CS130 : Computer Graphics Curves. Tamar Shinar Computer Science & Engineering UC Riverside

Transcription

1 CS130 : Computer Graphics Curves Tamar Shinar Computer Science & Engineering UC Riverside

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 join points or knots approximate out of a 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 Parametric (2D) (x,y) = f(t) (3D) (x,y,z) = f(t) map free parameter t to points on the curve t = 10 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 t = pi/2 Parametric (x,y) = f(t) = (cos t, sin t) 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 Parametric Form t = 10 t = 0 t = 5

21 Parametric Form Tangent Vector t = 10 t = 0 t = 5

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

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

24 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

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

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

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

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

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

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

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

32 piecewise parametric representation in other cases, not obvious

33 piecewise parametric representation strategy: break into simpler pieces

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

35 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

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

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

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

39 Finding a Parametric Representation

40 Polynomial Pieces

41 Polynomial Pieces coefficients n = degree

42 Polynomial Pieces coefficients n = degree canonical form (monomial basis)

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

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

45 Blending functions are more convenient basis than monomial basis

46 Blending functions are more convenient basis than monomial basis

47 Blending functions are more convenient basis than monomial basis Some examples <whiteboard>

48 Interpolating Polynomials

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

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