The Essentials of CAGD

Transcription

1 The Essentials of CAGD Chapter 6: Bézier Patches Gerald Farin & Dianne Hansford CRC Press, Taylor & Francis Group, An A K Peters Book c 2 Farin & Hansford The Essentials of CAGD 1 / 46

2 Outline 1 Introduction to Bézier Patches 2 Parametric Surfaces 3 Bilinear Patches 4 Bézier Patches 5 Properties of Bézier Patches 6 Derivatives 7 Higher Order Derivatives 8 The de Casteljau Algorithm 9 Normals 1 Changing Degrees 11 Subdivision 12 Ruled Bézier Patches 13 Functional Bézier Patches 14 Monomial Patches Farin & Hansford The Essentials of CAGD 2 / 46

3 Introduction to Bézier Patches The Utah teapot composed of Bézier patches Surfaces: Basic definitions Extend the concept of Bézier curves Farin & Hansford The Essentials of CAGD 3 / 46

4 Parametric Surfaces Parametric curve: mapping of the real line into 2- or 3-space Parametric surface: mapping of the real plane into 3-space R 2 is the domain of the surface A plane with a (u,v) coordinate system Corresponding 3D surface point: f(u,v) x(u,v) = g(u,v) h(u,v) Farin & Hansford The Essentials of CAGD 4 / 46

5 Parametric Surfaces Example: Parametric surface x(u,v) = u v u 2 +v 2 Only a portion of surface illustrated This is a functional surface Parametric surfaces may be rotated or moved around More general than z = f(x,y) Farin & Hansford The Essentials of CAGD 5 / 46

6 Bilinear Patches Typically interested in a finite piece of a parametric surface The image of a rectangle in the domain The finite piece of surface called a patch Let domain be the unit square {(u,v) : u,v 1} Map it to a surface patch defined by four points x(u,v) = [ 1 u u ][ ][ ] b, b,1 1 v b 1, b 1,1 v Surface patch is linear in both the u and v parameters bilinear patch Farin & Hansford The Essentials of CAGD 6 / 46

7 Bilinear Patches Bilinear patch: x(u,v) = [ 1 u u ][ ][ ] b, b,1 1 v b 1, b 1,1 v Geometric interpretation: rewrite as x(u,v) = (1 v)p u +vq u where p u = (1 u)b, +ub 1, q u = (1 u)b,1 +ub 1,1 Farin & Hansford The Essentials of CAGD 7 / 46

8 Bilinear Patches Example: Given four points b i,j and compute x(.25,.5) 1 1 b, = b 1, = b,1 = 1 b 1,1 = 1 1 p u = =.25 q u = = x(.25,.5) =.5p u +.5q u = Farin & Hansford The Essentials of CAGD 8 / 46

9 Bilinear Patches Rendered image of patch in previous example Farin & Hansford The Essentials of CAGD 9 / 46

10 Bilinear Patches Bilinear patch: Is equivalent to x(u,v) = (1 v)p u +vq u x(u,v) = (1 u)p v +uq v where p v = (1 v)b, +vb,1 q v = (1 v)b 1, +vb 1,1 Farin & Hansford The Essentials of CAGD 1 / 46

11 Bilinear Patches Bilinear patch also called a hyperbolic paraboloid Isoparametric curve: only one parameter is allowed to vary Isoparametric curves on a bilinear patch 2 families of straight lines (ū,v): line constant in u but varying in v (u, v): line constant in v but varying in u Four special isoparametric curves (lines): (u,) (u,1) (,v) (1,v) Farin & Hansford The Essentials of CAGD 11 / 46

12 Bilinear Patches A hyperbolic paraboloid also contains curves Consider the line u = v in the domain As a parametric line: u(t) = t, v(t) = t Domain diagonal mapped to the 3D curve on the surface d(t) = x(t,t) In more detail: d(t) = [ 1 t t ][ ][ ] b, b,1 1 t b 1, b 1,1 t Collecting terms now gives d(t) = (1 t) 2 b, +2(1 t)t[ 1 2 b, b 1,]+t 2 b 1,1 quadratic Bézier curve Farin & Hansford The Essentials of CAGD 12 / 46

