Bezier Curves and Surfaces (Redbook ch12)

Bezier Curves and Surfaces(Redbook ch12)27 March 2007CMPT370Dr. Sean HoTrinity Western University

Review last time● Bump mapping theory Creating a texture in OpenGL● Texture objects: glBindTexture()● Loading image data: glTexImage2D() Using the framebuffer as a texture Applying a texture in OpenGL● Blending modes: glTexEnvf()● Texture coordinates: glTexCoord2f() Auto-generated texcoords: glTexGen() Spherical environmental mapping● Mip-mapsCMPT370: Bezier curves 27 Mar 20072

What's on for today Polynomial curves and surfaces Cubic polynomial curves:● Interpolating (4 points)● Hermite (2 points + 2 derivatives)● Bezier (2 interpolating end points + 2midpoints)● Using Bezier evaluators in OpenGLCMPT370: Bezier curves 27 Mar 20073

Parametric representation Recall a 1D curve in 3D can be represented as:[ xup u= yuzu]xp u=[ ' 'y ' z u] ' p'(u) is the tangent (velocity) vector● Usually limit u to interval [0,1] for simplicity For surfaces we have two parameters (u, v):pu ,v=[xu , v]yu , vzu , v∂ p[ ∂ x/∂u]∂ u u , v= ∂ y/∂ u∂ z/∂ u∂ p[ ∂ x/∂v]∂ v u , v= ∂ y/∂ v∂ z/∂ vCMPT370: Bezier curves 27 Mar 20074

Polynomial curves Restrict the functions x(u), y(u), z(u) to bepolynomial (of degree n) in u:● Each coefficient c kis a 3-vector● u k are the n+1 basis functions● Often choose n=3: cubic polynomial k=0..3, (x,y,z): need 12 numbers Similarly for surfaces:np u ,v=∑j=0np u=∑k =0n∑k =0c ku kc jku j v kCMPT370: Bezier curves 27 Mar 20075

Interpolating Cubic Polynomials Simplest case, but rarely used in practice Four control points p 0, ..., p 3 Fit a cubic polynomial through them● Space u evenly: p 0= p(0), p 1=[= p(1/3), ...1p 0c 2 2 p 1]3c3 3 p 21 2 pp 3= p1=c 0c 1c 2c 33 2 23 3p 0 = p0=c 0p 1= p 1 3 =c 0 1 3 c 1 1 3 2c 2 1 3 3c 3p 2 = p 2 3 =c 0 2 3 c 1 2 3 2[0 0 01 1 3 1 3 2 1 3 3 2 3 31 1 1 1][c0c 1]c 2c 3CMPT370: Bezier curves 27 Mar 20076

Geometry matrix Invert this matrix to get the geometry matrix● Multiply the geometry matrix by the fourcontrol points to get the coefficients[c 0c 1] =[1 0 0 0] [p 0−5.5 9 −4.5 1 p 1]c 29 −22.5 18 −4.5 p 2c 3−4.5 13.5 −13.5 4.5 p 3 The coefficients define the cubic polynomial thatinterpolates these control points● Can render, e.g., by using many small linesegments (GL_LINE_STRIP)CMPT370: Bezier curves 27 Mar 20077

Blending functions We can also look at the contribution each controlpoint makes to the final curve For interpolating cubics:● p(u) = b 0(u) p 0+ b 1(u) p 1+ b 2(u) p 2+ b 3(u) p 3CMPT370: Bezier curves 27 Mar 20078

Hermite polynomial curves Another way of defining cubic polynomials Specify start+end position+velocity● Also 12 numbers In matrix form:[p 0'p 0] =[1p 3'p 30 0 00 1 0 01 1 1 10 1 2 3]● Invert to get Hermite geometry matrixfrom which we get the coefficients[c0c 1]c 2c 3CMPT370: Bezier curves 27 Mar 20079

Joining polynomial curves Each segment has 4 control points● {p 0, p 1, p 2, p 3}, {p 3, p 4, p 5, p 6}, ... Kinds of continuity: differential:● C 0 : touching but may have corner● C 1 : derivatives match (Hermite)● C 2 : curvatures match Geometric continuity:● G 1 : velocity vectors in samedirection but not necessarilysame magnitudeCMPT370: Bezier curves 27 Mar 200710

Bezier curves Widely used, provided in OpenGL Use control points to indicate tangent vectors● Does not interpolate middle control points!● p'(0) = 3(p 1–p 0), p'(1) = 3(p 3–p 2) p 0, p 3specify start+end position start+end velocity derived from control points Use Hermite form[c 0c C 0 but not C 11] =[1 0 0 0] [p−3 3 0 0c 23 −6 3 0c 3−1 3 −3 10p 1]p 2p 3CMPT370: Bezier curves 27 Mar 200711

Bezier blending functions Blending functions are smooth polynomials b 0(u) = (1-u) 3 b 1(u) = 3u(1-u) 2 b 2(u) = 3u 2 (1-u) b 3(u) = 3u 3CMPT370: Bezier curves 27 Mar 200712

PR EFACIO.cuentran continuamente en las actas del. Qoncilie'Tridentino.«Semper hæc fides in Ecclesia Dei fuit.» Sees. un. capitulo3. (1)«Ideo persuasum semper in Ecclesia Dei fuit, 'idque nunc

Bezier evaluators in OpenGL Specify array (1D or 2D) of control points: GLfloat ctrlpoints[4][3] = { {-4.0, -4.0, 0.0}, ... Create a Bezier evaluator: (type=GL_MAP1_VERTEX_3) glMap1f( type, u min, u max, stride, order, points ); Enable the evaluator: glEnable( type ); Evaluate the Bezier at a particular u/v: glEvalCoord1f( (GLfloat) u );● Use this instead of glVertex(), e.g., withinglBegin( GL_LINE_STRIP )CMPT370: Bezier curves 27 Mar 200714