Curves, Surfaces and Segments, Patches

Transcription

1 Curves, Surfaces and Segments, atches The University of Texas at Austin

2 Conics: Curves and Quadrics: Surfaces Implicit form arametric form Rational Bézier Forms and Join Continuity Recursive Subdivision of Bézier Curve segments Recursive Subdivision of Bézier Surface patches The University of Texas at Austin 2

3 Conic Curves Conic Sections (Implicit form) Ellipse Hyperbola arabola x 2 a 2 + y2 b 2 = a,b > 0 x 2 a 2 y2 b 2 = a,b > 0 y 2 = 4ax a > 0 The University of Texas at Austin 3

4 Conic Sections (arametric form) Ellipse Hyperbola x(t) = a t2 + t 2 y(t) = b 2t ( < t < + ) + t2 x(t) = a + t2 t 2 y(t) = b 2t ( < t < + ) t2 The University of Texas at Austin 4

5 arabola x(t) = at 2 y(t) = 2at ( < t < + ) The University of Texas at Austin 5

6 Constructing Curve Segments Linear blend: Line segment from an affine combination of points 0 (t) = ( t) 0 + t t ( _ t ) 0 0 The University of Texas at Austin 6

7 Quadratic blend: Quadratic segment from an affine combination of line segments 0 (t) = ( t) 0 + t (t) = ( t) + t (t) = ( t) 0 (t) + t (t) The University of Texas at Austin 7

8 Cubic blend: Cubic segment from an affine combination of quadratic segments 0 (t) = ( t) 0 + t (t) = ( t) + t (t) = ( t) 0 (t) + t (t) 2 (t) = ( t) 2 + t 3 2 (t) = ( t) (t) + t 2 (t) 3 0 (t) = ( t)2 0 (t) + t2 (t) The University of Texas at Austin 8

9 The pattern should be evident for higher degrees 3 The University of Texas at Austin 9

10 Geometric view (de Casteljau Algorithm): Join the points i by line segments Join the t : ( t) points of those line segments by line segments Repeat as necessary The t : ( t) point on the final line segment is a point on the curve The final line segment is tangent to the curve at t 0 t 0 ( _ ) t _ ) ( t 0 2 t ( _ t) t The University of Texas at Austin 0

11 Expanding Terms (Basis olynomials): The original points appear as coefficients of Bernstein polynomials 0 0 (t) = 0 0 (t) = ( t) 0 + t 2 0 (t) = ( t) ( t)t + t (t) = ( t) ( t) 2 t + 3( t)t t 3 3 n n 0 (t) = i B n i (t) i=0 where B n i (t) = n! (n i)!i! ( t)n i t i = ( n i ) ( t) n i t i The Bernstein polynomials of degree n form a basis for the space of all degree-n polynomials The University of Texas at Austin

12 Recursive evaluation schemes: To obtain curve points (upward diagram): Start with given points and form successive, pairwise, affine combinations 0 i j i = i = ( t) j i + t j i+ The generated points j i are the decasteljau points To obtain basis polynomials (downward diagram): Start with and form successive, pairwise, affine combinations B 0 0 = B j i = ( t)b j i + tb j i+ where B s r = 0 when r < 0 or r > s The University of Texas at Austin 2

13 0 3 ( _ t) t 0 2 ( _ t) t ( _ t) 2 t 0 2 ( _ t) t ( _ t) ( _ t t) t The University of Texas at Austin 3

14 ( _ t) B0 0 t B0 B ( _ t) t ( _ t) t 2 B0 2 B ( _ t) t ( _ t) ( _ t t) 2 B 2 t 3 B B B 2 B 3 The University of Texas at Austin 4

15 Bernstein-Bézier (BB) Splines Bernstein-Bézier (BB) Curve Segments and their roperties Definition: A degree n (order n + ) Bernstein-Bézier curve segment is (t) = n i=0 i B n i (t) where the i are k-dimensional control points. The University of Texas at Austin 5

16 Rational Quadratic BB Forms Quadratic Rational BB Form: Homogeneous form x(t) y(t) w(t) = x 0 y 0 w 0 B 2 0 (t) + x y w B 2 (t) + x 2 y 2 w 2 B 2 2 (t) = x 0 B 2 0 (t) + x B 2 (t) + x 2B 2 2 (t) y 0 B 2 0 (t) + y B 2 (t) + y 2B 2 2 (t) w 0 B 2 0 (t) + w B 2 (t) + w 2B 2 2 (t) The University of Texas at Austin 6

17 Rational (projected) form [ x(t) ȳ(t) ] = x 0 B 0 2 (t)+x B2 (t)+x 2 B2 2 (t) w 0 B 0 2(t)+w B2 (t)+w 2 B2 2 (t) y 0 B 0 2 (t)+y B2 (t)+y 2 B2 2 (t) w 0 B 0 2(t)+w B2 (t)+w 2 B2 2 (t) = [ x0 y 0 ] B 2 0 (t) + [ x y ] B 2 (t) + [ x2 y 2 w 0 B 2 0 (t) + w B 2 (t) + w 2B 2 2 (t) ] B 2 2 (t) The University of Texas at Austin 7

