Jianmin Zheng - Academia.edu (original) (raw)

Papers by Jianmin Zheng

Research paper thumbnail of Implicitizing rational surfaces with base points using the method of moving surfaces

Topics in Algebraic Geometry and Geometric Modeling, 2003

The method of moving planes and moving quadrics can express the implicit equation of a parametric... more The method of moving planes and moving quadrics can express the implicit equation of a parametric surface as the determinant of a matrix M. The rows of M correspond to moving planes or moving quadrics that follow the parametric surface. Previous papers on the method of moving surfaces have shown that a simple base point has the effect of converting one moving quadric to a moving plane. A much more general version of the method of moving surfaces is presented in this paper that is capable of dealing with multiple base points. For example, a double base point has the effect (in this new version) of converting two moving quadrics into moving planes, eliminating one additional moving quadric, and eliminating a column of the matrix (i.e., a blending function of the moving surfaces)-thereby dropping the degree of the implicit equation by four. Furthermore, this is a unifying approach whereby tensor product surfaces, pure degree surfaces, and "corner-cut" surfaces, can all be implicitized under the same framework and do not need to be treated as distinct cases. The central idea in this approach is that if a surface has a base point of multiplicity k, the moving surface blending functions must have the same base point, but of multiplicity k − 1. Thus, we draw moving surface blending functions from the derivative ideal I , where I is the ideal of the parametric equations. We explain the general outline of the method and show how it works in some specific cases. The paper concludes with a discussion of the method from the point of view of commutative algebra.

Research paper thumbnail of A Direct Approach to Computing theμ-basis of Planar Rational Curves

Journal of Symbolic Computation, 2001

This paper presents an O(n 2) algorithm, based on Gröbner basis techniques, to compute the µ-basi... more This paper presents an O(n 2) algorithm, based on Gröbner basis techniques, to compute the µ-basis of a degree n planar rational curve. The prior method involved solving a set of linear equations whose complexity by standard numerical methods was O(n 3). The µ-basis is useful in computing the implicit equation of a parametric curve and can express the implicit equation in the form of a determinant that is smaller than that obtained by taking the resultant of the parametric equations.

Research paper thumbnail of Interproximate curve subdivision

Journal of Computational and Applied Mathematics, 2013

This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright

Research paper thumbnail of Approximate Implicitization Using Monoid Curves and Surfaces

Graphical Models and Image Processing, 1999

This paper presents an approach to nding an approximate implicit equation and an approximate inve... more This paper presents an approach to nding an approximate implicit equation and an approximate inversion map of a planar rational parametric curve or a rational parametric surface. High accuracy of the approximation is achieved with a relatively small numberof low-degree curve segments or surface patches. By using monoid curves and surfaces, the method eliminates the undesirable singularities and phantom" branches, normally associated with implicit representation. The monoids are expressed in exact implicit and parametric equations simultaneously, and upper bounds are derived for the approximate errors of implicitization and inversion equations.

Research paper thumbnail of An alternative method for constructing interpolatory subdivision from approximating subdivision

Computer Aided Geometric Design, 2012

This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright Author's personal copy Computer Aided Geometric Design 29 (2012) 474-484 Contents lists available at SciVerse ScienceDirect

Research paper thumbnail of A conjecture on tangent intersections of surface patches

Computer Aided Geometric Design, 2004

This note conjectures that if two surface patches intersect with G 1 continuity along an entire c... more This note conjectures that if two surface patches intersect with G 1 continuity along an entire curve, the probability is one that the curve is rational. This idea has significance for surface intersection algorithms.

Research paper thumbnail of Gaussian and mean curvatures of rational Bézier patches

Computer Aided Geometric Design, 2003

This note derives formulae for Gaussian and mean curvatures for tensor-product and triangular rat... more This note derives formulae for Gaussian and mean curvatures for tensor-product and triangular rational Bézier patches in terms of the respective control meshes. These formulae provide more geometric intuition than the generic formulae from differential geometry.

