 p 4 p 5

80 80 Piecewise cubic Bézier curve Global parameter u, 0<=u<=3N x(u) x 0 ( 1 u), 0 u 3 3 x 1 ( 1 u 1), 3 u 6 3 x N 1 ( 1 3 u (N 1)), 3N 3 u 3N 1 x(u) x i u i 3, where i 1 u 3 x(8.75) x 0 (t) x 1 (t) x 2 (t) x 3 (t) u=0 x(3.5) u=12

81 81 Continuity Want smooth curves C 0 continuity No gaps Segments match at the endpoints C 1 continuity: first derivative is well defined No corners Tangents/normals are C 0 continuous (no jumps) C 2 continuity: second derivative is well defined Tangents/normals are C 1 continuous Important for high quality reflections on surfaces

82 82 Piecewise cubic Bézier curve C 0 continuous by construction C 1 continuous at segment endpoints p 3i if p 3i - p 3i-1 = p 3i+1 - p 3i C 2 is harder to get p 4 p 2 p 1 p 2 P 3 p p 1 P 6 p 3 6 p 4 p 5 p 0 p 0 C 0 continuous p 5 C 1 continuous

83 83 Piecewise cubic Bézier curves Used often in 2D drawing programs Inconveniences Must have 4 or 7 or 10 or 13 or (1 plus a multiple of 3) control points Some points interpolate (endpoints), others approximate (handles) Need to impose constraints on control points to obtain C 1 continuity C 2 continuity more difficult Solutions User interface using Bézier handles Generalization to B-splines, next time

84 Bézier handles Segment end points (interpolating) presented as curve control points Midpoints (approximating points) presented as handles Can have option to enforce C 1 continuity [ Adobe Illustrator 84

85 85 Next time B-splines and NURBS Extending curves to surfaces