Implicit Matrix Representations of Rational Bézier Curves and Surfaces
|
|
- Sydney Pearson
- 5 years ago
- Views:
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.
A Line/Trimmed NURBS Surface Intersection Algorithm Using Matrix Representations
A Line/Trimmed NURBS Surface Intersection Algorithm Using Matrix Representations Jingjing Shen, Laurent Busé, Pierre Alliez, Neil Dodgson To cite this version: Jingjing Shen, Laurent Busé, Pierre Alliez,
More informationCurve and Surface Basics
Curve and Surface Basics Implicit and parametric forms Power basis form Bezier curves Rational Bezier Curves Tensor Product Surfaces ME525x NURBS Curve and Surface Modeling Page 1 Implicit and Parametric
More informationGEOMETRIC TOOLS FOR COMPUTER GRAPHICS
GEOMETRIC TOOLS FOR COMPUTER GRAPHICS PHILIP J. SCHNEIDER DAVID H. EBERLY MORGAN KAUFMANN PUBLISHERS A N I M P R I N T O F E L S E V I E R S C I E N C E A M S T E R D A M B O S T O N L O N D O N N E W
More informationMA 323 Geometric Modelling Course Notes: Day 28 Data Fitting to Surfaces
MA 323 Geometric Modelling Course Notes: Day 28 Data Fitting to Surfaces David L. Finn Today, we want to exam interpolation and data fitting problems for surface patches. Our general method is the same,
More informationMultiple View Geometry in Computer Vision
Multiple View Geometry in Computer Vision Prasanna Sahoo Department of Mathematics University of Louisville 1 Projective 3D Geometry (Back to Chapter 2) Lecture 6 2 Singular Value Decomposition Given a
More informationABSTRACT TO BE PRESENTED COMPUTATIONAL METHODS FOR ALGEBRAIC SPLINE SURFACES COMPASS II. September 14-16th, 2005
ABSTRACT TO BE PRESENTED AT COMPUTATIONAL METHODS FOR ALGEBRAIC SPLINE SURFACES COMPASS II September 14-16th, 2005 CENTRE OF MATHEMATICS FOR APPLICATIONS GAIA II PROJECT IST--2002 35512 UNIVERSITY OF OSLO,
More informationMutation-linear algebra and universal geometric cluster algebras
Mutation-linear algebra and universal geometric cluster algebras Nathan Reading NC State University Mutation-linear ( µ-linear ) algebra Universal geometric cluster algebras The mutation fan Universal
More informationGEOMETRIC LIBRARY. Maharavo Randrianarivony
GEOMETRIC LIBRARY Maharavo Randrianarivony During the last four years, I have maintained a numerical geometric library. The constituting routines, which are summarized in the following list, are implemented
More informationClosest Points, Moving Surfaces, and Algebraic Geometry
Closest Points, Moving Surfaces, and Algebraic Geometry Jan B. Thomassen, Pål H. Johansen, and Tor Dokken Abstract. This paper presents a method for computing closest points to a given parametric surface
More informationCurves and Surface I. Angel Ch.10
Curves and Surface I Angel Ch.10 Representation of Curves and Surfaces Piece-wise linear representation is inefficient - line segments to approximate curve - polygon mesh to approximate surfaces - can
More informationCurves and Surfaces 1
Curves and Surfaces 1 Representation of Curves & Surfaces Polygon Meshes Parametric Cubic Curves Parametric Bi-Cubic Surfaces Quadric Surfaces Specialized Modeling Techniques 2 The Teapot 3 Representing
More informationMultiple View Geometry in Computer Vision
Multiple View Geometry in Computer Vision Prasanna Sahoo Department of Mathematics University of Louisville 1 Structure Computation Lecture 18 March 22, 2005 2 3D Reconstruction The goal of 3D reconstruction
More informationAn application of fast factorization algorithms in Computer Aided Geometric Design
Linear Algebra and its Applications 366 (2003) 121 138 www.elsevier.com/locate/laa An application of fast factorization algorithms in Computer Aided Geometric Design G. Casciola a,,f.fabbri b, L.B. Montefusco
More informationParameterization of triangular meshes
Parameterization of triangular meshes Michael S. Floater November 10, 2009 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to
More informationStructure from Motion
11/18/11 Structure from Motion Computer Vision CS 143, Brown James Hays Many slides adapted from Derek Hoiem, Lana Lazebnik, Silvio Saverese, Steve Seitz, and Martial Hebert This class: structure from
More informationCurves and Surfaces. Computer Graphics COMP 770 (236) Spring Instructor: Brandon Lloyd
Curves and Surfaces Computer Graphics COMP 770 (236) Spring 2007 Instructor: Brandon Lloyd 4/11/2007 Final projects Surface representations Smooth curves Subdivision Todays Topics 2 Final Project Requirements
More informationReading. Parametric surfaces. Surfaces of revolution. Mathematical surface representations. Required:
Reading Required: Angel readings for Parametric Curves lecture, with emphasis on 11.1.2, 11.1.3, 11.1.5, 11.6.2, 11.7.3, 11.9.4. Parametric surfaces Optional Bartels, Beatty, and Barsky. An Introduction
More informationChapter 3. Quadric hypersurfaces. 3.1 Quadric hypersurfaces Denition.