Research paper thumbnail of Knot intervals and multi-degree splines

Computer Aided Geometric Design, 2003

This paper studies the merits of using knot interval notation for B-spline curves, and presents f... more This paper studies the merits of using knot interval notation for B-spline curves, and presents formulae in terms of knot intervals for common B-spline operations such as knot insertion, differentiation, and degree elevation. Using knot interval notation, the paper introduces MD-splines, which are B-spline-like curves that are comprised of polynomial segments of various degrees (MD stands for "multi-degree"). MD-splines are a generalization of B-spline curves in that if all curve segments in an MD-spline have the same degree, it reduces to a B-spline curve. The paper focuses on MD-splines of degree 1, 2, and 3, as well as degree 1 and n. MD-splines have local support, obey the convex hull and variation diminishing properties, and are at least C n−1 , where n is the smaller of the degrees of two adjoining curve segments.

Research paper thumbnail of Linear perturbation methods for topologically consistent representations of free-form surface intersections

Computer Aided Geometric Design, 2004

By applying displacement maps to slightly perturb two free-form surfaces, one can ensure exact ag... more By applying displacement maps to slightly perturb two free-form surfaces, one can ensure exact agreement between the images in 3 of parameterdomain approximations to their curve of intersection. Thus, at the expense of slightly altering the surfaces in the vicinity of their intersection, a perfect matching of the surface trimming curves is guaranteed. This exact agreement of contiguous trimmed surfaces is essential to achieving topologically consistent solid model constructions through Boolean operations, and has a profound impact on the efficiency and reliability of applications such as meshing, rendering, and computing volumetric properties. Moreover, the control point perturbations require only the solution of a linear system for their determination. The basic principles of this approach to topologically consistent surface trimming curves are described, and example results from the implementation of a simple instance of the method are presented.

Research paper thumbnail of The mu-basis of a rational ruled surface

Computer Aided Geometric Design, 2001

The mu-basis of a planar rational curve is a polynomial ideal basis comprised of two polynomials ... more The mu-basis of a planar rational curve is a polynomial ideal basis comprised of two polynomials that greatly facilitates computing the implicit equation of the curve. This paper defines a mu-basis for a rational ruled surface, and presents a simple algorithm for computing the mu-basis. The mu-basis consists of two polynomials p(x, y, z, s) and q(x, y, z, s) that are linear in x, y, z and degree µ and m − µ in s respectively, where m is the degree of the implicit equation. The implicit equation of the surface is then obtained by merely taking the resultant of p and q with respect to s. This implicitization algorithm is faster and/or more robust than previous methods.

Research paper thumbnail of On linear independence of T-spline blending functions

Computer Aided Geometric Design, 2012

This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit:

Research paper thumbnail of T-spline simplification and local refinement

ACM Transactions on Graphics, 2004

A typical NURBS surface model has a large percentage of superfluous control points that significa... more A typical NURBS surface model has a large percentage of superfluous control points that significantly interfere with the design process. This paper presents an algorithm for eliminating such superfluous control points, producing a T-spline. The algorithm can remove substantially more control points than competing methods such as B-spline wavelet decomposition. The paper also presents a new T-spline local refinement algorithm and answers two fundamental open questions on T-spline theory.

Research paper thumbnail of T-splines and T-NURCCs

ACM Transactions on Graphics, 2003

This paper presents a generalization of non-uniform B-spline surfaces called T-splines. T-spline ... more This paper presents a generalization of non-uniform B-spline surfaces called T-splines. T-spline control grids permit T-junctions, so lines of control points need not traverse the entire control grid. T-splines support many valuable operations within a consistent framework, such as local refinement, and the merging of several B-spline surfaces that have different knot vectors into a single gap-free model. The paper focuses on T-splines of degree three, which are C 2 (in the absence of multiple knots). T-NURCCs (Non-Uniform Rational Catmull-Clark Surfaces with T-junctions) are a superset of both T-splines and Catmull-Clark surfaces. Thus, a modeling program for T-NURCCs can handle any NURBS or Catmull-Clark model as special cases. T-NURCCs enable true local refinement of a Catmull-Clark-type control grid: individual control points can be inserted only where they are needed to provide additional control, or to create a smoother tessellation, and such insertions do not alter the limit ...

