Implicit Matrix Representations of Rational Bézier Curves and Surfaces

Transcription

1 Implicit Matrix Representations of Rational Bézier Curves and Surfaces Laurent Busé INRIA Sophia Antipolis, France GD/SPM Conference, Denver, USA November 11, 2013

2 Overall motivation Intersection problems between parameterized curves and surfaces Reliability and accuracy are basic important prerequisites. Must deal with numerical errors due to finite precision computations.

3 Overall motivation Intersection problems between parameterized curves and surfaces Reliability and accuracy are basic important prerequisites. Must deal with numerical errors due to finite precision computations. Focus on Ray/Nurbs intersection: ray tracing meshing (via Delaunay refinement)

4 Purpose of this talk Implicitization as F (x, y, z) = 0 is very classical but fails in general Polynomial elimination, such as Gröbner basis, is not adapted. Resultant matrices and Sederberg s moving planes and quadrics method are very sensitive to the presence of base points. Turn to a more general matrix-based implicit representation : rank M(x, y, z) drops Overcome the difficulty of base points Enhance and exploit links with Linear Algebra Numerical computations by means of Numerical Linear Algebra methods (SVD, eigen-computations).

5 Purpose of this talk Implicitization as F (x, y, z) = 0 is very classical but fails in general Polynomial elimination, such as Gröbner basis, is not adapted. Resultant matrices and Sederberg s moving planes and quadrics method are very sensitive to the presence of base points. Turn to a more general matrix-based implicit representation : rank M(x, y, z) drops Overcome the difficulty of base points Enhance and exploit links with Linear Algebra Numerical computations by means of Numerical Linear Algebra methods (SVD, eigen-computations).

6 Moving planes of a Bézier patch Suppose given a tensor-product Bézier patch of bi-degree (d 1, d 2 ): φ(u, v) = d1 d2 i=0 d1 d2 i=0 j=0 w i,jb i,j B d1 i j=0 w i,jb d1 i (u)b d2 j (v) (u)b d2 j (v) b i,j s and w i,j s are the control points and weights.

7 Moving planes of a Bézier patch Suppose given a tensor-product Bézier patch of bi-degree (d 1, d 2 ): φ(u, v) = d1 d2 i=0 d1 d2 i=0 j=0 w i,jb i,j B d1 i j=0 w i,jb d1 i (u)b d2 j (u)b d2 j (v) (v) = b i,j s and w i,j s are the control points and weights. the f i s are polynomials of bi-degree (d 1, d 2 ). ( f1 (u, v) f 0 (u, v), f 2(u, v) f 0 (u, v), f ) 3(u, v) f 0 (u, v)

8 Moving planes of a Bézier patch Suppose given a tensor-product Bézier patch of bi-degree (d 1, d 2 ): φ(u, v) = d1 d2 i=0 d1 d2 i=0 j=0 w i,jb i,j B d1 i j=0 w i,jb d1 i (u)b d2 j (u)b d2 j (v) (v) = b i,j s and w i,j s are the control points and weights. the f i s are polynomials of bi-degree (d 1, d 2 ). ( f1 (u, v) f 0 (u, v), f 2(u, v) f 0 (u, v), f ) 3(u, v) f 0 (u, v) Definition (after Sederberg) A moving plane of bi-degree (ν 1, ν 2 ) is a polynomial a 0 (u, v) + a 1 (u, v)x + a 2 (u, v)y + a 3 (u, v)z such that deg(a i ) (ν 1, ν 2 ) (comp.-wise) and 3 i=0 a i(u, v)f i (u, v) 0. Planes parameterized by (u, v) passing through the point φ(u, v).