Chapter 3 Quadric hypersurfaces 3.1 Quadric hypersurfaces. 3.1.1 Denition. Denition 1. In an n-dimensional ane space A; given an ane frame fo;! e i g: A quadric hypersurface in A is a set S consisting
More informationLecture 3: Camera Calibration, DLT, SVD
Computer Vision Lecture 3 23--28 Lecture 3: Camera Calibration, DL, SVD he Inner Parameters In this section we will introduce the inner parameters of the cameras Recall from the camera equations λx = P
More informationCOMPUTER AIDED GEOMETRIC DESIGN. Thomas W. Sederberg
COMPUTER AIDED GEOMETRIC DESIGN Thomas W. Sederberg January 31, 2011 ii T. W. Sederberg iii Preface This semester is the 24 th time I have taught a course at Brigham Young University titled, Computer Aided
More information3D reconstruction class 11
3D reconstruction class 11 Multiple View Geometry Comp 290-089 Marc Pollefeys Multiple View Geometry course schedule (subject to change) Jan. 7, 9 Intro & motivation Projective 2D Geometry Jan. 14, 16
More informationInterlude: Solving systems of Equations
Interlude: Solving systems of Equations Solving Ax = b What happens to x under Ax? The singular value decomposition Rotation matrices Singular matrices Condition number Null space Solving Ax = 0 under
More informationCS6015 / LARP ACK : Linear Algebra and Its Applications - Gilbert Strang
Solving and CS6015 / LARP 2018 ACK : Linear Algebra and Its Applications - Gilbert Strang Introduction Chapter 1 concentrated on square invertible matrices. There was one solution to Ax = b and it was
More informationarxiv: v1 [math.na] 5 Jul 2017
APPROXIMATE IMPLICITIZATION OF TRIANGULAR BÉZIER SURFACES OLIVER J. D. BARROWCLOUGH AND TOR DOKKEN arxiv:707.0255v [math.na] 5 Jul 207 Abstract. We discuss how Dokken s methods of approximate implicitization
More informationcalibrated coordinates Linear transformation pixel coordinates
1 calibrated coordinates Linear transformation pixel coordinates 2 Calibration with a rig Uncalibrated epipolar geometry Ambiguities in image formation Stratified reconstruction Autocalibration with partial
More informationUntil now we have worked with flat entities such as lines and flat polygons. Fit well with graphics hardware Mathematically simple
Curves and surfaces Escaping Flatland Until now we have worked with flat entities such as lines and flat polygons Fit well with graphics hardware Mathematically simple But the world is not composed of
More informationSubdivision Surfaces
Subdivision Surfaces 1 Geometric Modeling Sometimes need more than polygon meshes Smooth surfaces Traditional geometric modeling used NURBS Non uniform rational B-Spline Demo 2 Problems with NURBS A single
More informationStructure from motion
Structure from motion Structure from motion Given a set of corresponding points in two or more images, compute the camera parameters and the 3D point coordinates?? R 1,t 1 R 2,t 2 R 3,t 3 Camera 1 Camera
More informationCS354 Computer Graphics Surface Representation IV. Qixing Huang March 7th 2018
CS354 Computer Graphics Surface Representation IV Qixing Huang March 7th 2018 Today s Topic Subdivision surfaces Implicit surface representation Subdivision Surfaces Building complex models We can extend
More informationLECTURE 13, THURSDAY APRIL 1, 2004
LECTURE 13, THURSDAY APRIL 1, 2004 FRANZ LEMMERMEYER 1. Parametrizing Curves of Genus 0 As a special case of the theorem that curves of genus 0, in particular those with the maximal number of double points,
More informationFitting Uncertain Data with NURBS
Fitting Uncertain Data with NURBS Wolfgang Heidrich, Richard Bartels, George Labahn Abstract. Fitting of uncertain data, that is, fitting of data points that are subject to some error, has important applications
