Parametric Surfaces. Michael Kazhdan ( /657) HB , FvDFH 11.2

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1 Parametric Surfaces Michael Kazhdan ( /657) HB , 10.1 FvDFH 11.2

2 Announcement OpenGL review session: When: Wednesday 7:00-9:00 PM Where: Olin 05

3 Cubic Splines Given a collection of control points, {p 0,, p n }, we define a collection of cubic polynomial functions {P 1 u,, P n 2 u } that jointly describe a curve approximating/interpolating the control points. P 1 (u) p 7 p 1 p 2 P 2 (u) p 6 P 5 (u) p 5 p 0 p p 4 P (u) P 4 (u) Each cubic function P k (u) is defined in the region 0 u 1 and is determined by the four points p k 1, p k, p k+1, and p k+2.

4 Cubic Blending Functions Blending functions provide a way for expressing the functions P k (u) as a weighted sum of the four control points p k 1, p k, p k+1, and p k+2 : 1 BF (u) BF 1 (u) BF 2 (u) BF (u) Cardinal Blending Functions (s = 1/2) 1 1 BF (u) BF 1 (u) BF 2 (u) BF (u) Uniform Cubic B-Spline Blending Functions (s = 1/2) 1 P k u = BF 0 u p k 1 + BF 1 u p k + BF 2 u p k+1 + BF u p k+2

5 Blending Functions Properties: Translation Commutativity: If we translate all the control points by the same vector q, the position of the new point at the value u will be the position of the old value at u, translated by q. Given control points {p k 1, p k, p k+1, p k+2 } and translation vector q. Let P k (u) be the curve defined by {p k 1, p k, p k+1, p k+2 }. Let Q k (u) be the curve defined by {q + p k 1, q + p k, q + p k+1, q + p k+2 }. We want: Q k u = q + P k (u) Writing out Q k u, we have: Q k u = BF 0 u q + p k 1 + BF 1 (u)(q + p k )+ BF 2 (u)(q + p k+1 )+ BF u q + p k+2 = BF 0 u + BF 1 u + BF 2 u + BF u q + P k u To satisfy translation commutativity, we must have: BF 0 u + BF 1 u + BF 2 u + BF u = 1 P k u = BF 0 u p k 1 + BF 1 u p k + BF 2 u p k+1 + BF u p k+2

6 Blending Functions Properties: Translation Commutativity: BF 0 u + BF 1 u + BF 2 u + BF u = 1 for all 0 u 1. Continuity: We need the curve P k+1 u to begin where P k (u) ended. Taking the difference, we get: 0 = P k+1 0 P k (1) Expanding this out, we get: 0 = BF 0 1 p k 1 + BF 0 0 BF 1 1 p k + BF 1 0 BF 2 1 p k+1 + BF 2 0 BF 1 p k+2 + BF 0 p k+ Since this is true for all control points p k 1, p k, p k+1, p k+2, p k+, we get: 0 = BF 0 1 BF 0 0 = BF 1 1 BF 1 0 = BF 2 1 BF 2 0 = BF 1 BF 0 = 0 P k u = BF 0 u p k 1 + BF 1 u p k + BF 2 u p k+1 + BF u p k+2

7 Blending Functions Properties: Translation Commutativity: BF 0 u + BF 1 u + BF 2 u + BF u = 1 for all 0 u 1. More Generally, for the spline to have continuous n-th order derivatives, the blending functions need to satisfy: Continuity: 0 = BF 0 (n) 1 We need the curve P k+1 u to begin where P k (u) ended. Taking the difference, we (n) get: (n) BF 0 = P k = BF1 P k (1) 1 Expanding this out, we get: (n) (n) BF 0 = 1 0 BF = BF2 0 1 p 1 k 1 + (n) BF 0 BF 1 1 (n) BF p k = BF 1 BF 1 BF 2 1 p k+1 + (n) BF BF 2 BF 1 p k = 0 BF 0 p k+ Since this is true for all control points p k 1, p k, p k+1, p k+2, p k+, we get: 0 = BF 0 1 BF 0 0 = BF 1 1 BF 1 0 = BF 2 1 BF 2 0 = BF 1 BF 0 = 0 P k u = BF 0 u p k 1 + BF 1 u p k + BF 2 u p k+1 + BF u p k+2