9 Building of matrix of moving planes Definition From φ, define the matrix M φ as follows: Its columns basis of moving planes of bi-degree (d 1 1, 2d 2 1), Each column is filled with the coeff. of a moving plane w.r.t. u, v. The entries of M φ are linear forms in R[X, Y, Z] : M φ (X, Y, Z). Extension of Sederberg s approach by taking all moving planes M φ is non-square in general, but have more columns than rows

10 Building of matrix of moving planes Definition From φ, define the matrix M φ as follows: Its columns basis of moving planes of bi-degree (d 1 1, 2d 2 1), Each column is filled with the coeff. of a moving plane w.r.t. u, v. The entries of M φ are linear forms in R[X, Y, Z] : M φ (X, Y, Z). Extension of Sederberg s approach by taking all moving planes M φ is non-square in general, but have more columns than rows

11 Building of matrix of moving planes Definition From φ, define the matrix M φ as follows: Its columns basis of moving planes of bi-degree (d 1 1, 2d 2 1), Each column is filled with the coeff. of a moving plane w.r.t. u, v. The entries of M φ are linear forms in R[X, Y, Z] : M φ (X, Y, Z). Extension of Sederberg s approach by taking all moving planes M φ is non-square in general, but have more columns than rows Example: from a ruled surface, (d 1, d 2 ) = (1, 2), we get 1 + X 1 X + Z 0 Y 1 + X + Z 2X Y 0 M φ (X, Y, Z) = 1 + X 2 2X + Z 0 Y 1. Z 2X Y 1 0

12 The key fact The drop of rank property Given P R 3, the matrix M φ (P) is not full rank (= number of rows) if and only if P belongs to the (algebraic closure of the) Bézier patch. The matrix M φ yields an implicit representation of φ. Remark: The same holds for triangular Bézier patches : for a degree d patch take degree 2(d 1) moving planes.

13 The key fact The drop of rank property Given P R 3, the matrix M φ (P) is not full rank (= number of rows) if and only if P belongs to the (algebraic closure of the) Bézier patch. The matrix M φ yields an implicit representation of φ. Remark: The same holds for triangular Bézier patches : for a degree d patch take degree 2(d 1) moving planes.

14 The key fact The drop of rank property Given P R 3, the matrix M φ (P) is not full rank (= number of rows) if and only if P belongs to the (algebraic closure of the) Bézier patch. The matrix M φ yields an implicit representation of φ. Remark: The same holds for triangular Bézier patches : for a degree d patch take degree 2(d 1) moving planes. Example of the sphere: Z Y 1 X 0 M φ = X Y Z 1 + X 0 Z Y

15 Computational aspects The matrix M φ is very easy to compute From b i,j s and w i,j s build directly a matrix (Sylvester-like) It is of size about 8d 1 d 2 (resp. 8d 2 ). M φ is then obtained after a single nullspace computation It is adapted to numerical computations through the Singular Value Decomposition (SVD) M φ can be computed numerically by means of the SVD It turns a geometric problem into a rank revealing problem. No polynomial computations, only classical linear algebra tools It can be easily added to an existing library on NURBS.

16 Computational aspects The matrix M φ is very easy to compute From b i,j s and w i,j s build directly a matrix (Sylvester-like) It is of size about 8d 1 d 2 (resp. 8d 2 ). M φ is then obtained after a single nullspace computation It is adapted to numerical computations through the Singular Value Decomposition (SVD) M φ can be computed numerically by means of the SVD It turns a geometric problem into a rank revealing problem. No polynomial computations, only classical linear algebra tools It can be easily added to an existing library on NURBS.

17 Example: the sphere A (triangular Bézier) parameterization of the sphere: ( 1 u 2 v 2 ) φ(u, v) = 1 + u 2 + v 2, 2u 1 + u 2 + v 2, 2v 1 + u 2 + v 2 Exact computation of matrix representation Z Y 1 X 0 M φ = X Y Z 1 + X 0 Z Y Approximate computation of matrix representation by SVD X 0.354Y Z X Y Z X Y Z X Y Z X 0.354Y Z X Y Z X Y 0.000Z X 0.000Y Z X Y Z X 0.707Y Z X Y Z X Y Z In a computer, it is stored as 4 numerical matrices: M φ (X, Y, Z) = M 0 + M 1 X + M 2 Y + M 3 Z.

18 Example: the sphere A (triangular Bézier) parameterization of the sphere: ( 1 u 2 v 2 ) φ(u, v) = 1 + u 2 + v 2, 2u 1 + u 2 + v 2, 2v 1 + u 2 + v 2 Exact computation of matrix representation Z Y 1 X 0 M φ = X Y Z 1 + X 0 Z Y Approximate computation of matrix representation by SVD X 0.354Y Z X Y Z X Y Z X Y Z X 0.354Y Z X Y Z X Y 0.000Z X 0.000Y Z X Y Z X 0.707Y Z X Y Z X Y Z In a computer, it is stored as 4 numerical matrices: M φ (X, Y, Z) = M 0 + M 1 X + M 2 Y + M 3 Z.