13 Bilinear Patches Example: Compute the curve on the surface for u(t) = t, v(t) = t c = b, = c 1 = [b 1, +b,1 ] =.5 1 c 2 = b 1,1 = 1 1 d(t) = c B 2 (t)+c 1 B 2 1(t)+c 2 B 2 2(t) Farin & Hansford The Essentials of CAGD 13 / 46

14 Bézier Patches Bilinear patch using linear Bernstein polynomials: x(u,v) = [ ][ ] B 1(u) B1 1 (u)][ b, b,1 B 1 (v) b 1, b 1,1 B1 1(v) Generalization: x(u,v) = [ B m(u)... Bm m(u) ] b,... b,n B n (v)... b m,... b m,n Bn(v) n =b, B m (u)b n (v)+...+b i,j B m i (u)b n j (v)+...+b m,n B m m(u)b n n(v) Examples: m = n = 1: bilinear m = n = 3: bicubic Farin & Hansford The Essentials of CAGD 14 / 46

15 Bézier Patches x(u,v) = [ b,... b,n B n B m(u)... Bm m (u)] (v)... b m,... b m,n Bn(v) n Abbreviated as x(u,v) = M T BN 2-stage explicit evaluation method at given (u, v) Step 1: generate c i C = M T B = [c,...,c n ] Step 2: generate point on surface x(u,v) = CN ( explicit because Bernstein polynomials evaluated) Farin & Hansford The Essentials of CAGD 15 / 46

16 Bézier Patches x(u,v) = M T BN x(u,v) = CN Control points c,...,c n of C do not depend on the parameter value v Curve CN: curve on surface Constant u Variable v isoparametric curve or isocurve Farin & Hansford The Essentials of CAGD 16 / 46

17 Bézier Patches Example: Evaluate the 2 3 control net at (u,v) = (.5,.5) B = Step 1) Compute quadratic Bernstein polynomials for u =.5: M T = [ ] Intermediate control points C = M T B = Bézier points of an isoparametric curve containing x(.5,.5) Farin & Hansford The Essentials of CAGD 17 / 46

18 Bézier Patches Step 2) Compute cubic Bernstein polynomials for v =.5:.125 N = x(.5,.5) = CN = Farin & Hansford The Essentials of CAGD 18 / 46

19 Bézier Patches Another approach to 2-stage explicit evaluation: x(u,v) = M T BN D = BN x = M T D v-isoparametric curve Farin & Hansford The Essentials of CAGD 19 / 46

20 Properties of Bézier Patches Bézier patches properties essentially the same as the curve ones 1 Endpoint interpolation: Patch passes through the four corner control points Control polygon boundaries define patch boundary curves 2 Symmetry: Shape of patch independent of corner selected to be b, 3 Affine invariance: Apply affine map to control net and then evaluate identical to applying affine map to the original patch 4 Convex hull property: x(u,v) in the convex hull of the control net for (u,v) [,1] [,1] 5 Bilinear precision: Sketch on next slide 6 Tensor product: evaluation via isoparametric curves Farin & Hansford The Essentials of CAGD 2 / 46

21 Interior control points evenly-spaced on lines connecting boundary control points on adjacent edges Farin & Hansford The Essentials of CAGD 21 / 46 Properties of Bézier Patches A degree 3 4 control net with bilinear precision Boundary control points evenly spaced on lines connecting the corner control points

22 Properties of Bézier Patches Tensor product property very powerful conceptual tool for understanding Bézier patches Shape as a record of the shape of a template moving through space Template can change shape as it moves Shape and position is guided by columns of Bézier control points Farin & Hansford The Essentials of CAGD 22 / 46

23 Derivatives A derivative is the tangent vector of a curve on the surface Called a partial derivative There are two isoparametric curves through a surface point The v = constant curve is a curve on the surface with parameter u Differentiate with respect to u x u (u,v) = x(u,v) u Called the u partial Farin & Hansford The Essentials of CAGD 23 / 46

24 Derivatives Example: Find partial x v (.5,.5) of B = Control polygon C for the u =.5 isoparametric curve Farin & Hansford The Essentials of CAGD 24 / 46