18 Conversions: Conic parameterization elements in BB form 2t = B 2 (t) + 2B2 2 (t) t 2 = B 2 0 (t) + B2 (t) + t 2 = B 2 0 (t) + B2 (t) + 2B2 2 (t) t 2 = B 2 2 (t) The University of Texas at Austin 8

19 Conics as Rational Bézier Curves Conics as NURBS (Ellipse) Rational Bézier [ x(t) y(t) ] = [ a( t 2 ) b(2t) + t 2 ] = [ ab 2 0 (t) + ab2 (t) + 0B2 2 (t) 0B 2 0 (t) + bb2 (t) + b2b2 2 (t) B 2 0 (t) + B2 (t) + 2B2 2 (t) ] = [ a 0 ] [ ] [ ] a 0 B 2 0 (t) B 2 b (t) B 2 2 2b (t) B 2 0 (t) + B2 (t) + 2B2 2 (t) The University of Texas at Austin 9

20 which implies w 0 = x 0 = a y 0 = 0 w = x = a y = b w 2 = 2 x 2 = 0 y 2 = 2b The University of Texas at Austin 20

21 Conics as NURBS (Hyperbola) Rational Bézier [ x(t) y(t) ] = [ a( + t 2 ) b(2t) t 2 ] = [ ab 2 0 (t) + ab2 (t) + a2b2 2 (t) 0B 2 0 (t) + bb2 (t) + b2b2 2 (t) ] B 2 0 (t) + B2 (t) = [ a 0 ] [ a B 2 0 (t) b ] B 2 (t) [ 2a 2b B 2 0 (t) + B2 (t) + 0B2 2 (t) ] B 2 2 (t) which implies w 0 = x 0 = a y 0 = 0 w = x = a y = b w 2 = 0 x 2 = 2a y 2 = 2b The University of Texas at Austin 2

22 Conics as NURBS (arabola) Rational Bézier [ x(t) y(t) ] = [ a(t 2 ) a(2t) ] = [ 0B 2 0 (t) + 0B2 (t) + ab2 2 (t) 0B 2 0 (t) + ab2 (t) + a2b2 2 (t) ] B 2 0 (t) + B2 (t) + B2 2 (t) = [ 0 0 ] [ ] [ 0 a B 2 0 (t) B 2 a (t) 2a B 2 0 (t) + B2 (t) + B2 2 (t) ] B 2 2 (t) which implies w 0 = x 0 = 0 y 0 = 0 w = x = 0 y = a w 2 = x 2 = a y 2 = 2a The University of Texas at Austin 22

23 Not Unique x,y,w are not unique Numerator and denominator can be multiplied by a common (positive) factor The following example is a common alternative form: [ x(t) y(t) ] = [ x0 y 0 which derives from rewriting ] w 0 B 2 0 (t) + [ x y ] w B 2 (t) + [ x2 y 2 w 0 B 2 0 (t) + w B 2 (t) + w 2B 2 2 (t) x y w w x ȳ ] w 2 B 2 2 (t) The University of Texas at Austin 23

24 Bernstein-Bézier Curve roperties Convex Hull: n B n i (t) =, Bn i i=0 (t) 0 for t [0,] = (t) is a convex combination of the i for t [0,] = (t) lies within convex hull of i for t [0,] The University of Texas at Austin 24

25 Affine Invariance: A Bernstein-Bézier curve is an affine combination of its control points Any affine transformation of a curve is the curve of the transformed control points T ( n ) i B n i (t) = i=0 n T( i )B n i (t) i=0 This property does not hold for projective transformations! Interpolation: B n 0 (0) =,Bn n () =, n B n i (t) =,Bn i i=0 (t) 0 for t [0,] = B n i (0) = 0 if i 0,Bn i () = 0 if i n = (0) = 0,() = n The University of Texas at Austin 25

26 Derivatives: d dt Bn i (t) = n( B n i (t) Bn i (t) ) = (0) = n( 0 ), () = n( n n ) The University of Texas at Austin 26

27 Smoothly Joined Curve Segments (G continuity) Let n, n be the last two control points of one segment Let Q 0, Q be the first two control points of the next segment n = Q 0 ( n n ) = β(q Q 0 ) for some β > 0 n- n Q 0 Q The University of Texas at Austin 27

28 Recurrence, Subdivision: B n i (t) = ( t)bn i + tb n i (t) = decasteljau s algorithm: (t) = n o (t) k i (t) = ( t)k i (t) + t k i+ (t) 0 i = i Use to evaluate point at t, or subdivide into two new curves: 0 0, 0,...n 0 0 n, n,...n 0 are the control points for the left half are the control points for the right half The University of Texas at Austin 28