19 Example: the sphere A (triangular Bézier) parameterization of the sphere: ( 1 u 2 v 2 ) φ(u, v) = 1 + u 2 + v 2, 2u 1 + u 2 + v 2, 2v 1 + u 2 + v 2 Exact computation of matrix representation Z Y 1 X 0 M φ = X Y Z 1 + X 0 Z Y Approximate computation of matrix representation by SVD X 0.354Y Z X Y Z X Y Z X Y Z X 0.354Y Z X Y Z X Y 0.000Z X 0.000Y Z X Y Z X 0.707Y Z X Y Z X Y Z In a computer, it is stored as 4 numerical matrices: M φ (X, Y, Z) = M 0 + M 1 X + M 2 Y + M 3 Z.

20 It also holds for curves! Suppose given a rational Bézier curve of degree d: φ(t) = d i=0 w ib i Bi d ( (t) d i=0 w ibi d (t) = f1 (t) f 0 (t), f 2(t) f 0 (t), f ) 3(t) f 0 (t) Consider moving planes of degree d 1 One can build M φ with the exact same definition M φ still have the same properties. Nice property: Curves and surfaces can be treated exactly in the same way!

21 It also holds for curves! Suppose given a rational Bézier curve of degree d: φ(t) = d i=0 w ib i Bi d ( (t) d i=0 w ibi d (t) = f1 (t) f 0 (t), f 2(t) f 0 (t), f ) 3(t) f 0 (t) Consider moving planes of degree d 1 One can build M φ with the exact same definition M φ still have the same properties. Nice property: Curves and surfaces can be treated exactly in the same way!

22 It also holds for curves! Suppose given a rational Bézier curve of degree d: φ(t) = d i=0 w ib i Bi d ( (t) d i=0 w ibi d (t) = f1 (t) f 0 (t), f 2(t) f 0 (t), f ) 3(t) f 0 (t) Consider moving planes of degree d 1 One can build M φ with the exact same definition M φ still have the same properties. Nice property: Curves and surfaces can be treated exactly in the same way!

23 It also holds for curves! Suppose given a rational Bézier curve of degree d: φ(t) = d i=0 w ib i Bi d ( (t) d i=0 w ibi d (t) = f1 (t) f 0 (t), f 2(t) f 0 (t), f ) 3(t) f 0 (t) Consider moving planes of degree d 1 One can build M φ with the exact same definition M φ still have the same properties. Nice property: Curves and surfaces can be treated exactly in the same way! Example of a matrix representation of a rational Bézier curve: 1 0 X + 3Y 0 Z X 3Y X + 3Y 2Z Z M φ = X X 3Y X 3Y 2Y 2Z 2Z 0 X X 3Y 0 2Y 2Z

24 Does a point belong to a Bézier patch? Suppose given: a rational Bézier curve or surface, a point P = (x, y, z) R 3. Evaluate the matrix M φ at P : M φ (P) := M 0 + M 1 x + M 2 y + M 3 z Compute the SVD of M φ (P) = U.S.V : if the ε-rank drops then P belongs to this rational Bézier curve or surface. Get the pre-image (inversion) for the same price: it can be extracted from the last row of U by ratio computations.

25 Does a point belong to a Bézier patch? Suppose given: a rational Bézier curve or surface, a point P = (x, y, z) R 3. Evaluate the matrix M φ at P : M φ (P) := M 0 + M 1 x + M 2 y + M 3 z Compute the SVD of M φ (P) = U.S.V : if the ε-rank drops then P belongs to this rational Bézier curve or surface. Get the pre-image (inversion) for the same price: it can be extracted from the last row of U by ratio computations.

26 Does a point belong to a Bézier patch? Suppose given: a rational Bézier curve or surface, a point P = (x, y, z) R 3. Evaluate the matrix M φ at P : M φ (P) := M 0 + M 1 x + M 2 y + M 3 z Compute the SVD of M φ (P) = U.S.V : if the ε-rank drops then P belongs to this rational Bézier curve or surface. Get the pre-image (inversion) for the same price: it can be extracted from the last row of U by ratio computations.

