Design considerations

Transcription

1 Curves

2 Design considerations local control of shape design each segment independently smoothness and continuity ability to evaluate derivatives stability small change in input leads to small change in output ease of rendering

3 Design considerations local control of shape design each segment independently smoothness and continuity ability to evaluate derivatives stability small change in input leads to small change in output ease of rendering approximate out of a number of wood strips

4 Design considerations local control of shape design each segment independently smoothness and continuity ability to evaluate derivatives stability small change in input leads to small change in output ease of rendering approximate out of a join points or knots number of wood strips

5 What is a curve? intuitive idea: draw with a pen set of points the pen traces may be 2D, like on paper or 3D, space curve

6 What is a curve? or be closed may have endpoints extend infinitely

7 How do we specify a curve?

8 How do we specify a curve? Implicit (2D) f(x,y) = 0 test if (x,y) is on the curve

9 How do we specify a curve? Implicit (2D) f(x,y) = 0 test if (x,y) is on the curve f(x,y) = 0 on curve

10 How do we specify a curve? Implicit (2D) f(x,y) = 0 test if (x,y) is on the curve f(x,y) 0 off curve

11 How do we specify a curve? Implicit (2D) f(x,y) = 0 test if (x,y) is on the curve Parametric (2D) (x,y) = f(t) (3D) (x,y,z) = f(t) map free parameter t to points on the curve

12 How do we specify a curve? Implicit (2D) f(x,y) = 0 test if (x,y) is on the curve t = 10 Parametric (2D) (x,y) = f(t) (3D) (x,y,z) = f(t) map free parameter t to points on the curve t = 5 t = 0

13 How do we specify a curve? Implicit (2D) f(x,y) = 0 test if (x,y) is on the curve Parametric (2D) (x,y) = f(t) (3D) (x,y,z) = f(t) map free parameter t to points on the curve Procedural e.g., fractals, subdivision schemes Fractal: Koch Curve [George Reese]

14 How do we specify a curve? Implicit (2D) f(x,y) = 0 test if (x,y) is on the curve Parametric (2D) (x,y) = f(t) (3D) (x,y,z) = f(t) map free parameter t to points on the curve Procedural e.g., fractals, subdivision schemes Bezier Curve

15 A curve may have multiple representations

16 A curve may have multiple representations Implicit f(x,y) = x 2 + y 2-1 = 0

17 A curve may have multiple representations Parametric (x,y) = f(t) = (cos t, sin t) t = pi/2 t = 0

18 A curve may have multiple representations t = pi/2 Parametric (x,y) = f(t) = (cos t, sin t), t in [0,2pi) t = 0 Same curve (set of points), but different mathematical representation!

19 A curve may have multiple representations t = pi/2 Parametric (x,y) = f(t) = (cos t, sin t), t in [0,2pi) t = 0 We will focus on parametric representations

20 Parameterization, re-parameterization t = 10 t = 0 f1(t) t = 5

21 Parameterization, re-parameterization s = 1 s = 0 f2(s) trace out the curve more quickly s = 0.5

22 Parameterization, re-parameterization s = 1 t = 10 t = 0 s = 0 relationship: t = 10*s f1(t) = f1(10*s) = f1(f(s)) = f2(s) s = 0.5 t = 5

23 Parameterization, re-parameterization t = 0 t = 10 f1(t) s = s0 f2(s) = f1(f(s)) s = s1

24 Parameterization, re-parameterization t = 0 t = 10 t = f(s) s = s0 f2(s) = f1(f(s)) s = s1

25 Natural parameterization note: points uneven t = 10 t = 0 t = 5

26 Natural parameterization pen moves at a constant velocity: evenly spaced points s = 10 s = 0 s = 5

27 Natural parameterization pen moves at a constant velocity: evenly spaced points s = 10 s = 0 also called arc-length parameterization s = 5

28 Natural parameterization pen moves at a constant velocity: evenly spaced points s = 10 s = 0 also called arc-length parameterization s = 5

29 piecewise parametric representation sometimes easy to find a parametric representation e.g., circle, line segment

30 piecewise parametric representation in other cases, not obvious

31 piecewise parametric representation strategy: break into simpler pieces

32 piecewise parametric representation strategy: break into simpler pieces switch between functions that represent pieces:

33 piecewise parametric representation strategy: break into simpler pieces switch between functions that represent pieces: map the inputs to f1 and f2 to be from 0 to 1

34 Curve Properties Local properties: continuity position direction curvature Global properties (examples): closed curve curve crosses itself Interpolating vs. non-interpolating

35 Continuity: stitching curve segments together knot parametric continuity geometric continuity

36 Finding a Parametric Representation

37 Polynomial Pieces <whiteboard>

38 Blending Functions

39 Blending functions are more convenient basis than monomial basis canonical form (monomial basis) geometric form (blending functions)

40 Interpolating Polynomials

41 Interpolating polynomials Given n+1 data points, can find a unique interpolating polynomial of degree n Different methods: Vandermonde matrix Lagrange interpolation Newton interpolation

42 higher order interpolating polynomials are rarely used overshoots non-local effects 4th order (gray) to 5th order (black)

43 Piecewise Polynomial Curves

44 Example: blending functions for two line segments

45 Cubics Allow up to C2 continuity at knots need 4 control points may be 4 points on the curve, combination of points and derivatives,... good smoothness and computational properties

46 We can get any 3 of 4 properties 1.piecewise cubic 2.curve interpolates control points 3.curve has local control 4.curves has C2 continuity at knots

47 Cubics Natural cubics C2 continuity n points -> n-1 cubic segments control is non-local :( ill-conditioned x(

48 Cubic Hermite Curves C1 continuity specify both positions and derivatives

49 Cubic Hermite Curves Specify endpoints and derivatives construct curve with C^1 continuity / /

50 Hermite blending functions [Wikimedia Commons]

51 Example: keynote curve tool

52 Interpolating vs. Approximating Curves Interpolating Approximating (non-interpolating)

53 Cubic Bezier Curves

54 Cubic Bezier Curves

55 Cubic Bezier Curve Examples

56 Cubic Bezier blending functions

57 Bezier Curves Degrees 2-6

58 Bernstein Polynomials The blending functions are a special case of the Bernstein polynomials These polynomials give the blending polynomials for any degree Bezier form All roots at 0 and 1 For any degree they all sum to 1 They are all between 0 and 1 inside (0,1) 58

59 n = 3 n = 5 n = 4 n = 6 59

60 Bezier Curve Properties curve lies in the convex hull of the data variation diminishing symmetry affine invariant efficient evaluation and subdivision

61 Joining Cubic Bezier Curves

62 Joining Cubic Bezier Curves for C1 continuity, the vectors must line up and be the same length for G1 continuity, the vectors need only line up

63 Evaluating p(u) geometrically

64 Evaluating p(u) geometrically De Casteljau algorithm

65 Bezier subdivision

66 Recursive Subdivision for Rendering

67 Cubic B-Splines

68 Cubic B-Splines

69 Spline blending functions

70 General Splines Defined recursively by Cox-de Boor recursion formula

71 Spline properties Basis functions convexity

72 Surfaces

73 Parametric Surface

74 Parametric Surface - tangent plane

75 Bicubic Surface Patch

76 Bezier Surface Patch Patch lies in convex hull