8 Blending Functions Properties: Translation Commutativity: BF 0 u + BF 1 u + BF 2 u + BF u = 1 for all 0 u 1. Continuity: 0 = BF 0 1, BF 0 0 = BF 1 1, BF 1 0 = BF 2 1, BF 2 0 = BF 1, BF 0 = 0 Convex Hull Containment: A point is inside the convex hull of a collection of points if and only if it can be expressed as the weighted average of the points, where all the weights are non-negative. BF 0 u, BF 1 u, BF 2 u, BF u 0, for all 0 u 1. P k u = BF 0 u p k 1 + BF 1 u p k + BF 2 u p k+1 + BF u p k+2

9 Blending Functions Properties: Translation Commutativity: BF 0 u + BF 1 u + BF 2 u + BF u = 1 for all 0 u 1. Continuity: 0 = BF 0 1, BF 0 0 = BF 1 1, BF 1 0 = BF 2 1, BF 2 0 = BF 1, BF 0 = 0 Convex Hull Containment: BF 0 u, BF 1 u, BF 2 u, BF u 0, for all 0 u 1. Interpolation: We want the spline segments to satisfy: P k 0 = p k and P k 1 = p k+1 At the end-points, the blending functions satisfy: BF 0 (0) BF 1 (0) BF 2 (0) BF (0) = and BF 0 (1) BF 1 (1) BF 2 (1) BF (1) = P k u = BF 0 u p k 1 + BF 1 u p k + BF 2 u p k+1 + BF u p k+2

10 Overview From Curves to surfaces Spline Curves and Blending Functions Weighted Averaging Spline Surfaces Spline Surface Properties

11 Weighted Averaging Suppose we have an array of values: v 1, v 2, v, and v 4, and we have weights: α 1, α 2, α, and α 4, with α 1 + α 2 + α + α 4 = 1, β 1, β 2, β, and β 4, with β 1 + β 2 + β + β 4 = 1. We can express the weighted average of the v i in matrix form: 4 i=1 α i v i = α 1 α 2 α α 4 v 2 v v 1 v 4 4 i=1 β i v i = v 1 v 2 v v 4 β 2 β β 1 β 4

12 Weighted Averaging If we have a matrix of values: v 11 v 21 v 1 v 41 v 12 v 22 v 2 v 42 v 1 v 2 v v 4 v 14 v 24 v 4 v 44 multiplying on the left by (α 1 α 2 α α 4 ) gives: t α i v 1i = α 1 α 2 α α 4 v 12 v 1 v 22 v 2 v 2 v v 42 v 4 v 11 v 21 v 1 v 41 v 14 v 24 v 4 v 44 α i v 2i α i v i α i v 4i

13 Weighted Averaging If we have a matrix of values: v 11 v 21 v 1 v 41 v 12 v 22 v 2 v 42 v 1 v 2 v v 4 v 14 v 24 v 4 v 44 multiplying on the left by (α 1 α 2 α α 4 ) gives: t α i v 1i = α 1 α 2 α α 4 v 12 v 1 v 22 v 2 v 2 v v 42 v 4 v 11 v 21 v 1 v 41 v 14 v 24 v 4 v 44 A row vector with entries that are the weighted average of the columns of the matrix. α i v 2i α i v i α i v 4i

14 Weighted Averaging Similarly, if we have a matrix of values: v 11 v 21 v 1 v 41 v 12 v 22 v 2 v 42 v 1 v 2 v v 4 v 14 v 24 v 4 v 44 multiplying on the right by β 1 β 2 β β 4 t gives: β j v j1 v 11 v 21 v 1 v 41 v 12 v 22 v 2 v 42 v 1 v 2 v v 4 v 14 v 24 v 4 v 44 A column vector with entries that are the weighted average of the rows of the matrix. β 1 β 2 β β 4 = β j v j2 β j v j β j v j4

15 Weighted Averaging Simultaneously multiplying on the left by (α 1 α 2 α α 4 ) and on the right by β 1 β 2 β β t 4 gives: α 1 α 2 α α 4 v 12 v 1 v 22 v 2 v 2 v v 42 v 4 v 11 v 21 v 1 v 41 v 14 v 24 v 4 v 44 β 1 β 2 β β 4 4 = i,j=1 α i β j v ij