More informationVisualizing Quaternions
Visualizing Quaternions Andrew J. Hanson Computer Science Department Indiana University Siggraph 1 Tutorial 1 GRAND PLAN I: Fundamentals of Quaternions II: Visualizing Quaternion Geometry III: Quaternion
More informationEnvelopes Computational Theory and Applications
Envelopes Computational Theory and Applications Category: survey Abstract for points, whose tangent plane maps to a line under the projection. These points form the so-called Based on classical geometric
More informationOn the classification of real algebraic curves and surfaces
On the classification of real algebraic curves and surfaces Ragni Piene Centre of Mathematics for Applications and Department of Mathematics, University of Oslo COMPASS, Kefermarkt, October 2, 2003 1 Background
More informationManifold T-spline. Ying He 1 Kexiang Wang 2 Hongyu Wang 2 Xianfeng David Gu 2 Hong Qin 2. Geometric Modeling and Processing 2006
Ying He 1 Kexiang Wang 2 Hongyu Wang 2 Xianfeng David Gu 2 Hong Qin 2 1 School of Computer Engineering Nanyang Technological University, Singapore 2 Center for Visual Computing (CVC) Stony Brook University,
More informationCS-184: Computer Graphics. Today
CS-84: Computer Graphics Lecture #5: Curves and Surfaces Prof. James O Brien University of California, Berkeley V25F-5-. Today General curve and surface representations Splines and other polynomial bases
More informationReal time Ray-Casting of Algebraic Surfaces
Real time Ray-Casting of Algebraic Surfaces Martin Reimers Johan Seland Center of Mathematics for Applications University of Oslo Workshop on Computational Method for Algebraic Spline Surfaces Thursday
More informationSubdivision Curves and Surfaces
Subdivision Surfaces or How to Generate a Smooth Mesh?? Subdivision Curves and Surfaces Subdivision given polyline(2d)/mesh(3d) recursively modify & add vertices to achieve smooth curve/surface Each iteration
More informationProjective 2D Geometry
Projective D Geometry Multi View Geometry (Spring '08) Projective D Geometry Prof. Kyoung Mu Lee SoEECS, Seoul National University Homogeneous representation of lines and points Projective D Geometry Line
More informationGeneralized barycentric coordinates
Generalized barycentric coordinates Michael S. Floater August 20, 2012 In this lecture, we review the definitions and properties of barycentric coordinates on triangles, and study generalizations to convex,
More informationimplicit surfaces, approximate implicitization, B-splines, A- patches, surface fitting
24. KONFERENCE O GEOMETRII A POČÍTAČOVÉ GRAFICE ZBYNĚK ŠÍR FITTING OF PIECEWISE POLYNOMIAL IMPLICIT SURFACES Abstrakt In our contribution we discuss the possibility of an efficient fitting of piecewise
More informationCS3621 Midterm Solution (Fall 2005) 150 points
CS362 Midterm Solution Fall 25. Geometric Transformation CS362 Midterm Solution (Fall 25) 5 points (a) [5 points] Find the 2D transformation matrix for the reflection about the y-axis transformation (i.e.,
More informationKnow it. Control points. B Spline surfaces. Implicit surfaces
Know it 15 B Spline Cur 14 13 12 11 Parametric curves Catmull clark subdivision Parametric surfaces Interpolating curves 10 9 8 7 6 5 4 3 2 Control points B Spline surfaces Implicit surfaces Bezier surfaces
More informationCOMP 558 lecture 19 Nov. 17, 2010
COMP 558 lecture 9 Nov. 7, 2 Camera calibration To estimate the geometry of 3D scenes, it helps to know the camera parameters, both external and internal. The problem of finding all these parameters is
More informationSummer School: Mathematical Methods in Robotics
