# CSE 167: Introduction to Computer Graphics Lecture #11: Bezier Curves. Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2016

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1 CSE 167: Introduction to Computer Graphics Lecture #11: Bezier Curves Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2016

2 Announcements Project 3 due tomorrow Midterm 2 next Thursday 2

3 Lecture Overview Polynomial Curves Introduction Polynomial functions Bézier Curves Introduction Drawing Bézier curves Piecewise Bézier curves 3

4 Bézier Curves Are a higher order extension of linear interpolation p 1 p 1 p 1 p 2 p 3 p 0 p 0 p 0 p 2 Linear Quadratic Cubic 4

5 Bézier Curves Give intuitive control over curve with control points Endpoints are interpolated, intermediate points are approximated Convex Hull property Demo: 5

6 Cubic Bézier Curve Most commonly used case Defined by four control points: Two interpolated endpoints (points are on the curve) Two points control the tangents at the endpoints Points x on curve defined as function of parameter t p 1 p 0 x(t) p 2 6 p 3

7 Algorithmic Construction Algorithmic construction De Casteljau algorithm, developed at Citroen in 1959, named after its inventor Paul de Casteljau (pronounced Cast-all- Joe ) Developed independently from Bézier s work: Bézier created the formulation using blending functions, Casteljau devised the recursive interpolation algorithm 7

8 De Casteljau Algorithm A recursive series of linear interpolations Works for any order Bezier function, not only cubic Not very efficient to evaluate Other forms more commonly used But: Gives intuition about the geometry Useful for subdivision 8

9 De Casteljau Algorithm Given: Four control points A value of t (here t 0.25) p 1 p 0 p 2 p 3 9

10 De Casteljau Algorithm p 1 q 1 q 0 (t) = Lerp( t,p 0,p ) 1 p 0 q 0 q 1 (t) = Lerp( t,p 1,p ) 2 p 2 q 2 (t) = Lerp( t,p 2,p ) 3 q 2 p 3 10

11 De Casteljau Algorithm q 0 r 0 q 1 r 0 (t) = Lerp( t,q 0 (t),q 1 (t)) r 1 (t) = Lerp( t,q 1 (t),q 2 (t)) r 1 q 2 11

12 De Casteljau Algorithm r 0 x r 1 x(t) = Lerp( t,r 0 (t),r 1 (t)) 12

13 De Casteljau Algorithm p 1 p 0 x p 2 Demo p 3 13

14 Recursive Linear Interpolation x = Lerp t,r 0,r 1 ( ) r 0 = Lerp( t,q 0,q ) 1 ( ) r 1 = Lerp t,q 1,q 2 q 0 = Lerp( t,p 0,p ) 1 q 1 = Lerp( t,p 1,p ) 2 q 2 = Lerp( t,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 14

15 Expand the LERPs q 0 (t) = Lerp( t,p 0,p 1 )= ( 1 t)p 0 + tp 1 q 1 (t) = Lerp( t,p 1,p 2 )= ( 1 t)p 1 + tp 2 q 2 (t) = Lerp( t,p 2,p 3 )= ( 1 t)p 2 + tp 3 r 0 (t) = Lerp( t,q 0 (t),q 1 (t))= 1 t r 1 (t) = Lerp( t,q 1 (t),q 2 (t))= 1 t ( ) ( )p 0 + tp 1 )+ t ( 1 t)p 1 + tp 2 ( )p 1 + tp 2 )+ t ( 1 t)p 2 + tp 3 ( ) 1 t ( ) 1 t x(t) = Lerp t,r 0 (t),r 1 (t) = ( 1 t) ( 1 t) ( 1 t)p 0 + tp 1 )+ t ( 1 t)p 1 + tp 2 +t ( 1 t) ( 1 t)p 1 + tp 2 )+ t ( 1 t)p 2 + tp 3 15 ( ) ( ) ( ) ( ) ( ) ( )

16 Weighted Average of Control Points Regroup for p: x(t) = 1 t ( )p 0 + tp 1 )+ t ( 1 t)p 1 + tp 2 ( )p 1 + tp 2 )+ t ( 1 t)p 2 + tp 3 ( ) ( 1 t) 1 t +t 1 t ( ) ( ) ( ) ( ) 1 t ( ) x(t) = ( 1 t) 3 p 0 + 3( 1 t) 2 tp 1 + 3( 1 t)t 2 p 2 + t 3 p 3 B 0 (t ) B 1 (t ) x(t) = t 3 + 3t 2 3t + 1 p 0 + 3t 3 6t 2 + 3t ( ) ( ) + 3t 3 + 3t 2 p + t 3 2 p 3 B 2 (t ) ( ) B 3 (t ) ( ) p 1 16

17 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 Weights B i (t) add up to 1 for any value of t 17

18 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 18 B i n ( t)= n i ( 1 t )n i t i = n! i! ( n i)! B n i ( t) = 1 n! = factorial of n ( ) i n (n+1)! = n! x (n+1)

19 Any order Bézier Curves nth-order Bernstein polynomials form nth-order Bézier curves B i n ( t)= n i ( 1 t)n i t n i=0 x( t)= B n i ( t)p i ( ) i 19

20 Demo 20

21 Bézier Curve Properties Overview: Convex Hull property Affine Invariance 21

22 Convex Hull Property A Bézier curve is always inside the convex hull Makes curve predictable Allows culling, intersection testing, adaptive tessellation p 1 p 3 p 0 p 2 22

