This paper shows the usefulness of discrete differential geometry in global analysis. Using the discrete differential geometry of triangles, we could consider the global structure of closed trajectories (of triangles)...This paper shows the usefulness of discrete differential geometry in global analysis. Using the discrete differential geometry of triangles, we could consider the global structure of closed trajectories (of triangles) on a triangular mesh consisting of congruent isosceles triangles. As an example, we perform global analysis of an Escher-style trick art, i.e., a simpler version of “Ascending and Descending”. After defining the local structure on the trick art, we analyze its global structure and attribute its paradox to a singular point (i.e., a singular triangle) at the center. Then, the endless “Penrose stairs” is described as a closed trajectory around the isolated singular point. The approach fits well with graphical projection and gives a simple and intuitive example of the interaction between global and local structures. We could deal with higher dimensional objects as well by considering n-simplices (n > 2) instead of triangles.展开更多
An algebraic differential variety is defined as the zero-set of a differentialpolynomial set,and algebraic differential geometry is devoted to the study of such varieties.We give various decomposition formulas for the...An algebraic differential variety is defined as the zero-set of a differentialpolynomial set,and algebraic differential geometry is devoted to the study of such varieties.We give various decomposition formulas for the structures of such zero-sets which imply inparticular,the unique decomposition of an algebraic differential variety into its irreduciblecomponents.These formulas will find applications in various directions including mechanicaltheorem-proving of differential geometries.展开更多
This research is concerned with coordinated standoff tracking, and a guidance law against a moving target is proposed by using differential geometry. We first present the geometry between the unmanned aircraft(UA) and...This research is concerned with coordinated standoff tracking, and a guidance law against a moving target is proposed by using differential geometry. We first present the geometry between the unmanned aircraft(UA) and the target to obtain the convergent solution of standoff tracking when the speed ratio of the UA to the target is larger than one. Then, the convergent solution is used to guide the UA onto the standoff tracking geometry. We propose an improved guidance law by adding a derivative term to the relevant algorithm. To keep the phase angle difference of multiple UAs, we add a second derivative term to the relevant control law. Simulations are done to demonstrate the feasibility and performance of the proposed approach. The proposed algorithm can achieve coordinated control of multiple UAs with its simplicity and stability in terms of the standoff distance and phase angle difference.展开更多
Quantum electron states, in the case of an improved Dirac equation, are linked to the Christoffel symbols of the connection of space-time geometry. Each solution of the wave equation, in the case of the hydrogen atom ...Quantum electron states, in the case of an improved Dirac equation, are linked to the Christoffel symbols of the connection of space-time geometry. Each solution of the wave equation, in the case of the hydrogen atom induces a connection which is completely calculated. This allows us to discover the global and chiral properties of the space-time connection, with spin 2.展开更多
Proteins perform a variety of functions in living organisms and their functions are largely determined by their shape. In this paper, we propose a novel mathematical method for designing protein-like molecules of a gi...Proteins perform a variety of functions in living organisms and their functions are largely determined by their shape. In this paper, we propose a novel mathematical method for designing protein-like molecules of a given shape. In the mathematical model, molecules are represented as loops of n-simplices (2-simplices are triangles and 3-simplices are tetrahedra). We design a new molecule of a given shape by patching together a set of smaller molecules that cover the shape. The covering set of small molecules is defined using a binary relation between sets of molecules. A new molecule is then obtained as a sum of the smaller molecules, where addition of molecules is defined using transformations acting on a set of (n + 1)-dimensional cones. Due to page limitations, only the two-dimensional case (i.e., loops of triangles) is considered. No prior knowledge of Sheaf Theory, Category Theory, or Protein Science is required. The author hopes that this paper will encourage further collaboration between Mathematics and Protein Science.展开更多
We present a tensor description of Euclidean spaces that emphasizes the use of geometric vectors which leads to greater geometric insight and a higher degree of organization in analytical expressions. We demonstrate t...We present a tensor description of Euclidean spaces that emphasizes the use of geometric vectors which leads to greater geometric insight and a higher degree of organization in analytical expressions. We demonstrate the effectiveness of the approach by proving a number of integral identities with vector integrands. The presented approach may be aptly described as absolute vector calculus or as vector tensor calculus.展开更多
