The problems of geometric continuity for rational Bezter surfaces are discussed. Concise conditions of first order and second order geometric continuity for rational triangular Bézier surfaces are given. Meanwhil...The problems of geometric continuity for rational Bezter surfaces are discussed. Concise conditions of first order and second order geometric continuity for rational triangular Bézier surfaces are given. Meanwhile,a geometric condition for smoothness between adjacent rational Bézier surfaces and the transformation formulae between rational triangular patches and rational rectangular patches are obtained.展开更多
This paper introduces the algebraic property of bivariate orthonormal Jacobi polynomials into geometric approximation. Based on the latest results on the transformation formulae between bivariate Bernstein polynomials...This paper introduces the algebraic property of bivariate orthonormal Jacobi polynomials into geometric approximation. Based on the latest results on the transformation formulae between bivariate Bernstein polynomials and Jacobi polynomials, we naturally deduce a novel algorithm for multi-degree reduction of triangular B^zier surfaces. This algorithm possesses four characteristics: ability of error forecast, explicit expression, less time consumption, and best precision. That is, firstly, whether there exists a multi-degree reduced surface within a prescribed tolerance is judged beforehand; secondly, all the operations of multi-degree reduction are just to multiply the column vector generated by sorting the series of the control points of the original surface in lexicographic order by a matrix; thirdly, this matrix can be computed at one time and stored in an array before processing degree reduction; fourthly, the multi-degree reduced surface achieves an optimal approximation in the norm L2. Some numerical experiments are presented to validate the effectiveness of this algorithm, and to show that the algorithm is applicable to information processing of products in CAD system.展开更多
This paper proposes and applies a method to sort two-dimensional control points of triangular Bezier surfaces in a row vector. Using the property of bivariate Jacobi basis functions, it further presents two algorithms...This paper proposes and applies a method to sort two-dimensional control points of triangular Bezier surfaces in a row vector. Using the property of bivariate Jacobi basis functions, it further presents two algorithms for multi-degree reduction of triangular Bezier surfaces with constraints, providing explicit degree-reduced surfaces. The first algorithm can obtain the explicit representation of the optimal degree-reduced surfaces and the approximating error in both boundary curve constraints and corner constraints. But it has to solve the inversion of a matrix whose degree is related with the original surface. The second algorithm entails no matrix inversion to bring about computational instability, gives stable degree-reduced surfaces quickly, and presents the error bound. In the end, the paper proves the efficiency of the two algorithms through examples and error analysis.展开更多
We implemented accurate FFD in terms of triangular Bezier surfaces as matrix multiplications in CUDA and rendered them via OpenGL. Experimental results show that the proposed algorithm is more efficient than the previ...We implemented accurate FFD in terms of triangular Bezier surfaces as matrix multiplications in CUDA and rendered them via OpenGL. Experimental results show that the proposed algorithm is more efficient than the previous GPU acceleration algorithm and tessel- lation shader algorithms.展开更多
Seismic ray tracing in anisotropic media with irregular surface is crucial for the exploration of the fine crustal structure. Elliptically anisotropic medium is the type of anisotropic media with only four independent...Seismic ray tracing in anisotropic media with irregular surface is crucial for the exploration of the fine crustal structure. Elliptically anisotropic medium is the type of anisotropic media with only four independent elastic parameters. Usually, this medium can be described by only the vertical phase velocity and the horizontal phase velocity for seismic wave propagation. Model parameteri- zation in this study is described by flexible triangular grids, which is beneficial for the description of irregular surface with high degree of approximation. Both the vertical and horizontal phase velocities are defined in the triangular grids, respectively, which are used for the description of phase velocity distribution everywhere in the model by linear interpolation. We develop a shooting ray tracing method of turning wave in the elliptically anisotropic media with irregular surface. Runge-Kutta method is applied to solve the partial differential equation of seismic ray in elliptically anisotropic media. Linearly modified method is used for adjusting emergent phase angles in the shooting scheme. Numerical tests demonstrate that ray paths coincide well with analytical trajectories in trans- versely homogeneous elliptically anisotropic media. Seis- mic ray tracing results in transversely inhomogeneous elliptically anisotropic media demonstrate that our method is effective for further first-arrival tomography in ellipti- cally anisotropic media with an irregular surface.展开更多