Summer School: Mathematical Methods in Robotics Part IV: Projective Geometry Harald Löwe TU Braunschweig, Institute Computational Mathematics 2009/07/16 Löwe (TU Braunschweig) Math. robotics 2009/07/16
More informationCurves & Surfaces. MIT EECS 6.837, Durand and Cutler
Curves & Surfaces Schedule Sunday October 5 th, * 3-5 PM * Review Session for Quiz 1 Extra Office Hours on Monday Tuesday October 7 th : Quiz 1: In class 1 hand-written 8.5x11 sheet of notes allowed Wednesday
More informationCS231A Course Notes 4: Stereo Systems and Structure from Motion
CS231A Course Notes 4: Stereo Systems and Structure from Motion Kenji Hata and Silvio Savarese 1 Introduction In the previous notes, we covered how adding additional viewpoints of a scene can greatly enhance
More informationLinear Precision for Parametric Patches
Department of Mathematics Texas A&M University March 30, 2007 / Texas A&M University Algebraic Geometry and Geometric modeling Geometric modeling uses polynomials to build computer models for industrial
More informationRay-casting Algebraic Surfaces using the Frustum Form. Eurographics 2008 Crete, Thursday April 17.
Ray-casting Algebraic Surfaces using the Frustum Form Martin Reimers Johan Seland Eurographics 2008 Crete, Thursday April 17. Algebraic Surfaces Zero set of polynomial f : R 3 R f (x, y, z) = f ijk x i
More informationThe Geometry Behind the Numerical Reconstruction of Two Photos
The Geometry Behind the Numerical Reconstruction of Two Photos Hellmuth Stachel stachel@dmg.tuwien.ac.at http://www.geometrie.tuwien.ac.at/stachel ICEGD 2007, The 2 nd Internat. Conf. on Eng g Graphics
More informationCS231A Midterm Review. Friday 5/6/2016
CS231A Midterm Review Friday 5/6/2016 Outline General Logistics Camera Models Non-perspective cameras Calibration Single View Metrology Epipolar Geometry Structure from Motion Active Stereo and Volumetric
More informationComputer Graphics CS 543 Lecture 13a Curves, Tesselation/Geometry Shaders & Level of Detail
Computer Graphics CS 54 Lecture 1a Curves, Tesselation/Geometry Shaders & Level of Detail Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) So Far Dealt with straight lines
More informationDM545 Linear and Integer Programming. Lecture 2. The Simplex Method. Marco Chiarandini
DM545 Linear and Integer Programming Lecture 2 The Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1. 2. 3. 4. Standard Form Basic Feasible Solutions
More informationComputer Vision I - Algorithms and Applications: Multi-View 3D reconstruction
Computer Vision I - Algorithms and Applications: Multi-View 3D reconstruction Carsten Rother 09/12/2013 Computer Vision I: Multi-View 3D reconstruction Roadmap this lecture Computer Vision I: Multi-View
More informationJorg s Graphics Lecture Notes Coordinate Spaces 1
Jorg s Graphics Lecture Notes Coordinate Spaces Coordinate Spaces Computer Graphics: Objects are rendered in the Euclidean Plane. However, the computational space is better viewed as one of Affine Space
More informationDon t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary?
Don t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case?
More informationChapter 6. Curves and Surfaces. 6.1 Graphs as Surfaces
Chapter 6 Curves and Surfaces In Chapter 2 a plane is defined as the zero set of a linear function in R 3. It is expected a surface is the zero set of a differentiable function in R n. To motivate, graphs
More informationRay scene intersections
Ray scene intersections 1996-2018 Josef Pelikán CGG MFF UK Praha pepca@cgg.mff.cuni.cz http://cgg.mff.cuni.cz/~pepca/ Intersection 2018 Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 1 / 26 Ray scene intersection
More informationC / 35. C18 Computer Vision. David Murray. dwm/courses/4cv.