This study proposes a scheme for state estimation and,consequently,fault diagnosis in nonlinear systems.Initially,an optimal nonlinear observer is designed for nonlinear systems subject to an actuator or plant fault.B...This study proposes a scheme for state estimation and,consequently,fault diagnosis in nonlinear systems.Initially,an optimal nonlinear observer is designed for nonlinear systems subject to an actuator or plant fault.By utilizing Lyapunov's direct method,the observer is proved to be optimal with respect to a performance function,including the magnitude of the observer gain and the convergence time.The observer gain is obtained by using approximation of Hamilton-Jacobi-Bellman(HJB)equation.The approximation is determined via an online trained neural network(NN).Next a class of affine nonlinear systems is considered which is subject to unknown disturbances in addition to fault signals.In this case,for each fault the original system is transformed to a new form in which the proposed optimal observer can be applied for state estimation and fault detection and isolation(FDI).Simulation results of a singlelink flexible joint robot(SLFJR)electric drive system show the effectiveness of the proposed methodology.展开更多
This paper proposes a novel category theoretic approach to describe protein’s shape, <i>i.e.</i>, a description of their shape by a set of algebraic equations. The focus of the approach is on the relation...This paper proposes a novel category theoretic approach to describe protein’s shape, <i>i.e.</i>, a description of their shape by a set of algebraic equations. The focus of the approach is on the relations between proteins, rather than on the proteins themselves. Knowledge of category theory is not required as mathematical notions are defined concretely. In this paper, proteins are represented as closed trajectories (<i>i.e.</i>, loops) of flows of triangles. The relations between proteins are defined using the fusion and fission of loops of triangles, where allostery occurs naturally. The shape of a protein is then described with quantities that are measurable with unity elements called “unit loops”. That is, protein’s shape is described with the loops that are obtained by the fusion of unit loops. Measurable loops are called “integral”. In the approach, the unit loops play a role similar to the role “1” plays in the set Z of integers. In particular, the author considers two categories of loops, the “integral” loops and the “rational” loops. Rational loops are then defined using algebraic equations with “integral loop” coefficients. Because of the approach, our theory has some similarities to quantum mechanics, where only observable quantities are admitted in physical theory. The author believes that this paper not only provides a new perspective on protein engineering, but also promotes further collaboration between biology and other disciplines.展开更多
This paper proposes a novel application of cohomology to protein structure analysis. Since proteins interact each other by forming transient protein complexes, their shape (e.g., shape complementarity) plays an import...This paper proposes a novel application of cohomology to protein structure analysis. Since proteins interact each other by forming transient protein complexes, their shape (e.g., shape complementarity) plays an important role in their functions. In our mathematical toy models, proteins are represented as a loop of triangles (2D model) or tetrahedra (3D model), where their interactions are defined as fusion of loops. The purpose of this paper is to describe the conditions for loop fusion using the language of cohomology. In particular, this paper uses cohomology to describe the conditions for “allosteric regulation”, which has been attracted attention in safer drug discovery. I hope that this paper will provide a new perspective on the mechanism of allosteric regulation. Advantages of the model include its topological nature. That is, we can deform the shape of loops by deforming the shape of triangles (or tetrahedra) as long as their folded structures are preserved. Another advantage is the simplicity of the “allosteric regulation” mechanism of the model. Furthermore, the effect of the “post-translational modification” can be understood as a resolution of singularities of a flow of triangles (or tetrahedra). No prior knowledge of either protein science, exterior calculus, or cohomology theory is required. The author hopes that this paper will facilitate the interaction between mathematics and protein science.展开更多
Along with the construction of non-Lorentz-invariant effective field theories, recent studies which are based on geometric models of Finsler space-time become more and more popular. In this respect, the Finslerian app...Along with the construction of non-Lorentz-invariant effective field theories, recent studies which are based on geometric models of Finsler space-time become more and more popular. In this respect, the Finslerian approach to the problem of Lorentz symmetry violation is characterized by the fact that the violation of Lorentz symmetry is not accompanied by a violation of relativistic symmetry. That means, in particular, that preservation of relativistic symmetry can be considered as a rigorous criterion of the viability for any non-Lorentz-invariant effective field theory. Although this paper has a review character, it contains (with few exceptions) only those results on Finsler extensions of relativity theory, that were obtained by the authors.展开更多