Voronoi diagrams on triangulated surfaces based on the geodesic metric play a key role in many applications of computer graphics.Previous methods of constructing such Voronoi diagrams generally depended on having an e...Voronoi diagrams on triangulated surfaces based on the geodesic metric play a key role in many applications of computer graphics.Previous methods of constructing such Voronoi diagrams generally depended on having an exact geodesic metric.However,exact geodesic computation is time-consuming and has high memory usage,limiting wider application of geodesic Voronoi diagrams(GVDs).In order to overcome this issue,instead of using exact methods,we reformulate a graph method based on Steiner point insertion,as an effective way to obtain geodesic distances.Further,since a bisector comprises hyperbolic and line segments,we utilize Apollonius diagrams to encode complicated structures,enabling Voronoi diagrams to encode a medial-axis surface for a dense set of boundary samples.Based on these strategies,we present an approximation algorithm for efficient Voronoi diagram construction on triangulated surfaces.We also suggest a measure for evaluating similarity of our results to the exact GVD.Although our GVD results are constructed using approximate geodesic distances,we can get GVD results similar to exact results by inserting Steiner points on triangle edges.Experimental results on many 3D models indicate the improved speed and memory requirements compared to previous leading methods.展开更多
Degree reduction of parametric curves and surfaces is an important process in the exchange of product model data between various CAD systems. In this paper the degenerate conditions of triangular Bezier surface patch...Degree reduction of parametric curves and surfaces is an important process in the exchange of product model data between various CAD systems. In this paper the degenerate conditions of triangular Bezier surface patches are derived. The degenerate conditions and constrained optimization methods are used to develop a degree reduction method for triangular Bezier surface patches. The error in the degree reduction of a triangular Bezier surface is also shown to depend on some geometric invariants which decrease exponentially in the subdivision process. Therefore, the degree reduction method can be combined with a subdivision algorithm to generate lower degree approximations which are within some preset error tolerance.展开更多
A new type of bivariate generalized Ball basis function on a triangle is presented for free-form surface design. Some properties of the basis function are given, then degree elevation, recursive evaluation and some ot...A new type of bivariate generalized Ball basis function on a triangle is presented for free-form surface design. Some properties of the basis function are given, then degree elevation, recursive evaluation and some other properties of the generalized Ball surfaces are also derived. It is shown that the proposed recursive evaluation algorithm is more efficient than those of the old surfaces.展开更多
This paper describes practical approaches on how to construct bounding pyramids and bounding cones for triangular Bezier surfaces. Examples are provided to illustrate the process of construction and comparison is made...This paper describes practical approaches on how to construct bounding pyramids and bounding cones for triangular Bezier surfaces. Examples are provided to illustrate the process of construction and comparison is made between various surface bounding volumes. Furthermore, as a starting point for the construction, we provide a way to compute hodographs of triangular Bezier surfaces and improve the algorithm for computing the bounding cone of a set of vectors. [ABSTRACT FROM AUTHOR]展开更多
文摘The problems of geometric continuity for rational Bezter surfaces are discussed. Concise conditions of first order and second order geometric continuity for rational triangular Bézier surfaces are given. Meanwhile,a geometric condition for smoothness between adjacent rational Bézier surfaces and the transformation formulae between rational triangular patches and rational rectangular patches are obtained.
基金Supported by the National Grand Fundamental Research 973 Program of China (Grant No. 2004CB719400)the National Natural Science Foun-dation of China (Grant Nos. 60673031 and 60333010)the National Natural Science Foundation for Innovative Research Groups (Grant No. 60021201)
文摘This paper introduces the algebraic property of bivariate orthonormal Jacobi polynomials into geometric approximation. Based on the latest results on the transformation formulae between bivariate Bernstein polynomials and Jacobi polynomials, we naturally deduce a novel algorithm for multi-degree reduction of triangular B^zier surfaces. This algorithm possesses four characteristics: ability of error forecast, explicit expression, less time consumption, and best precision. That is, firstly, whether there exists a multi-degree reduced surface within a prescribed tolerance is judged beforehand; secondly, all the operations of multi-degree reduction are just to multiply the column vector generated by sorting the series of the control points of the original surface in lexicographic order by a matrix; thirdly, this matrix can be computed at one time and stored in an array before processing degree reduction; fourthly, the multi-degree reduced surface achieves an optimal approximation in the norm L2. Some numerical experiments are presented to validate the effectiveness of this algorithm, and to show that the algorithm is applicable to information processing of products in CAD system.
基金Supported by the National Natural Science Foundation of China (6087311160933007)
文摘This paper proposes and applies a method to sort two-dimensional control points of triangular Bezier surfaces in a row vector. Using the property of bivariate Jacobi basis functions, it further presents two algorithms for multi-degree reduction of triangular Bezier surfaces with constraints, providing explicit degree-reduced surfaces. The first algorithm can obtain the explicit representation of the optimal degree-reduced surfaces and the approximating error in both boundary curve constraints and corner constraints. But it has to solve the inversion of a matrix whose degree is related with the original surface. The second algorithm entails no matrix inversion to bring about computational instability, gives stable degree-reduced surfaces quickly, and presents the error bound. In the end, the paper proves the efficiency of the two algorithms through examples and error analysis.