Research paper thumbnail of Algebraic Methods for Computer Aided Geometric Design

Handbook of Computer Aided Geometric Design, 2002

The concepts and methods of algebra and algebraic geometry have found significant applications in... more The concepts and methods of algebra and algebraic geometry have found significant applications in many disciplines. This chapter presents a collection of gleanings from algebra or algebraic geometry that hold practical value for the field of computer aided geometric design. We focus on the insights, algorithm enhancements and practical capabilities that algebraic methods have contributed to CAGD. Specifically, we examine resultants and Gröbner basis, and discuss their applications in implicitization, inversion, parametrization and intersection algorithms. Other topics of CAGD research work using algebraic methods are also outlined.

Research paper thumbnail of Estimating tessellation parameter intervals for rational curves and surfaces

ACM Transactions on Graphics, 2000

This paper presents a method for determining a priori a constant parameter interval with which a ... more This paper presents a method for determining a priori a constant parameter interval with which a rational curve or surface can be tessellated such that the deviation of the curve or surface from its piecewise linear approximation is within a specified tolerance. The parameter interval is estimated based on information about the second order derivatives in the homogeneous coordinates, instead of using affine coordinates directly. This new step size can be found with roughly the same amount of computation as the step size presented in [Cheng 1992], though it can be proven to always be larger than Cheng's step size. In fact, numerical experiments show the new step is typically orders of magnitude larger than the step size in [Cheng 1992]. Furthermore, for rational cubic and quartic curves, the new step size is generally twice as large as the step size found by computing bounds on the Bernstein polynomial coefficients of the second derivatives function.

Research paper thumbnail of T-spline simplification and local refinement

ACM Transactions on Graphics, 2004

A typical NURBS surface model has a large percentage of superfluous control points that significa... more A typical NURBS surface model has a large percentage of superfluous control points that significantly interfere with the design process. This paper presents an algorithm for eliminating such superfluous control points, producing a T-spline. The algorithm can remove substantially more control points than competing methods such as B-spline wavelet decomposition. The paper also presents a new T-spline local refinement algorithm and answers two fundamental open questions on T-spline theory.

Research paper thumbnail of Implicitizing rational surfaces with base points using the method of moving surfaces

Topics in Algebraic Geometry and Geometric Modeling, 2003

The method of moving planes and moving quadrics can express the implicit equation of a parametric... more The method of moving planes and moving quadrics can express the implicit equation of a parametric surface as the determinant of a matrix M. The rows of M correspond to moving planes or moving quadrics that follow the parametric surface. Previous papers on the method of moving surfaces have shown that a simple base point has the effect of converting one moving quadric to a moving plane. A much more general version of the method of moving surfaces is presented in this paper that is capable of dealing with multiple base points. For example, a double base point has the effect (in this new version) of converting two moving quadrics into moving planes, eliminating one additional moving quadric, and eliminating a column of the matrix (i.e., a blending function of the moving surfaces)-thereby dropping the degree of the implicit equation by four. Furthermore, this is a unifying approach whereby tensor product surfaces, pure degree surfaces, and "corner-cut" surfaces, can all be implicitized under the same framework and do not need to be treated as distinct cases. The central idea in this approach is that if a surface has a base point of multiplicity k, the moving surface blending functions must have the same base point, but of multiplicity k − 1. Thus, we draw moving surface blending functions from the derivative ideal I , where I is the ideal of the parametric equations. We explain the general outline of the method and show how it works in some specific cases. The paper concludes with a discussion of the method from the point of view of commutative algebra.

Research paper thumbnail of A Direct Approach to Computing theμ-basis of Planar Rational Curves

Journal of Symbolic Computation, 2001

This paper presents an O(n 2) algorithm, based on Gröbner basis techniques, to compute the µ-basi... more This paper presents an O(n 2) algorithm, based on Gröbner basis techniques, to compute the µ-basis of a degree n planar rational curve. The prior method involved solving a set of linear equations whose complexity by standard numerical methods was O(n 3). The µ-basis is useful in computing the implicit equation of a parametric curve and can express the implicit equation in the form of a determinant that is smaller than that obtained by taking the resultant of the parametric equations.