C18 2015 1 / 35 C18 Computer Vision David Murray david.murray@eng.ox.ac.uk www.robots.ox.ac.uk/ dwm/courses/4cv Michaelmas 2015 C18 2015 2 / 35 Computer Vision: This time... 1. Introduction; imaging geometry;
More informationInteractive Computer Graphics A TOP-DOWN APPROACH WITH SHADER-BASED OPENGL
International Edition Interactive Computer Graphics A TOP-DOWN APPROACH WITH SHADER-BASED OPENGL Sixth Edition Edward Angel Dave Shreiner Interactive Computer Graphics: A Top-Down Approach with Shader-Based
More informationSurface Approximation and Interpolation via Matrix SVD
Surface Approximation and Interpolation via Matrix SVD Andrew E. Long and Clifford A. Long Andy Long (longa@nku.edu) is at Northern Kentucky University. He received his Ph.D. in applied mathematics at
More informationThe real voyage of discovery consists not in seeking new landscapes, but in having new eyes.
The real voyage of discovery consists not in seeking new landscapes, but in having new eyes. - Marcel Proust University of Texas at Arlington Camera Calibration (or Resectioning) CSE 4392-5369 Vision-based
More informationRendering Curves and Surfaces. Ed Angel Professor of Computer Science, Electrical and Computer Engineering, and Media Arts University of New Mexico
Rendering Curves and Surfaces Ed Angel Professor of Computer Science, Electrical and Computer Engineering, and Media Arts University of New Mexico Objectives Introduce methods to draw curves - Approximate
More informationThe exam begins at 5:10pm and ends at 8:00pm. You must turn your exam in when time is announced or risk not having it accepted.
CS 184: Foundations of Computer Graphics page 1 of 11 Student Name: Student ID: Instructions: Read them carefully! The exam begins at 5:10pm and ends at 8:00pm. You must turn your exam in when time is
More informationSurface Reconstruction from Unorganized Points
Survey of Methods in Computer Graphics: Surface Reconstruction from Unorganized Points H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, W. Stuetzle SIGGRAPH 1992. Article and Additional Material at: http://research.microsoft.com/en-us/um/people/hoppe/proj/recon/
More informationCHAPTER 1 Graphics Systems and Models 3
?????? 1 CHAPTER 1 Graphics Systems and Models 3 1.1 Applications of Computer Graphics 4 1.1.1 Display of Information............. 4 1.1.2 Design.................... 5 1.1.3 Simulation and Animation...........
More informationSung-Eui Yoon ( 윤성의 )
CS480: Computer Graphics Curves and Surfaces Sung-Eui Yoon ( 윤성의 ) Course URL: http://jupiter.kaist.ac.kr/~sungeui/cg Today s Topics Surface representations Smooth curves Subdivision 2 Smooth Curves and
More informationIntroduction to the Mathematical Concepts of CATIA V5
CATIA V5 Training Foils Introduction to the Mathematical Concepts of CATIA V5 Version 5 Release 19 January 2009 EDU_CAT_EN_MTH_FI_V5R19 1 About this course Objectives of the course Upon completion of this
More informationRobust Geometry Estimation from two Images
Robust Geometry Estimation from two Images Carsten Rother 09/12/2016 Computer Vision I: Image Formation Process Roadmap for next four lectures Computer Vision I: Image Formation Process 09/12/2016 2 Appearance-based
More informationCIS 580, Machine Perception, Spring 2014: Assignment 4 Due: Wednesday, April 10th, 10:30am (use turnin)
CIS 580, Machine Perception, Spring 2014: Assignment 4 Due: Wednesday, April 10th, 10:30am (use turnin) Solutions (hand calculations, plots) have to be submitted electronically as a single pdf file using
More informationAn introduction to interpolation and splines
An introduction to interpolation and splines Kenneth H. Carpenter, EECE KSU November 22, 1999 revised November 20, 2001, April 24, 2002, April 14, 2004 1 Introduction Suppose one wishes to draw a curve
More informationInverse and Implicit functions