16 Weighted Averaging Simultaneously multiplying on the left by (α 1 α 2 α α 4 ) and on the right by β 1 β 2 β β t 4 gives: α 1 α 2 α α 4 v 12 v 1 v 22 v 2 v 2 v v 42 v 4 v 11 v 21 v 1 v 41 v 14 v 24 v 4 v 44 β 1 β 2 β β 4 = The weighted sum of the v ij, weighted by α j β i. This is a weighted average of the v ij : 4 i,j=1 To show this, we have to show that the total sum of the weights α i β j is equal to 1. α i β j v ij

17 Weighted Averaging Simultaneously multiplying on the left by (α 1 α 2 α α 4 ) and on the right by β 1 β 2 β β t 4 gives: α 1 α 2 α α 4 v 12 v 1 v 22 v 2 v 2 v v 42 v 4 v 11 v 21 v 1 v 41 v 14 v 24 v 4 v 44 β 1 β 2 β β 4 4 = The weighted sum of the v ij, weighted by α j β i. This is a weighted average of the v ij : 4 i,j=1 4 α i β j = i=1 4 α i 4 j=1 = α i = 1 i=1 β j i,j=1 α i β j v ij

18 Overview From Curves to surfaces Spline Curves and Blending Functions Weighted Averaging Spline Surfaces Spline Surface Properties

19 Spline Surfaces A parametric curve is a function in one variable Φ(u) associating a position to every value of u. (1/2) (5/8) (/8) (/4) (1/4) (7/8) (1/8) (0) (1)

20 Spline Surfaces A parametric curve is a function in one variable Φ(u) associating a position to every value of u. A parametric patch is a function in two variables Φ(u, v) that associates a position to every pair of values of (u, v). (0,1) (1/,1) (0,2/) (2/,1) (1/,2/) (2/,2/) (1,1) (1,2/) (0,1/) (0,0) (1/,1/) (2/,1/) (1/,0) (2/,0) (1,0) (1,1/)

21 Spline Surfaces When considering spline curves, we use four control points to define a cubic polynomial P k (u) in one variable (0 u 1). P k (u) p k p k+1 p k-1 p k+2

22 Spline Surfaces When considering spline curves, we use four control points to define a cubic polynomial P k (u) in one variable (0 u 1). When considering spline surfaces, we use 4 4 control points to define a bi-cubic polynomial P k,l (u, v) in two variables (0 u, v 1). p k-1,l+2 p k,l+2 p k+1,l+2 p k+2,l+2 p k,l+1 p p k+2,l+1 k-1,l+1 P k,l (u,v) p k+1,l+1 p k,l p k-1,l p k+1,l p k+2,l p k-1,l-1 p k,l-1 p k+1,l-1 p k+2,l-1

23 Spline Surfaces When considering spline curves, we use four control points to define a cubic polynomial P k (u) in one variable (0 u 1). When considering spline surfaces, we use 4 4 control points to define a bi-cubic polynomial P k,l (u, v) in two variables (0 u, v 1). A bi-cubic polynomial is a polynomial which is cubic in each variable: P u, v = au v + p k,l+1 p p k+2,l+1 k-1,l+1 P k,l (u,v) p k+1,l+1 p k-1,l p k-1,l+2 p k,l+2 p k+1,l+2 p k+2,l+2 p k,l +bu v 2 + cu 2 v + p k+2,l +du 2 v 2 + eup 1 k+1,l v + fu v p k-1,l-1 p k,l-1 p k+1,l-1 p k+2,l-1

24 Spline Surfaces Given n points, we generate a piecewise cubic curve consisting of n segments that approximate/interpolate the points. p 1 p 2 P 0 (u) p 1 p 2 p 1 p 2 P 1 (u) p p 4 p p 4 p p 4 p 0 p 0 p 0 p 1 p 2 p p 4 p 0

25 Spline Surfaces Given n points, we generate a piecewise cubic curve consisting of n segments that approximate/interpolate the points. Given n m points, we generate a piecewise bicubic curve, consisting of n (m ) patches that approximate/interpolate the points. P 0,0 (u,v) P 1,0 (u,v) P 0,1 (u,v) P 1,1 (u,v)

26 Spline Surfaces We generate spline curves by using the blending function to compute the weighted average of the control points. We do the same for surfaces.