A charming feature of symplectic geometry is that it is at the crossroad of many other mathematical disciplines. In this article we review the basic notions with examples of symplectic structures and show the connecti...A charming feature of symplectic geometry is that it is at the crossroad of many other mathematical disciplines. In this article we review the basic notions with examples of symplectic structures and show the connections of symplectic geometry with the various branches of differential geometry using important theorems.展开更多
In this work, we discuss the topological transformation of quantum dynamics by showing the wave dynamics of a quantum particle on different types of topological structures in various dimensions from the fundamental po...In this work, we discuss the topological transformation of quantum dynamics by showing the wave dynamics of a quantum particle on different types of topological structures in various dimensions from the fundamental polygons of the corresponding universal covering spaces. This is not the view from different perspectives of an observer who simply uses different coordinate systems to describe the same physical phenomenon but rather possible geometric and topological structures that quantum particles are endowed with when they are identified with differentiable manifolds that are embedded or immersed in Euclidean spaces of higher dimension. We present our discussions in the form of Bohr model in one, two and three dimensions using linear wave equations. In one dimension, the fundamental polygon is an interval and the universal covering space is the straight line and in this case the standing wave on a finite string is transformed into the standing wave on a circle which can be applied into the Bohr model of the hydrogen atom. In two dimensions, the fundamental polygon is a square and the universal covering space is the plane and in this case, the standing wave on the square is transformed into the standing wave on different surfaces that can be formed by gluing opposite sides of the square, which include a 2-sphere, a 2-torus, a Klein bottle and a projective plane. In three dimensions, the fundamental polygon is a cube and the universal covering space is the three-dimensional Euclidean space. It is shown that a 3-torus and the manifold K?× S1?defined as the product of a Klein bottle and a circle can be constructed by gluing opposite faces of a cube. Therefore, in three-dimensions, the standing wave on a cube is transformed into the standing wave on a 3-torus or on the manifold K?× S1. We also suggest that the mathematical degeneracy may play an important role in quantum dynamics and be associated with the concept of wavefunction collapse in quantum mechanics.展开更多
A detailed geometric analysis of spherical triboelectric nanogenerators is presented.In comparison with earlier works on spherical triboelectric generators,the general case where the moving dielectric rolls on the ins...A detailed geometric analysis of spherical triboelectric nanogenerators is presented.In comparison with earlier works on spherical triboelectric generators,the general case where the moving dielectric rolls on the inside surface of the larger sphere of the TENG is discussed in terms of maximum energy harvesting.An optimization analysis of geometrical parameters allows various cases of electrode geometry,either in the form of a spherical circle,spherical ellipse,spherical rectangle,or spherical isosceles trapezium,to be solved.The analytical insight and computational effective models provided by differential geometry make the mathematical model superior compared to standard three-dimensional(3D)numerical methods.展开更多
The authors study rotational hypersurfaces with constant Gauss-Kronecker curvature in R^(n).They solve the ODE associated with the generating curve of such hypersurface using integral expressions and obtain several ge...The authors study rotational hypersurfaces with constant Gauss-Kronecker curvature in R^(n).They solve the ODE associated with the generating curve of such hypersurface using integral expressions and obtain several geometric properties of such hypersurfaces.In particular,they discover a class of non-compact rotational hypersurfaces with constant and negative Gauss-Kronecker curvature and finite volume,which can be seen as the higher-dimensional generalization of the pseudo-sphere.展开更多
Existing shape models with spherical topology are typically designed either in the discrete domain using interpolating polygon meshes or in the continuous domain using smooth but non-interpolating schemes such as subd...Existing shape models with spherical topology are typically designed either in the discrete domain using interpolating polygon meshes or in the continuous domain using smooth but non-interpolating schemes such as subdivision or NURBS. Both polygon models and subdivision methods require a large number of parameters to model smooth surfaces.NURBS need fewer parameters but have a complicated rational expression and non-uniform shifts in their formulation. We present a new method to construct deformable closed surfaces, which includes exact spheres, by combining the best of two worlds: a smooth, interpolating model with a continuously varying tangent plane and well-defined curvature at every point on the surface. Our formulation is considerably simpler than NURBS and requires fewer parameters than polygon meshes. We demonstrate the generality of our method with applications including intuitive user-interactive shape modeling,continuous surface deformation, shape morphing,reconstruction of shapes from parameterized point clouds, and fast iterative shape optimization for image segmentation. Comparisons with discrete methods and non-interpolating approaches highlight the advantages of our framework.展开更多