23 Affine Invariance Transforming Bézier curves Two ways to transform: First transform control points, then compute spline points First compute spline points, then transform them Results are identical! Invariant under affine transformations (i.e., translation, scale, rotation, shear) Convex hull property remains true 23

24 Cubic Polynomial Form Start with Bernstein form: x(t) = ( t 3 + 3t 2 3t + 1)p 0 + ( 3t 3 6t 2 + 3t)p 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 24

25 Cubic Matrix Form x(t) = a t 3 b c t 2 d 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 ( ) ( ) t x(t) = [ p 0 p 1 p 2 p 3 ] t t G Bez B Bez T Other types of cubic splines use different basis matrices B Bez 25

26 Cubic Matrix Form In 3D: 3 equations for x, y and z: 26 [ ] 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

27 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 Pre-compute C Take advantage of existing 4x4 matrix hardware support 27

28 Lecture Overview Polynomial Curves Introduction Polynomial functions Bézier Curves Introduction Drawing Bézier curves Piecewise Bézier curves 28

29 Drawing Bézier Curves Draw line segments or individual pixels Approximate the curve as a series of line segments (tessellation) Uniform sampling Adaptive sampling Recursive subdivision 29

30 Uniform Sampling Approximate curve with N straight segments N chosen in advance Evaluate x i = x t i ( ) where t i = i N for i = 0, 1,, N Connect the points with lines Too few points? x i = a i3 N + b i2 3 N + c i 2 N + d x(t) x 2 x4 Poor approximation Curve is faceted Too many points? x 0 x 1 x 3 Slow to draw too many line segments Segments may draw on top of each other 30

31 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 x(t) 31

32 Recursive Subdivision Any cubic curve segment can be expressed as a Bézier curve Any piece of a cubic curve is itself a cubic curve Therefore: Any Bézier curve can be broken down into smaller Bézier curves 32

33 De Casteljau Subdivision p 1 p 0 q r 0 0 r x 1 p 2 De Casteljau construction points are the control points of two Bézier sub-segments q 2 p 3 33

34 Adaptive Subdivision Algorithm Use De Casteljau construction to split Bézier segment in half For each half If flat enough : draw line segment Else: recurse Curve is flat enough if hull is flat enough Test how far the approximating control points are from a straight segment If less than one pixel, the hull is flat enough 34

35 Drawing Bézier Curves With OpenGL Indirect OpenGL support for drawing curves: Define evaluator map (glmap) Draw line strip by evaluating map (glevalcoord) Optimize by pre-computing coordinate grid (glmapgrid and glevalmesh) More details about OpenGL implementation: /opengl_nurbs.pdf 35

36 Lecture Overview Polynomial Curves Introduction Polynomial functions Bézier Curves Introduction Drawing Bézier curves Longer curves 36

37 More Control Points Cubic Bézier curve limited to 4 control points Cubic curve can only have one inflection (point where curve changes direction of bending) 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 37 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

38 Piecewise Curves Sequence of line segments Piecewise linear curve Sequence of cubic curve segments Piecewise cubic curve (here piecewise Bézier) 38

39 Overview Piecewise Bezier curves Bezier surfaces 39

40 Global Parameterization Given N curve segments x 0 (t), x 1 (t),, x N-1 (t) Each is parameterized for t from 0 to 1 Define a piecewise curve Global parameter u from 0 to N x(u) = x 0 (u), 0 u 1 x 1 (u 1), 1 u 2 x N 1 (u ( N 1)), N 1 u N x(u) = x i (u i), where i = u (and x(n) = x N 1 (1)) Alternate solution: u defined from 0 to 1 x(u) = x i (Nu i), where i = Nu 40

41 Piecewise Bézier curve Given 3N + 1 points p 0,p 1,,p 3N Define N Bézier segments: 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 7 p 8 p 0 p 1 x 0 (t) p 2 p 3 x 1 (t) p 6 x 2 (t) p 9 x 3 (t) p 10 p 11 p 12 p 4 p 5 41

42 Piecewise Bézier Curve Parameter in 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 42

43 Parametric Continuity C 0 continuity: Curve segments are connected C 1 continuity: C 0 & 1st-order derivatives agree Curves have same tangents Relevant for smooth shading C 2 continuity: C 1 & 2nd-order derivatives agree Curves have same tangents and curvature Relevant for high quality reflections

44 Piecewise Bézier Curve 3N+1 points define N Bézier segments x(3i)=p 3i C 0 continuous by construction C 1 continuous at p 3i when p 3i - p 3i-1 = p 3i+1 - p 3i C 2 is harder to achieve and rarely necessary p 1 p 2 p 4 p 6 p 1 p 2 P 3 p 6 p 4 P 3 p 5 p 0 p 5 p 0 44 C 1 discontinuous C 1 continuous

45 Piecewise 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, others approximate Need to impose constraints on control points to obtain C 1 continuity Solutions User interface using Bézier handles to ascertain C 1 continuity Generalization to B-splines or NURBS 45

46 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 46

47 Demo ml 47

48 Rational Curves Weight causes point to pull more (or less) Can model circles with proper points and weights, Below: rational quadratic Bézier curve (three control points) pull less 48

49 B-Splines B as in Basis-Splines Basis is blending function Difference to Bézier blending function: B-spline blending function can be zero outside a particular range (limits scope over which a control point has influence) B-Spline is defined by control points and range in which each control point is active. 49

50 NURBS Non Uniform Rational B-Splines Generalization of Bézier curves Non uniform: Combine B-Splines (limited scope of control points) and Rational Curves (weighted control points) Can exactly model conic sections (circles, ellipses) OpenGL support: see glunurbscurve Demos: html 50

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