CHAPTER 3 Inverse and Implicit functions. Inverse Functions and Coordinate Changes Let U R d be a domain. Theorem. (Inverse function theorem). If ϕ : U R d is differentiable at a and Dϕ a is invertible,
More informationNear-Optimal Parameterization of the Intersection of Quadrics: III. Parameterizing Singular Intersections
Near-Optimal Parameterization of the Intersection of Quadrics: III. Parameterizing Singular Intersections Laurent Dupont, Daniel Lazard, Sylvain Lazard, Sylvain Petitjean To cite this version: Laurent
More informationEpipolar Geometry and the Essential Matrix
Epipolar Geometry and the Essential Matrix Carlo Tomasi The epipolar geometry of a pair of cameras expresses the fundamental relationship between any two corresponding points in the two image planes, and
More informationApproximating Point Clouds by Generalized Bézier Surfaces
WAIT: Workshop on the Advances in Information Technology, Budapest, 2018 Approximating Point Clouds by Generalized Bézier Surfaces Péter Salvi 1, Tamás Várady 1, Kenjirō T. Miura 2 1 Budapest University
More informationAdvanced Modeling 2. Katja Bühler, Andrej Varchola, Eduard Gröller. March 24, x(t) z(t)
Advanced Modeling 2 Katja Bühler, Andrej Varchola, Eduard Gröller March 24, 2014 1 Parametric Representations A parametric curve in E 3 is given by x(t) c : c(t) = y(t) ; t I = [a, b] R z(t) where x(t),
More informationINF3320 Computer Graphics and Discrete Geometry
INF3320 Computer Graphics and Discrete Geometry More smooth Curves and Surfaces Christopher Dyken, Michael Floater and Martin Reimers 10.11.2010 Page 1 More smooth Curves and Surfaces Akenine-Möller, Haines
More informationRational Bezier Surface
Rational Bezier Surface The perspective projection of a 4-dimensional polynomial Bezier surface, S w n ( u, v) B i n i 0 m j 0, u ( ) B j m, v ( ) P w ij ME525x NURBS Curve and Surface Modeling Page 97
More informationFinding All Real Points of a Complex Algebraic Curve
Finding All Real Points of a Complex Algebraic Curve Charles Wampler General Motors R&D Center In collaboration with Ye Lu (MIT), Daniel Bates (IMA), & Andrew Sommese (University of Notre Dame) Outline
More informationCS-9645 Introduction to Computer Vision Techniques Winter 2019
Table of Contents Projective Geometry... 1 Definitions...1 Axioms of Projective Geometry... Ideal Points...3 Geometric Interpretation... 3 Fundamental Transformations of Projective Geometry... 4 The D
More informationCS-184: Computer Graphics
CS-184: Computer Graphics Lecture #12: Curves and Surfaces Prof. James O Brien University of California, Berkeley V2007-F-12-1.0 Today General curve and surface representations Splines and other polynomial
More informationAdaptive and Smooth Surface Construction by Triangular A-Patches
Adaptive and Smooth Surface Construction by Triangular A-Patches Guoliang Xu Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, China Abstract
More informationPhysically-Based Modeling and Animation. University of Missouri at Columbia
Overview of Geometric Modeling Overview 3D Shape Primitives: Points Vertices. Curves Lines, polylines, curves. Surfaces Triangle meshes, splines, subdivision surfaces, implicit surfaces, particles. Solids
More informationConic Duality. yyye
Conic Linear Optimization and Appl. MS&E314 Lecture Note #02 1 Conic Duality Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/
More informationParameterization. Michael S. Floater. November 10, 2011
Parameterization Michael S. Floater November 10, 2011 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to generate from point
More informationECE 470: Homework 5. Due Tuesday, October 27 in Seth Hutchinson. Luke A. Wendt
ECE 47: Homework 5 Due Tuesday, October 7 in class @:3pm Seth Hutchinson Luke A Wendt ECE 47 : Homework 5 Consider a camera with focal length λ = Suppose the optical axis of the camera is aligned with
More informationCurves and Surfaces Computer Graphics I Lecture 10