Elliptical splats are used to represent and render the isosurface of volume data. The method consists of two steps. The first step is to extract points on the isosurface by looking up the case table. In the second ste...Elliptical splats are used to represent and render the isosurface of volume data. The method consists of two steps. The first step is to extract points on the isosurface by looking up the case table. In the second step, properties of splats are computed based on local geometry. Rendering is achieved using surface splatting algorithm. The obtained results show that the extraction time of isosurfaces can be reduced by a factor of three. So our approach is more appropriate for interactive visualization of large medical data than the classical marching cubes (MC) technique.展开更多
We present a coupled thermal-fluid model for Benard-Marangoni convection in a three-dimensional fluid layer.The governing equations are derived in detail for two reasons:first,we do not assume a flat free surface as c...We present a coupled thermal-fluid model for Benard-Marangoni convection in a three-dimensional fluid layer.The governing equations are derived in detail for two reasons:first,we do not assume a flat free surface as commonly done;and second,we prepare for the use of flexible discretizations.The governing equations are discretized using spectral elements in space and an operator splitting approach in time.Since we are here primarily interested in steady state solutions,the focus is on the spatial discretization.The overall computational approach is very attractive to use for several reasons:(i)the solution can be expected to have a high degree of regularity,and rapid convergence can be expected;(ii)the spectral element decomposition automatically gives a convenient parameterization of the free surface that allows powerful results from differential geometry to easily be exploited;(iii)free surface deformation can readily be included;(iv)both normal and tangential stresses are conveniently accounted for through a single surface integral;(v)no differentiation of the surface tension is necessary in order to include thermocapillary effects(due to integration-byparts twice);(vi)the geometry representation of the free surface need only be C0 across element boundaries even though curvature effects are included.Three-dimensional simulation results are presented,including the free surface deflection due to buoyancy and thermocapillary effects.展开更多
This paper develops a novel approach and some main results on the varying-speed missile guided by pure proportional navigation(PPN)against a stationary target in the planar interception problem.The missile kinematic e...This paper develops a novel approach and some main results on the varying-speed missile guided by pure proportional navigation(PPN)against a stationary target in the planar interception problem.The missile kinematic equation is established in the arc-length domain based on the differential geometry theory,which eliminates the influence of time-varying missile speed.Then,the closed-form solutions of line-of-sight(LOS)rate,leading angle,closing speed,and curvature command are derived in the arc-length domain.The performance of the varyingspeed missile is analyzed,including the maximum relative distance,maximum curvature command,accurate path-to-go,and curvature increment.Additionally,the capture region is obtained considering the missile maneuvering acceleration limit.These new theoretical results could be extended to improve the performance of existing guidance laws designed under the constant-speed assumption.展开更多
文摘This paper shows the usefulness of discrete differential geometry in global analysis. Using the discrete differential geometry of triangles, we could consider the global structure of closed trajectories (of triangles) on a triangular mesh consisting of congruent isosceles triangles. As an example, we perform global analysis of an Escher-style trick art, i.e., a simpler version of “Ascending and Descending”. After defining the local structure on the trick art, we analyze its global structure and attribute its paradox to a singular point (i.e., a singular triangle) at the center. Then, the endless “Penrose stairs” is described as a closed trajectory around the isolated singular point. The approach fits well with graphical projection and gives a simple and intuitive example of the interaction between global and local structures. We could deal with higher dimensional objects as well by considering n-simplices (n > 2) instead of triangles.
文摘An algebraic differential variety is defined as the zero-set of a differentialpolynomial set,and algebraic differential geometry is devoted to the study of such varieties.We give various decomposition formulas for the structures of such zero-sets which imply inparticular,the unique decomposition of an algebraic differential variety into its irreduciblecomponents.These formulas will find applications in various directions including mechanicaltheorem-proving of differential geometries.