29 The University of Texas at Austin 29

30 Quadric Surfaces Implicit form Ellipsoid Hyperboloid Hyperbolic araboloid arabolic x 2 a 2 + y2 b 2 + z2 c 2 = a,b,c > 0 x 2 a 2 + y2 b 2 z2 c 2 = a,b,c > 0 x 2 a 2 y2 b 2 z2 c 2 = a,b,c > 0 y 2 = 4ax a > 0 arametric form The University of Texas at Austin 30

31 Ellipsoid x(s,t) = a s2 t 2 + s 2 + t 2 2s y(s, t) = b ( < s < + ) + s 2 + t2 2t z(s, t) = b ( < t < + ) + s 2 + t2 araboloid x(s,t) = a(s 2 + t 2 ) y(s,t) = as ( < s < + ) z(s,t) = at ( < t < + ) The University of Texas at Austin 3

32 Tensor roduct Bernstein-Bézier atches Tensor atches: The control polygon is the polygonal mesh with vertices i,j The patch basis functions are products of Bézier curve basis functions (s,t) = n i=0 n i,j B n i,j (s,t) j=0 where B n i,j (s,t) = Bn i (s)bn j (t) The University of Texas at Austin 32

33 The University of Texas at Austin 33

34 roperties: Tensor Bernstein-Bézier atch basis functions sume to one n i=0 n B n i (s)bn j (t) = j=0 atch basis functions are nonnegative on [0, ] [0, ] B n i (s)bn j (t) 0 for 0 s,t = Surface patch is in the convex hull of the control points = Surface patch is affinely invariant (Transform the patch by transforming the control points) Subdivision, Recursion, Evaluation: As for curves in each variable separately and independently Normals must be computed from partial derivatives The University of Texas at Austin 34

35 artial Derivatives: Ordinary derivative in each variable separately : s (s,t) = t (s,t) = n i=0 n i=0 n [ d i,j ds Bn i ]B (s) n j (t) j=0 n [ ] d i,j B n i (s) dt Bn j (t) j=0 Each of the above is a tangent vector in a parametric direction Surface is regular at each (s,t) where these two vectors are linearly independent The (unnormalized) surface normal is given at any regular point by ± [ s (s,t) ] t (s,t) (the sign dictates what is the outward pointing normal) The University of Texas at Austin 35

36 In particular, the cross-boundary tangent is given by (e.g., for the s = 0 boundary): n n i=0 n (,j 0,j )B n j (t) j=0 (and similarly for the other boundaries) The University of Texas at Austin 36

37 Smoothly Joined Tensor Bernstein-Bézier atches: 2 3 Q 02 0 Q Q 0 Q Q 20 Q 2 Q Q 3 32 Can be achieved by ensuring that ( i,n i,n ) = β(q i, Qi,0) for β > 0 (and correspondingly for other boundaries) The University of Texas at Austin 37

38 Rendering via Subdivision: Divide up into polygons:. By stepping s = 0,δ,2δ,..., t =,γ,2γ,..., and joining up sides and diagonals to produce a triangular mesh 2. By subdividing and rendering the control polygon The University of Texas at Austin 38

39 Barycentric Triangular Bernstein-Bézier atches de Casteljau Revisited Barycentrically: Linear blend expressed in barycentric terms ( t) 0 + t = r 0 + t where r + t = Higher powers and a symmetric form of the Bernstein polynomials: n ( ) n! (t) = i ( t) n i t i i!(n i)! i=0 = ( ) n! i t i r j where r + t = i!j! i+j=n i 0,j 0 = i+j=n i 0,j 0 ij B n ij (r,t) The University of Texas at Austin 39

40 Examples {B 0 00 (r,t)} = {} {B 0 (r,t),b 0 (r,t)} = {r,t} {B 2 02 (r,t),b2 (r,t),b2 20 (r,t)} = {r2,2rt,t 2 } {B 3 03 (r,t),b3 2 (r,t),b3 2 (r,t),b3 30 (r,t)} = {r3,3r 2 t,3rt 2,t 3 } The University of Texas at Austin 40

41 Surfaces Barycentric Blends on Triangles: Formulas (r,s,t) = i+j+k=n i 0,j 0,k 0 B n n! ijk (r,s,t) = i!j!k! ri s j t k ijk B n ijk (r,s,t) The University of Texas at Austin 4

42 Triangular Bézier Surface atches Triangular decasteljau: Join adjacently indexed ijk by triangles Find r : s : t barycentric point in each triangle Join adjacent points by triangles Repeat Final point is the surface point (r,s,t) final triangle is tangent to the surface at (r,s,t) Triangle up/down schemes become tetrahedral up/down schemes roperties: The University of Texas at Austin 42

43 Each boundary curve is a Bézier curve atches will be joined smoothly if pairs of boundary triangles are planar as shown Q Q Q 02 The University of Texas at Austin 43

44 Reading Assignment and News Before the next class please review Chapter 0 and its practice exercises, of the recommended text. (Recommended Text: Interactive Computer Graphics, by Edward Angel, Dave Shreiner, 6th edition, Addison-Wesley) lease track Blackboard for the most recent Announcements and roject postings related to this course. ( The University of Texas at Austin 44