15-462 Computer Graphics I Lecture 10 Curves and Surfaces Parametric Representations Cubic Polynomial Forms Hermite Curves Bezier Curves and Surfaces [Angel 10.1-10.6] September 30, 2003 Doug James Carnegie
More informationSurfaces for CAGD. FSP Tutorial. FSP-Seminar, Graz, November
Surfaces for CAGD FSP Tutorial FSP-Seminar, Graz, November 2005 1 Tensor Product Surfaces Given: two curve schemes (Bézier curves or B splines): I: x(u) = m i=0 F i(u)b i, u [a, b], II: x(v) = n j=0 G
More informationSolving Polynomial Equations in Geometric Problems
Solving Polynomial Equations in Geometric Problems B. Mourrain INRIA, BP 93, 06902 Sophia Antipolis mourrain@sophia.inria.fr December 9, 2004 SHAPES 1 Semialgebraic models: Bezier parameterisation, NURBS,
More informationCurves D.A. Forsyth, with slides from John Hart
Curves D.A. Forsyth, with slides from John Hart Central issues in modelling Construct families of curves, surfaces and volumes that can represent common objects usefully; are easy to interact with; interaction
More informationIsometries. 1 Identifying Isometries
Isometries 1 Identifying Isometries 1. Modeling isometries as dynamic maps. 2. GeoGebra files: isoguess1.ggb, isoguess2.ggb, isoguess3.ggb, isoguess4.ggb. 3. Guessing isometries. 4. What can you construct
More informationB-spline Curves. Smoother than other curve forms
Curves and Surfaces B-spline Curves These curves are approximating rather than interpolating curves. The curves come close to, but may not actually pass through, the control points. Usually used as multiple,
More information09 - Designing Surfaces. CSCI-GA Computer Graphics - Fall 16 - Daniele Panozzo
9 - Designing Surfaces Triangular surfaces A surface can be discretized by a collection of points and triangles Each triangle is a subset of a plane Every point on the surface can be expressed as an affine
More informationProjection of a Smooth Space Curve:
Projection of a Smooth Space Curve: Numeric and Certified Topology Computation Marc Pouget Joint work with Rémi Imbach Guillaume Moroz 1 Projection and apparent contour 3D curve = intersection of 2 implicit
More informationTwo-View Geometry (Course 23, Lecture D)
Two-View Geometry (Course 23, Lecture D) Jana Kosecka Department of Computer Science George Mason University http://www.cs.gmu.edu/~kosecka General Formulation Given two views of the scene recover the
More informationInstitutionen för matematik, KTH.
Institutionen för matematik, KTH. Chapter 10 projective toric varieties and polytopes: definitions 10.1 Introduction Tori varieties are algebraic varieties related to the study of sparse polynomials.
More informationCurves & Surfaces. Last Time? Progressive Meshes. Selective Refinement. Adjacency Data Structures. Mesh Simplification. Mesh Simplification
Last Time? Adjacency Data Structures Curves & Surfaces Geometric & topologic information Dynamic allocation Efficiency of access Mesh Simplification edge collapse/vertex split geomorphs progressive transmission
More informationApproximate Algebraic Methods for Curves and Surfaces and their Applications
Approximate Algebraic Methods for Curves and Surfaces and their Applications Bert Jüttler, Pavel Chalmovianský, Mohamed Shalaby and Elmar Wurm Institute of Applied Geometry, Johannes Kepler University,
More informationMathematical Tools in Computer Graphics with C# Implementations Table of Contents
Mathematical Tools in Computer Graphics with C# Implementations by Hardy Alexandre, Willi-Hans Steeb, World Scientific Publishing Company, Incorporated, 2008 Table of Contents List of Figures Notation
More informationIntroduction p. 1 What Is Geometric Modeling? p. 1 Computer-aided geometric design Solid modeling Algebraic geometry Computational geometry
Introduction p. 1 What Is Geometric Modeling? p. 1 Computer-aided geometric design Solid modeling Algebraic geometry Computational geometry Representation Ab initio design Rendering Solid modelers Kinematic
More information