MAE 152 Computer Graphics for Scientists and Engineers

1. MAE 152Computer Graphics for Scientists and Engineers Splines and Bezier Curves

2. Introduction • |Representing Curves • A number of small line-segments joined • Interpolation • Parametric equations • Types of curve we study • Natural Cubic Splines and Bezier Curves • Derivation • Implementation

3. “Computers can’t draw curves.” The more points/line segments that are used, the smoother the curve.

4. Why have curves ? • Representation of “irregular surfaces” • Example: Auto industry (car body design) • Artist’s representation • Clay / wood models • Digitizing • Surface modeling (“body in white”) • Scaling and smoothening • Tool and die Manufacturing

5. Curve representation • Problem: How to represent a curve easily and efficiently • “Brute force and ignorance” approaches: • storing a curve as many small straight line segments • doesn’t work well when scaled • inconvenient to have to specify so many points • need lots of points to make the curve look smooth • working out the equation that represents the curve • difficult for complex curves • moving an individual point requires re-calculation of the entire curve

6. Define a small number of points • Use a technique called “interpolation” to invent the extra points for us. • Join the points with a series of (short) straight lines Solution - Interpolation

7. The need for smoothness • So far, mostly polygons • Can approximate any geometry, but • Only approximate • Need lots of polygons to hide discontinuities • Storage problems • Math problems • Normal direction • Texture coordinates • Not very convenient as modeling tool • Gets even worse in animation • Almost always need smooth motion

8. Geometric modeling • Representing world geometry in the computer • Discrete vs. continuous again • At least some part has to be done by a human • Will see some automatic methods soon • Reality: humans are discrete in their actions • Specify a few points, have computer create something which “makes sense” • Examples: goes through points, goes “near” points

9. Requirements • Want mathematical smoothness • Some number of continuous derivatives of P • Local control • Local data changes have local effect • Continuous with respect to the data • No wiggling if data changes slightly • Low computational effort

10. A solution • Use SEVERAL polynomials • Complete curve consists of several pieces • All pieces are of low order • Third order is the most common • Pieces join smoothly • This is the idea of spline curves • or just “splines”

11. Parametric Equations - linear Walk from P0 to P1 at a constant speed. Start from P0 at t=0 Arrive at P1 at t=1 dx = x1 - x0 dy = y1 - y0 Where are you at a general time t? x(t) = x0 + t.dx y(t) = y0 + t.dy Equation(s) of a straight line as a function of an arbitrary parameter t.

12. Why use parametric equations? • One, two, three or n-dimensional representation is possible • Can handle “infinite slope” of tangents • Can represent multi-valued functions

13. For this class … • Natural Cubic Spline • Bezier Curves

14. Splines • Define the “knots” • (x0,y0) - (x3,y3) • Calculate all other points

15. Continuity • Parametric continuity Cx • Only P is continuous: C0 • Positional continuity • P and first derivative dP/du are continuous: C1 • Tangential continuity • P + first + second: C2 • Curvature continuity • Geometric continuity Gx • Only directions have to match

16. Order of continuity (a) (b) (c)

17. Wiggling effect • Example: • Four data points • Third degree polynomial • Might look something like: • This the ONLY third degree polynomial which fits the data • Wiggling gets much worse with higher degree

18. Polynomial parametric equations • To represent a straight line - linear parametric equation (i.e. one where the highest power of t was 1). • For curves, we need polynomial equations. Why? Because the graphs of polynomial equations wiggle!

19. Natural Cubic Spline • spline, n. 1A long narrow and relatively thin piece or strip of wood, metal, etc. 2A flexible strip of wood or rubber used by draftsmen in laying out broad sweeping curves, as in railroad work. • A natural spline defines the curve that minimizes the potential energy of an idealized elastic strip.

20. A natural cubic spline defines a curve, in which the points P(u)that define each segment of the spline are represented as a cubic P(u) =a0 + a1u + a2u2 + a3u3 • Where u is between 0 and 1 and a0, a1, a2 and a3 are (as yet) undetermined parameters. • We assume that the positions of n+1 control points Pk, where k = 0, 1,…, n are given, and the 1st and 2nd derivatives of P(u) are continuous at each interior control point.

21. Why cubic? • Undesirable wiggles and oscillations for higher orders • The lowest order polynomials to satisfy the following conditions Caution! Heavy math ahead!

22. A parametrically defined curve Parameterization

23. “Re-parameterization” non-uniform

24. Interpolation Approximation Interpolation versus Approximation

25. Hermite Interpolation 1 2 3 4 5

26. Hermite Spline • Say the user provides • A cubic spline has degree 3, and is of the form: • For some constants a, b, c and d derived from the control points, but how? • We have constraints: • The curve must pass through x0 when t=0 • The derivative must be x’0 when t=0 • The curve must pass through x1 when t=1 • The derivative must be x’1 when t=1

27. How to specify slope Di

28. Hermite Spline A Hermite spline is a curve for which the user provides: • The endpoints of the curve • The parametric derivatives of the curve at the endpoints • The parametric derivatives are dx/dt, dy/dt, dz/dt • That is enough to define a cubic Hermite spline, more derivatives are required for higher order curves

29. Hermite Spline • Solving for the unknowns gives: • Rearranging gives: or

30. Hermite curves in 2D and 3D • We have defined only 1D splines: x = f(t:x0,x1,x’0,x’1) • For higher dimensions, define the control points in higher dimensions (that is, as vectors)

31. Basis Functions • A point on a Hermite curve is obtained by multiplying each control point by some function and summing • The functions are called basis functions

32. An interpolating cubic spline Hermite and Cubic Spline Interpolation

33. need two more Eq’s to determine all coefficients Spline Interpolation Natural Spline

34. Deriving Natural Cubic Splines

35. Tri-Diagonal Matrix for Natural Cubic Spline

36. Closed Curves Homework

37. Bezier Curves • An alternative to splines • M. Bezier was a French mathematician who worked for the Renault motor car company. • He invented his curves to allow his firm’s computers to describe the shape of car bodies.

38. Bezier Approximation

39. Bezier curves • Typically, cubic polynomials • Similar to Hermite interpolation • Special way of specifying end tangents • Requires special set of coefficients • Need four points • Two at the ends of a segment • Two control tangent vectors

40. Bezier curves • Control polygon: control points connected to each other • Easy to generalize to higher order • Insert more control points • Hermite – third order only

41. De Casteljau algorithm • Can compute any point on the curve in a few iterations • No polynomials, pure geometry • Repeated linear interpolation • Repeated order of the curve times • The algorithm can be used as definition of the curve

42. Third order, u=0.75 De Casteljau algorithm

43. De Casteljau algorithm

44. De Casteljau algorithm

45. De Casteljau algorithm

46. De Casteljau Algorithm

47. What does it mean ? 1 0 1

48. Bezier Polynomial Function

49. Parametric equations for x(t),y(t)

50. Some Bezier Curves