25 Derivatives Example con t: Derivative curve x v (.5,v) = 3( c B 2 (v)+ c 1B 2 1 (v)+ c 2B 2 2 (v)) c = 3 c 1 = 3 c Evaluate at v =.5 x v (.5,.5) = u partials differentiate the isoparametric curve with control points D = BN Farin & Hansford The Essentials of CAGD 25 / 46

26 Derivatives Computing derivatives via a closed-form expression x u (u,v) = m [ 1, b,... 1, b,n B n B m 1 (u)... Bm 1 m 1 (u)] (v)... 1, b m 1,... 1, b m 1,n Bn n(v) 1, b i,j denote forward differences: 1, b i,j = b i+1,j b i,j Closed-form u-partial derivative expression is a degree (m 1) n patch with a control net consisting of vectors rather than points Farin & Hansford The Essentials of CAGD 26 / 46

27 Derivatives u-partial formed from differences of control points of the original patch in the u-direction x u (u,v) = 2 [ B 3 B 1(u) B1 1 (u)] (v) B B1 3 (v) B2 3(v) B3 3(v) 3 B = x u (.5,.5) = Farin & Hansford The Essentials of CAGD 27 / 46

28 Derivatives Closed-form v-partial derivative x v (u,v) = n [,1 b,...,1 b,n 1 B m(u)... Bm m (u)]..,1 b m,...,1 b m,n 1 (v) B n 1. B n 1 n 1 (v),1 b i,j = b i,j+1 b i,j Closed-form v-partial derivative is a degree m (n 1) patch Farin & Hansford The Essentials of CAGD 28 / 46

29 Higher Order Derivatives A Bézier patch may be differentiated several times Derivatives of order k or k th partials v partials: x (k) v (u,v) = n! (n k)! [ B m (u)... Bm(u) ],k b,...,k b,n 1 m..,k b m,...,k b m,n 1 k th forward differences,k b i,j Acting only on the second subscripts B n k (v). Bn 1 n k (v) Farin & Hansford The Essentials of CAGD 29 / 46

30 Higher Order Derivatives Mixed partial or twist vector x u,v (u,v) = x u,v (u,v) = x u(u,v) v or x v (u,v) u mn [ 1,1 b,... 1,1 b,n 1 B n 1 (v) B m 1 (u)... Bm 1 m 1 (u)] ,1 b m 1,... 1,1 b m 1,n 1 Bn 1 n 1 (v) Farin & Hansford The Essentials of CAGD 3 / 46

31 Higher Order Derivatives 1,1 b i,j =,1 (b i+1,j b i,j ) =,1 b i+1,j,1 b i,j = b i+1,j+1 b i+1,j b i,j+1 +b i,j Farin & Hansford The Essentials of CAGD 31 / 46

32 Higher Order Derivatives Example: Bilinear patch 1 1 b, = b 1, = b,1 = 1 b 1,1 = 1 1 x u,v (u,v) = B (u) 1,1 b, B (v) = 1,1 b, = 1 B (u) = 1 for all u a bilinear patch has a constant twist vector Farin & Hansford The Essentials of CAGD 32 / 46

33 Higher Order Derivatives The Bernstein basis functions property: B() n = 1 and Bi n () = for i = 1,n Bn(1) n = 1 and Bi n (1) = for i =,n 1 Simple form of the twist at the corners of the patch x u,v (,) = mn 1,1 b, x u,v (1,) = mn 1,1 b m 1, x u,v (,1) = mn 1,1 b,n 1 x u,v (1,1) = mn 1,1 b m 1,n 1 Farin & Hansford The Essentials of CAGD 33 / 46

34 The de Casteljau Algorithm Evaluation of a Bézier patch: x(u,v) = M T BN Define an intermediate set of points C = M T B c = B m (u)b, +...+Bm(u)b m m, c 1 = B m (u)b, Bm m (u)b m,1... c n = B m (u)b,n +...+Bm m (u)b m,n Evaluate n degree m curves with the de Casteljau algorithm Farin & Hansford The Essentials of CAGD 34 / 46