27 Cubic Blending Functions Recall: For a cubic segment of a spline curve, we can express the spline curve in matrix form as: t BF 0 (u) p k 1 BF P k u = 1 (u) p k BF 2 (u) p k+1 BF (u) p k+2 Since the sum of the BF i (u) is always equal to 1, this is a weighted average of the control points. P k u = BF 0 u p k 1 + BF 1 u p k + BF 2 u p k+1 + BF u p k+2

28 Cubic Blending Functions If we are given a 4 4 array of control points, we can define a bi-cubic spline patch similarly: P k,l u, v = BF 0 (u) BF 1 (u) BF 2 (u) BF (u) t p k 1,l 1 p k,l 1 p k+1,l 1 p k+2,l 1 p k 1,l p k,l p k+1,l p k+2,l p k 1,l+1 p k,l+1 p k+1,l+1 p k+2,l+1 p k 1,l+2 p k,l+2 p k+1,l+2 p k+2,l+2 BF 0 (v) BF 1 (v) BF 2 (v) BF (v) Again, the sum of the BF i (u) equals 1, so P k,l (u, v) is a weighted average of the control points.

29 Cubic Spline Patches We can choose our favorite spline curve (Cardinal, uniform cubic-b, etc.) and use its blending functions to define a spline patch: P k,l u, v = BF 0 (u) BF 1 (u) BF 2 (u) BF (u) t p k 1,l 1 p k,l 1 p k+1,l 1 p k+2,l 1 p k 1,l p k,l p k+1,l p k+2,l p k 1,l+1 p k,l+1 p k+1,l+1 p k+2,l+1 p k 1,l+2 p k,l+2 p k+1,l+2 p k+2,l+2 BF 0 (v) BF 1 (v) BF 2 (v) BF (v)

30 Cubic Spline Patches Computing the value of the patch at some point (u 0, v 0 ) amounts to: 1. Averaging the rows using the weights BF i v 0 2. Averaging the result using the weights BF i (u 0 ). P k,l u, v = BF 0 (u) BF 1 (u) BF 2 (u) BF (u) t p k 1,l 1 p k,l 1 p k+1,l 1 p k+2,l 1 p k 1,l p k,l p k+1,l p k+2,l p k 1,l+1 p k,l+1 p k+1,l+1 p k+2,l+1 p k 1,l+2 p k,l+2 p k+1,l+2 p k+2,l+2 BF 0 (v) BF 1 (v) BF 2 (v) BF (v)

31 Cubic Spline Patches Computing the value of the patch at some point (u 0, v 0 ) amounts to: 1. Averaging the rows using the weights BF i v 0 2. Averaging the result using the weights BF i (u 0 ). (u 0, v 0 ) (u 0, v 0 )

32 Cubic Spline Patches P k,l u, v = BF 0 (u) BF 1 (u) BF 2 (u) BF (u) t p k 1,l 1 p k,l 1 p k+1,l 1 p k+2,l 1 p k 1,l p k,l p k+1,l p k+2,l p k 1,l+1 p k,l+1 p k+1,l+1 p k+2,l+1 p k 1,l+2 p k,l+2 p k+1,l+2 p k+2,l+2 BF 0 (v) BF 1 (v) BF 2 (v) BF (v) Multiplying out the matrices we get: P k,l u, v = BF 0 u BF 0 v p k 1,l 1 + BF 0 u BF 1 v p k 1,l + + BF 1 u BF 0 v p k,l 1 + BF 1 u BF 1 v p k,l + + Or, if we set BF ij u, v = BF i u BF j (v) we get: P k,l u, v = BF 00 u, v p k 1,l 1 + BF 01 u, v p k 1,l + + BF 10 u, v p k,l 1 + BF 11 u, v p k,l + +

33 Cubic Spline Patches P k,l u, v = BF 0 (u) BF 1 (u) BF 2 (u) BF (u) t p k 1,l 1 p k,l 1 p k+1,l 1 p k+2,l 1 p k 1,l p k,l p k+1,l p k+2,l p k 1,l+1 p k,l+1 p k+1,l+1 p k+2,l+1 p k 1,l+2 p k,l+2 p k+1,l+2 p k+2,l+2 BF 0 (v) BF 1 (v) BF 2 (v) BF (v) Recall that we can write out blending functions as: BF 0 (u) BF 1 (u) BF 2 (u) BF (u) = UM Spline with U = (u u 2 u 1) and M Spline the spline matrix. This gives: p k 1,l 1 p k,l 1 p k+1,l 1 p k+2,l 1 p k 1,l p k,l p k+1,l p k+2,l t P k,l u, v = UM Spline p k 1,l+1 p k,l+1 p k+1,l+1 p M k+2,l+1 Spline V t p k 1,l+2 p k,l+2 p k+1,l+2 p k+2,l+2 with V = (v v 2 v 1).