Research paper thumbnail of Interproximate curve subdivision

Journal of Computational and Applied Mathematics, 2013

This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright

Research paper thumbnail of Approximate Implicitization Using Monoid Curves and Surfaces

Graphical Models and Image Processing, 1999

This paper presents an approach to nding an approximate implicit equation and an approximate inve... more This paper presents an approach to nding an approximate implicit equation and an approximate inversion map of a planar rational parametric curve or a rational parametric surface. High accuracy of the approximation is achieved with a relatively small numberof low-degree curve segments or surface patches. By using monoid curves and surfaces, the method eliminates the undesirable singularities and phantom" branches, normally associated with implicit representation. The monoids are expressed in exact implicit and parametric equations simultaneously, and upper bounds are derived for the approximate errors of implicitization and inversion equations.

Research paper thumbnail of An alternative method for constructing interpolatory subdivision from approximating subdivision

Computer Aided Geometric Design, 2012

This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright Author's personal copy Computer Aided Geometric Design 29 (2012) 474-484 Contents lists available at SciVerse ScienceDirect

Research paper thumbnail of A conjecture on tangent intersections of surface patches

Computer Aided Geometric Design, 2004

This note conjectures that if two surface patches intersect with G 1 continuity along an entire c... more This note conjectures that if two surface patches intersect with G 1 continuity along an entire curve, the probability is one that the curve is rational. This idea has significance for surface intersection algorithms.

Research paper thumbnail of Gaussian and mean curvatures of rational Bézier patches

Computer Aided Geometric Design, 2003

This note derives formulae for Gaussian and mean curvatures for tensor-product and triangular rat... more This note derives formulae for Gaussian and mean curvatures for tensor-product and triangular rational Bézier patches in terms of the respective control meshes. These formulae provide more geometric intuition than the generic formulae from differential geometry.

Research paper thumbnail of Knot intervals and multi-degree splines

Computer Aided Geometric Design, 2003

This paper studies the merits of using knot interval notation for B-spline curves, and presents f... more This paper studies the merits of using knot interval notation for B-spline curves, and presents formulae in terms of knot intervals for common B-spline operations such as knot insertion, differentiation, and degree elevation. Using knot interval notation, the paper introduces MD-splines, which are B-spline-like curves that are comprised of polynomial segments of various degrees (MD stands for "multi-degree"). MD-splines are a generalization of B-spline curves in that if all curve segments in an MD-spline have the same degree, it reduces to a B-spline curve. The paper focuses on MD-splines of degree 1, 2, and 3, as well as degree 1 and n. MD-splines have local support, obey the convex hull and variation diminishing properties, and are at least C n−1 , where n is the smaller of the degrees of two adjoining curve segments.

Research paper thumbnail of Linear perturbation methods for topologically consistent representations of free-form surface intersections

Computer Aided Geometric Design, 2004

By applying displacement maps to slightly perturb two free-form surfaces, one can ensure exact ag... more By applying displacement maps to slightly perturb two free-form surfaces, one can ensure exact agreement between the images in 3 of parameterdomain approximations to their curve of intersection. Thus, at the expense of slightly altering the surfaces in the vicinity of their intersection, a perfect matching of the surface trimming curves is guaranteed. This exact agreement of contiguous trimmed surfaces is essential to achieving topologically consistent solid model constructions through Boolean operations, and has a profound impact on the efficiency and reliability of applications such as meshing, rendering, and computing volumetric properties. Moreover, the control point perturbations require only the solution of a linear system for their determination. The basic principles of this approach to topologically consistent surface trimming curves are described, and example results from the implementation of a simple instance of the method are presented.

Research paper thumbnail of The mu-basis of a rational ruled surface

Computer Aided Geometric Design, 2001

The mu-basis of a planar rational curve is a polynomial ideal basis comprised of two polynomials ... more The mu-basis of a planar rational curve is a polynomial ideal basis comprised of two polynomials that greatly facilitates computing the implicit equation of the curve. This paper defines a mu-basis for a rational ruled surface, and presents a simple algorithm for computing the mu-basis. The mu-basis consists of two polynomials p(x, y, z, s) and q(x, y, z, s) that are linear in x, y, z and degree µ and m − µ in s respectively, where m is the degree of the implicit equation. The implicit equation of the surface is then obtained by merely taking the resultant of p and q with respect to s. This implicitization algorithm is faster and/or more robust than previous methods.