27 Does a point belong to a Bézier patch? Suppose given: a rational Bézier curve or surface, a point P = (x, y, z) R 3. Evaluate the matrix M φ at P : M φ (P) := M 0 + M 1 x + M 2 y + M 3 z Compute the SVD of M φ (P) = U.S.V : if the ε-rank drops then P belongs to this rational Bézier curve or surface. Get the pre-image (inversion) for the same price: it can be extracted from the last row of U by ratio computations.

28 Drop of rank and singularities of φ Suppose given a point P R 3 and a rational Bézier curve or surface φ: What is the rank of M φ (P)? M φ (P) is not full rank P belongs to Im(φ). The rank of M φ (P) drops by exactly one P is a regular point and one can get its pre-image easily (inversion). The rank M φ (P) drops by more than one P is a singularity of φ. More precisely: Theorem If φ 1 (P) is finite, then the drop of rank is equal to the number of pre-images (counting multiplicities), if φ 1 (P) is a curve, then the drop of rank is related to the degree and genus of this curve.

29 Drop of rank and singularities of φ Suppose given a point P R 3 and a rational Bézier curve or surface φ: What is the rank of M φ (P)? M φ (P) is not full rank P belongs to Im(φ). The rank of M φ (P) drops by exactly one P is a regular point and one can get its pre-image easily (inversion). The rank M φ (P) drops by more than one P is a singularity of φ. More precisely: Theorem If φ 1 (P) is finite, then the drop of rank is equal to the number of pre-images (counting multiplicities), if φ 1 (P) is a curve, then the drop of rank is related to the degree and genus of this curve.

30 Drop of rank and singularities of φ Suppose given a point P R 3 and a rational Bézier curve or surface φ: What is the rank of M φ (P)? M φ (P) is not full rank P belongs to Im(φ). The rank of M φ (P) drops by exactly one P is a regular point and one can get its pre-image easily (inversion). The rank M φ (P) drops by more than one P is a singularity of φ. More precisely: Theorem If φ 1 (P) is finite, then the drop of rank is equal to the number of pre-images (counting multiplicities), if φ 1 (P) is a curve, then the drop of rank is related to the degree and genus of this curve.

31 Drop of rank and singularities of φ Suppose given a point P R 3 and a rational Bézier curve or surface φ: What is the rank of M φ (P)? M φ (P) is not full rank P belongs to Im(φ). The rank of M φ (P) drops by exactly one P is a regular point and one can get its pre-image easily (inversion). The rank M φ (P) drops by more than one P is a singularity of φ. More precisely: Theorem If φ 1 (P) is finite, then the drop of rank is equal to the number of pre-images (counting multiplicities), if φ 1 (P) is a curve, then the drop of rank is related to the degree and genus of this curve.

32 Drop of rank and singularities of φ Suppose given a point P R 3 and a rational Bézier curve or surface φ: What is the rank of M φ (P)? M φ (P) is not full rank P belongs to Im(φ). The rank of M φ (P) drops by exactly one P is a regular point and one can get its pre-image easily (inversion). The rank M φ (P) drops by more than one P is a singularity of φ. More precisely: Theorem If φ 1 (P) is finite, then the drop of rank is equal to the number of pre-images (counting multiplicities), if φ 1 (P) is a curve, then the drop of rank is related to the degree and genus of this curve.

33 Curve/Curve and Curve/Surface intersection Suppose given: a rational Bézier curve ψ(t) (e.g. a ray), a rational Bézier curve or surface and its matrix representation M φ. Compute the intersection points Evaluate M φ at the point ψ(t): M φ (ψ(t)) is a polynomial matrix whose rank drops at the values of t corresponding to the intersection. If ψ(t) is a ray, then M φ (ψ(t)) = A tb. If ψ(t) is a curve, then linearization (compagnon matrices) yields a similar pencil A tb Generalized eigenvalue problem