基金Supported by the National Natural Science Foundation of China(61170138 and 61472349)
文摘We implemented accurate FFD in terms of triangular Bezier surfaces as matrix multiplications in CUDA and rendered them via OpenGL. Experimental results show that the proposed algorithm is more efficient than the previous GPU acceleration algorithm and tessel- lation shader algorithms.
基金financial support for this work contributed by the National Key Research and Development Program of China(Grants Nos.2016YFC0600101,2016YFC0600201 and 2016YFC0600302)the National Natural Science Foundation of China(Grants Nos.41522401 and 41474068)
文摘Seismic ray tracing in anisotropic media with irregular surface is crucial for the exploration of the fine crustal structure. Elliptically anisotropic medium is the type of anisotropic media with only four independent elastic parameters. Usually, this medium can be described by only the vertical phase velocity and the horizontal phase velocity for seismic wave propagation. Model parameteri- zation in this study is described by flexible triangular grids, which is beneficial for the description of irregular surface with high degree of approximation. Both the vertical and horizontal phase velocities are defined in the triangular grids, respectively, which are used for the description of phase velocity distribution everywhere in the model by linear interpolation. We develop a shooting ray tracing method of turning wave in the elliptically anisotropic media with irregular surface. Runge-Kutta method is applied to solve the partial differential equation of seismic ray in elliptically anisotropic media. Linearly modified method is used for adjusting emergent phase angles in the shooting scheme. Numerical tests demonstrate that ray paths coincide well with analytical trajectories in trans- versely homogeneous elliptically anisotropic media. Seis- mic ray tracing results in transversely inhomogeneous elliptically anisotropic media demonstrate that our method is effective for further first-arrival tomography in ellipti- cally anisotropic media with an irregular surface.
基金supported in part by the Youth Teacher Development Foundation of Harbin Institute of Technology(IDGA10002143)the National Natural Science Foundation of China(62072139,62272277,62072284)+1 种基金the National Key R&D Program of China(2021YFB1715900)the Joint Funds of the National Natural Science Foundation of China(U22A2033).
文摘Voronoi diagrams on triangulated surfaces based on the geodesic metric play a key role in many applications of computer graphics.Previous methods of constructing such Voronoi diagrams generally depended on having an exact geodesic metric.However,exact geodesic computation is time-consuming and has high memory usage,limiting wider application of geodesic Voronoi diagrams(GVDs).In order to overcome this issue,instead of using exact methods,we reformulate a graph method based on Steiner point insertion,as an effective way to obtain geodesic distances.Further,since a bisector comprises hyperbolic and line segments,we utilize Apollonius diagrams to encode complicated structures,enabling Voronoi diagrams to encode a medial-axis surface for a dense set of boundary samples.Based on these strategies,we present an approximation algorithm for efficient Voronoi diagram construction on triangulated surfaces.We also suggest a measure for evaluating similarity of our results to the exact GVD.Although our GVD results are constructed using approximate geodesic distances,we can get GVD results similar to exact results by inserting Steiner points on triangle edges.Experimental results on many 3D models indicate the improved speed and memory requirements compared to previous leading methods.
文摘Degree reduction of parametric curves and surfaces is an important process in the exchange of product model data between various CAD systems. In this paper the degenerate conditions of triangular Bezier surface patches are derived. The degenerate conditions and constrained optimization methods are used to develop a degree reduction method for triangular Bezier surface patches. The error in the degree reduction of a triangular Bezier surface is also shown to depend on some geometric invariants which decrease exponentially in the subdivision process. Therefore, the degree reduction method can be combined with a subdivision algorithm to generate lower degree approximations which are within some preset error tolerance.
基金This work was supported by the National Natural Science Foundation of China and China Postdoctoral Science Foundation. The secon
文摘A new type of bivariate generalized Ball basis function on a triangle is presented for free-form surface design. Some properties of the basis function are given, then degree elevation, recursive evaluation and some other properties of the generalized Ball surfaces are also derived. It is shown that the proposed recursive evaluation algorithm is more efficient than those of the old surfaces.
基金NKBRSF on Mathematics Mechanics! (grant G1998030600)the National Natural Science Foundation of China! (grants 69603009 and 1
文摘This paper describes practical approaches on how to construct bounding pyramids and bounding cones for triangular Bezier surfaces. Examples are provided to illustrate the process of construction and comparison is made between various surface bounding volumes. Furthermore, as a starting point for the construction, we provide a way to compute hodographs of triangular Bezier surfaces and improve the algorithm for computing the bounding cone of a set of vectors. [ABSTRACT FROM AUTHOR]