Research paper thumbnail of On linear independence of T-spline blending functions

Computer Aided Geometric Design, 2012

This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit:

Research paper thumbnail of T-spline simplification and local refinement

ACM Transactions on Graphics, 2004

A typical NURBS surface model has a large percentage of superfluous control points that significa... more A typical NURBS surface model has a large percentage of superfluous control points that significantly interfere with the design process. This paper presents an algorithm for eliminating such superfluous control points, producing a T-spline. The algorithm can remove substantially more control points than competing methods such as B-spline wavelet decomposition. The paper also presents a new T-spline local refinement algorithm and answers two fundamental open questions on T-spline theory.

Research paper thumbnail of T-splines and T-NURCCs

ACM Transactions on Graphics, 2003

This paper presents a generalization of non-uniform B-spline surfaces called T-splines. T-spline ... more This paper presents a generalization of non-uniform B-spline surfaces called T-splines. T-spline control grids permit T-junctions, so lines of control points need not traverse the entire control grid. T-splines support many valuable operations within a consistent framework, such as local refinement, and the merging of several B-spline surfaces that have different knot vectors into a single gap-free model. The paper focuses on T-splines of degree three, which are C 2 (in the absence of multiple knots). T-NURCCs (Non-Uniform Rational Catmull-Clark Surfaces with T-junctions) are a superset of both T-splines and Catmull-Clark surfaces. Thus, a modeling program for T-NURCCs can handle any NURBS or Catmull-Clark model as special cases. T-NURCCs enable true local refinement of a Catmull-Clark-type control grid: individual control points can be inserted only where they are needed to provide additional control, or to create a smoother tessellation, and such insertions do not alter the limit ...

Research paper thumbnail of Algebraic Methods for Computer Aided Geometric Design

Handbook of Computer Aided Geometric Design, 2002

The concepts and methods of algebra and algebraic geometry have found significant applications in... more The concepts and methods of algebra and algebraic geometry have found significant applications in many disciplines. This chapter presents a collection of gleanings from algebra or algebraic geometry that hold practical value for the field of computer aided geometric design. We focus on the insights, algorithm enhancements and practical capabilities that algebraic methods have contributed to CAGD. Specifically, we examine resultants and Gröbner basis, and discuss their applications in implicitization, inversion, parametrization and intersection algorithms. Other topics of CAGD research work using algebraic methods are also outlined.

Research paper thumbnail of Estimating tessellation parameter intervals for rational curves and surfaces

ACM Transactions on Graphics, 2000

This paper presents a method for determining a priori a constant parameter interval with which a ... more This paper presents a method for determining a priori a constant parameter interval with which a rational curve or surface can be tessellated such that the deviation of the curve or surface from its piecewise linear approximation is within a specified tolerance. The parameter interval is estimated based on information about the second order derivatives in the homogeneous coordinates, instead of using affine coordinates directly. This new step size can be found with roughly the same amount of computation as the step size presented in [Cheng 1992], though it can be proven to always be larger than Cheng's step size. In fact, numerical experiments show the new step is typically orders of magnitude larger than the step size in [Cheng 1992]. Furthermore, for rational cubic and quartic curves, the new step size is generally twice as large as the step size found by computing bounds on the Bernstein polynomial coefficients of the second derivatives function.

Research paper thumbnail of T-spline simplification and local refinement

ACM Transactions on Graphics, 2004

A typical NURBS surface model has a large percentage of superfluous control points that significa... more A typical NURBS surface model has a large percentage of superfluous control points that significantly interfere with the design process. This paper presents an algorithm for eliminating such superfluous control points, producing a T-spline. The algorithm can remove substantially more control points than competing methods such as B-spline wavelet decomposition. The paper also presents a new T-spline local refinement algorithm and answers two fundamental open questions on T-spline theory.