34 Curve/Curve and Curve/Surface intersection Suppose given: a rational Bézier curve ψ(t) (e.g. a ray), a rational Bézier curve or surface and its matrix representation M φ. Compute the intersection points Evaluate M φ at the point ψ(t): M φ (ψ(t)) is a polynomial matrix whose rank drops at the values of t corresponding to the intersection. If ψ(t) is a ray, then M φ (ψ(t)) = A tb. If ψ(t) is a curve, then linearization (compagnon matrices) yields a similar pencil A tb Generalized eigenvalue problem

35 Curve/Curve and Curve/Surface intersection Suppose given: a rational Bézier curve ψ(t) (e.g. a ray), a rational Bézier curve or surface and its matrix representation M φ. Compute the intersection points Evaluate M φ at the point ψ(t): M φ (ψ(t)) is a polynomial matrix whose rank drops at the values of t corresponding to the intersection. If ψ(t) is a ray, then M φ (ψ(t)) = A tb. If ψ(t) is a curve, then linearization (compagnon matrices) yields a similar pencil A tb Generalized eigenvalue problem

36 Curve/Curve and Curve/Surface intersection Suppose given: a rational Bézier curve ψ(t) (e.g. a ray), a rational Bézier curve or surface and its matrix representation M φ. Compute the intersection points Evaluate M φ at the point ψ(t): M φ (ψ(t)) is a polynomial matrix whose rank drops at the values of t corresponding to the intersection. If ψ(t) is a ray, then M φ (ψ(t)) = A tb. If ψ(t) is a curve, then linearization (compagnon matrices) yields a similar pencil A tb Generalized eigenvalue problem

37 Ray tracing of Rational Bézier Surface Classical approach: implicitization of the ray and solving in the parameters of the surface, With matrix representation, implicitization of the surface and solving in the parameter of the ray. Inversion is very important here: normal computations (reflexion of the light) Bézier patch of a NURBS, trimmed patch.

38 Ray tracing of Rational Bézier Surface Classical approach: implicitization of the ray and solving in the parameters of the surface, With matrix representation, implicitization of the surface and solving in the parameter of the ray. Inversion is very important here: normal computations (reflexion of the light) Bézier patch of a NURBS, trimmed patch.

39 Ray tracing of Rational Bézier Surface Classical approach: implicitization of the ray and solving in the parameters of the surface, With matrix representation, implicitization of the surface and solving in the parameter of the ray. Inversion is very important here: normal computations (reflexion of the light) Bézier patch of a NURBS, trimmed patch.

40 Matrix representations yield distance-like functions Let M φ be a matrix representation of a Bézier curve or surface Definition P R 3 : δ Mφ (P) := i σ i (M φ (P)) where σ i (M φ (P)) are the singular values of M φ (P). δ Mφ (P) vanishes exactly on the Bézier curve or surface The growth of δ Mφ (P) is comparable to the usual Euclidean distance. This squared distance is algebraic: δ Mφ (P) 2 = i σ i (M ν (P)) 2 = det(m φ (P)M φ (P) T ). det(m φ M t φ ) is an implicit equation of this Bézier curve or surface over the real numbers.

41 Matrix representations yield distance-like functions Let M φ be a matrix representation of a Bézier curve or surface Definition P R 3 : δ Mφ (P) := i σ i (M φ (P)) where σ i (M φ (P)) are the singular values of M φ (P). δ Mφ (P) vanishes exactly on the Bézier curve or surface The growth of δ Mφ (P) is comparable to the usual Euclidean distance. This squared distance is algebraic: δ Mφ (P) 2 = i σ i (M ν (P)) 2 = det(m φ (P)M φ (P) T ). det(m φ M t φ ) is an implicit equation of this Bézier curve or surface over the real numbers.

42 Matrix representations yield distance-like functions Let M φ be a matrix representation of a Bézier curve or surface Definition P R 3 : δ Mφ (P) := i σ i (M φ (P)) where σ i (M φ (P)) are the singular values of M φ (P). δ Mφ (P) vanishes exactly on the Bézier curve or surface The growth of δ Mφ (P) is comparable to the usual Euclidean distance. This squared distance is algebraic: δ Mφ (P) 2 = i σ i (M ν (P)) 2 = det(m φ (P)M φ (P) T ). det(m φ M t φ ) is an implicit equation of this Bézier curve or surface over the real numbers.