35 The de Casteljau Algorithm Final evaluation step: x(u,v) = CN Evaluate this degree n Bézier curve with the de Casteljau algorithm The 2-stage de Casteljau evaluation method Repeated calls to the de Casteljau algorithm for curves Advantage of this geometric approach: Allows computation of a point and derivative Control polygon C: Evaluate point x(u,v) = CN and tangent x v Control polygon D = BN: Evaluate point x = M T D and tangent x u Farin & Hansford The Essentials of CAGD 35 / 46

36 Normals The normal vector or normal is a fundamental geometric concept Used throughout computer graphics and CAD/CAM At a given point x(u,v) the normal is perpendicular to the surface Tangent plane at x(u,v) Defined by x,x u,x v A point and two vectors The normal n is a unit vector defined by n = x u x v x u x v Farin & Hansford The Essentials of CAGD 36 / 46

37 Normals 3-stage de Casteljau evaluation method Ingredients for n are x,x u, and x v 1 For all m+1 rows Compute n 1 levels of dca v parameter triangles 2 Compute m 1 levels of dca parameter u squares 3 Four points (squares) form a bilinear patch Its tangent plane is surface s tangent plane Evaluate and compute the partials Vectors must be scaled for original patch Farin & Hansford The Essentials of CAGD 37 / 46

38 Normals Example: 3-stage de Casteljau evaluation method at (u,v) = (.5,.5) Results in a bilinear patch Bilinear patch shares the same tangent plane as the original patch x.124 n =.9922 Farin & Hansford The Essentials of CAGD 38 / 46

39 Changing Degrees Bézier patch degrees: m in u direction and n in v direction Degree elevation for curves used to degree elevate patch Example: Raise m to m+1 then resulting control net will have n+1 columns of control points Each column contains m+2 control points Still describes same surface Degree reduction performed on a row-by-row or column-by-column basis Repeatedly applying the curve algorithm Farin & Hansford The Essentials of CAGD 39 / 46

40 Changing Degrees Degree elevation of a bilinear patch Elevate to degree 2 in u Farin & Hansford The Essentials of CAGD 4 / 46

41 Subdivision Curve subdivision: Splitting one curve segment into two segments Patch subdivision: split into two patches Example: u splits the domain unit square into two rectangles Patch split along an isoparametric curve Method: Perform curve subdivision for each degree m column of the control net at parameter u Farin & Hansford The Essentials of CAGD 41 / 46

42 Ruled Bézier Patches Ruled surface is linear in one isoparametric direction v-direction linear: x(u,v) = (1 v)x(u,)+vx(u,1) u-direction linear: x(u,v) = (1 u)x(,v)+ux(1,v) Simple method to fit a surface between two curves Two curves the same degree Example: A bilinear surface Farin & Hansford The Essentials of CAGD 42 / 46

43 Ruled Bézier Patches Let the two curves be given u = : b,,...,b,n and u = 1 : b,1,...,b m,1 Ruled surface: b, b,1 [ ] x(u,v) = [B m (u),...,bm(u)] m B 1.. (v) B1 1 b m, b (v) m,1 A developable surface is a special ruled surface Important in manufacturing Bending a piece of sheet metal without tearing or stretching Special conditions for a ruled surface to be developable (Gaussian curvature must be zero everywhere) Farin & Hansford The Essentials of CAGD 43 / 46

44 Functional Bézier Patches Functional or nonparametric Bézier patches are analogous to their curve counterparts The graph of a functional surface is a parametric surface of the form: x x(u) u y = y(v) = v z z(u,v) f(u,v) Important feature: single-valued Useful in some applications such as sheet metal stamping Farin & Hansford The Essentials of CAGD 44 / 46

45 Functional Bézier Patches Control points for a functional Bézier patch defined over [,1] [,1] i/m b i,j = j/n b i,j Over an arbitrary rectangular region [a,b] [c,d]: (Direct generalization of functional Bézier curves over an arbitrary interval) Farin & Hansford The Essentials of CAGD 45 / 46

46 Monomial Patches x(u,v) = [ a,... a,n 1 u... u m].. a m,... a m,n = U T AV Analogous to curves: a, represents a point on the patch at (u,v) = (,) All other a i,j are partial derivatives Conversion between monomial and the Bézier forms: Analogous to curves ( )( ) m n a i,j = i,j b, i j 1 v. v n Farin & Hansford The Essentials of CAGD 46 / 46