34 Cubic Spline Patches P k,l u, v = BF 0 (u) BF 1 (u) BF 2 (u) BF (u) t p k 1,l 1 p k,l 1 p k+1,l 1 p k+2,l 1 p k 1,l p k,l p k+1,l p k+2,l p k 1,l+1 p k,l+1 p k+1,l+1 p k+2,l+1 p k 1,l+2 p k,l+2 p k+1,l+2 p k+2,l+2 BF 0 (v) BF 1 (v) BF 2 (v) BF (v) Recall that we can write out blending functions as: BF 0 (u) BF 1 (u) BF 2 (u) BF (u) = UM Spline Surface splines that are obtained from curve splines in this way are often referred to as tensor product splines. with U = (u u 2 u 1) and M Spline the spline matrix. This gives: p k 1,l 1 p k,l 1 p k+1,l 1 p k+2,l 1 p k 1,l p k,l p k+1,l p k+2,l t P k,l u, v = UM Spline p k 1,l+1 p k,l+1 p k+1,l+1 p M k+2,l+1 Spline V t p k 1,l+2 p k,l+2 p k+1,l+2 p k+2,l+2 with V = (v v 2 v 1).

35 Overview From Curves to surfaces Spline Curves and Blending Functions Weighted Averaging Spline Surfaces Spline Surface Properties

36 Blending Functions For spline curves, we want: Translation Commutativity: BF 0 u + BF 1 u + BF 2 u + BF u = 1 for all 0 u 1. n-th Order Continuity: 0 = BF 0 n 1 BF 0 n 0 = BF 1 n 1 BF 1 n 0 = BF 2 n 1 BF 2 n 0 = BF n 1 BF n 0 = 0 Convex Hull Containment: BF 0 u, BF 1 u, BF 2 u, BF u 0, for all 0 u 1. Interpolation: BF 0 (0) BF 1 (0) = BF 2 (0) BF (0) and BF 0 (1) BF 1 (1) = BF 2 (1) BF (1) Do the tensor product splines satisfy these conditions?

37 Surface Spline Properties Translation commutativity: As in the curve case, we need the sum of the blending functions BF ij (u, v) to be equal to one. But we have already shown that since BF ij u, v = B i u B j v if the BF i (u) are weighting function that sum to 1, then the tensor product functions BF ij (u, v) also sum to 1.

38 Surface Spline Properties Continuity: Consider the continuity along the yellow edge: 0 = P k,l 1, v P k+1,l (0, v) (0,) (1,) (2,) (,) (0,2) (1,2) (2,2) (,2) (4,) (4,2) P 1,1 (u, v) P 2,1 (u, v) (0,1) (1,1) (2,1) (,1) (0,0) (1,0) (2,0) (,0) (4,1) (4,0)

39 Surface Spline Properties Continuity: Consider the continuity along the yellow edge: 0 = P k,l 1, v P k+1,l 0, v 0 = B ij 1, v p i,j B ij 0, v p i+1,j i=0 j=0 i=0 j=0 0 = B 0j 1, v p 0,j + B ij 1, v p i,j B ij 0, v p i+1,j B j 0, v p 4,j j=0 i=1 j=0 2 i=0 j=0 0 = B 0j 1, v p 0,j + B ij 1, v p i,j B i 1j 0, v p i,j B j 0, v p 4,j j=0 i=1 j=0 i=1 j=0 0 = B 0j 1, v p 0,j B j 0, v p 4,j + j=0 j=0 i=1 j=0 j=0 j=0 B ij 1, v B i 1j 0, v p i,j

40 Surface Spline Properties Continuity: Consider the continuity along the yellow edge: 0 = B 0j 1, v p 0,j B j 0, v p 4,j + j=0 j=0 i=1 j=0 B ij 1, v B i 1j 0, v For this to be true for all control points p ij, we require: B 0j 1, v = B j 0, v = 0 B 1j 1, v = B 0j (0, v) B 2j 1, v = B 1j (0, v) B j 1, v = B 2j (0, v) p i,j (0,) (1,) (2,) (,) (0,2) (1,2) (2,2) (,2) (4,) (4,2) P 1,1 (u, v) P 2,1 (u, v) (0,1) (1,1) (2,1) (,1) (0,0) (1,0) (2,0) (,0) (4,1) (4,0)

41 Surface Spline Properties Continuity: Consider the continuity along the yellow edge: 0 = B 0j 1, v p 0,j B j 0, v p 4,j + j=0 j=0 i=1 j=0 B ij 1, v B i+1j 0, v For this to be true for all control points p ij, we require: B 0j 1, v = B j 0, v = 0 B 1j 1, v = B 0j (0, v) B 2j 1, v = B 1j (0, v) B j 1, v = B 2j (0, v) Similarly, for continuity along horizontal edges, we need: B i0 u, 1 = B i u, 0 = 0 B i1 u, 1 = B i0 (u, 0) B i2 u, 1 = B i1 (u, 0) B i u, 1 = B i2 (u, 0) p i,j

42 Surface Spline Properties Continuity: Consider the continuity along the yellow edge: 0 = B 0j 1, v p 0,j B j 0, v p 4,j + j=0 j=0 i=1 j=0 B ij 1, v B i+1j 0, v For this to be true for all control points p ij, we require: B 0j 1, v = B j 0, v = 0 B 1j 1, v = B 0j (0, v) B 2j 1, v = B 1j (0, v) B B j 1, v = B 2j (0, v) 0j 1, v = B j 0, v = 0 Similarly, for continuity along horizontal edges, we need: B 0 1 B j v = B 0 B j v = 0 B i0 u, 1 = B i u, 0 = 0 B i1 u, 1 = B i0 (u, 0) B B i2 u, 1 = B i1 (u, 0) 0 1 = B 0 = 0 Which is satisfied if the 1D B-spline is continuous! B i u, 1 = B i2 (u, 0) p i,j

43 Surface Spline Properties Continuity: Consider the continuity along the yellow edge: 0 = B 0j 1, v p 0,j B j 0, v p 4,j + j=0 j=0 i=1 j=0 B ij 1, v B i+1j 0, v Similarly, For this to the be other true continuity for all control conditions points pfor ij, 2D we B-spline require: are B 0j satisfied 1, v = Bif j they 0, v are = 0satisfied by the 1D B-spline! B 1j 1, More v = Bgenerally, 0j (0, v) if the 1D B-spline gives B 2j 1, continuous v = B 1j (0, v) n-th order derivatives, then B j 1, v = Bso 2j (0, will v) the tensor-product. Similarly, for continuity along horizontal edges, we need: B i0 u, 1 = B i u, 0 = 0 B i1 u, 1 = B i0 (u, 0) B i2 u, 1 = B i1 (u, 0) B i u, 1 = B i2 (u, 0) p i,j

44 Surface Spline Properties Convex hull containment: For convexity we need the weights of the blending function to be non-negative. If the BF i (u) are non-negative, then since BF ij u, v = BF i u BF j (v) the BF ij (u, v) will also be non-negative.

45 Surface Spline Properties Interpolation: For the spline surface to interpolate, it must satisfy:» BF 11 0,0 = BF 12 0,1 = BF 21 1,0 = BF 22 1,1 = 1.» All the other blending functions are 0 at the endpoints. The spline curve is interpolating if:» BF 0 0 = BF 2 0 = BF 0 = 0» BF 0 1 = BF 1 1 = BF 1 = 0» BF 1 0 = 1» BF 2 1 = 1 (0,) (1,) (2,) (,) (0,2) (1,2) (2,2) (,2) P 1,1 (u, v) BF 12 0,1 = BF 1 0 BF 2 (1) (0,1) (1,1) (2,1) (,1) (0,0) (1,0) (2,0) (,0)

46 Surface Spline Properties We began by describing some of the properties that we would like spline curves to satisfy: Translation commutativity Continuity Convex hull containment Interpolation If the curve spline satisfies these properties, then so will the tensor product spline! Spline Demo