基金Project supported by the National Natural Science Foundation of China(Nos.61273327 and 71201076)the Key Pre-research Fund of the PLA General Armament Department(No.9140A06050213BQX)the Natural Science Foundation of Jiangsu Province,China(No.BK2011564)
文摘This research is concerned with coordinated standoff tracking, and a guidance law against a moving target is proposed by using differential geometry. We first present the geometry between the unmanned aircraft(UA) and the target to obtain the convergent solution of standoff tracking when the speed ratio of the UA to the target is larger than one. Then, the convergent solution is used to guide the UA onto the standoff tracking geometry. We propose an improved guidance law by adding a derivative term to the relevant algorithm. To keep the phase angle difference of multiple UAs, we add a second derivative term to the relevant control law. Simulations are done to demonstrate the feasibility and performance of the proposed approach. The proposed algorithm can achieve coordinated control of multiple UAs with its simplicity and stability in terms of the standoff distance and phase angle difference.
文摘Quantum electron states, in the case of an improved Dirac equation, are linked to the Christoffel symbols of the connection of space-time geometry. Each solution of the wave equation, in the case of the hydrogen atom induces a connection which is completely calculated. This allows us to discover the global and chiral properties of the space-time connection, with spin 2.
文摘Proteins perform a variety of functions in living organisms and their functions are largely determined by their shape. In this paper, we propose a novel mathematical method for designing protein-like molecules of a given shape. In the mathematical model, molecules are represented as loops of n-simplices (2-simplices are triangles and 3-simplices are tetrahedra). We design a new molecule of a given shape by patching together a set of smaller molecules that cover the shape. The covering set of small molecules is defined using a binary relation between sets of molecules. A new molecule is then obtained as a sum of the smaller molecules, where addition of molecules is defined using transformations acting on a set of (n + 1)-dimensional cones. Due to page limitations, only the two-dimensional case (i.e., loops of triangles) is considered. No prior knowledge of Sheaf Theory, Category Theory, or Protein Science is required. The author hopes that this paper will encourage further collaboration between Mathematics and Protein Science.
文摘We present a tensor description of Euclidean spaces that emphasizes the use of geometric vectors which leads to greater geometric insight and a higher degree of organization in analytical expressions. We demonstrate the effectiveness of the approach by proving a number of integral identities with vector integrands. The presented approach may be aptly described as absolute vector calculus or as vector tensor calculus.
文摘This study proposes a scheme for state estimation and,consequently,fault diagnosis in nonlinear systems.Initially,an optimal nonlinear observer is designed for nonlinear systems subject to an actuator or plant fault.By utilizing Lyapunov's direct method,the observer is proved to be optimal with respect to a performance function,including the magnitude of the observer gain and the convergence time.The observer gain is obtained by using approximation of Hamilton-Jacobi-Bellman(HJB)equation.The approximation is determined via an online trained neural network(NN).Next a class of affine nonlinear systems is considered which is subject to unknown disturbances in addition to fault signals.In this case,for each fault the original system is transformed to a new form in which the proposed optimal observer can be applied for state estimation and fault detection and isolation(FDI).Simulation results of a singlelink flexible joint robot(SLFJR)electric drive system show the effectiveness of the proposed methodology.
文摘This paper proposes a novel category theoretic approach to describe protein’s shape, <i>i.e.</i>, a description of their shape by a set of algebraic equations. The focus of the approach is on the relations between proteins, rather than on the proteins themselves. Knowledge of category theory is not required as mathematical notions are defined concretely. In this paper, proteins are represented as closed trajectories (<i>i.e.</i>, loops) of flows of triangles. The relations between proteins are defined using the fusion and fission of loops of triangles, where allostery occurs naturally. The shape of a protein is then described with quantities that are measurable with unity elements called “unit loops”. That is, protein’s shape is described with the loops that are obtained by the fusion of unit loops. Measurable loops are called “integral”. In the approach, the unit loops play a role similar to the role “1” plays in the set Z of integers. In particular, the author considers two categories of loops, the “integral” loops and the “rational” loops. Rational loops are then defined using algebraic equations with “integral loop” coefficients. Because of the approach, our theory has some similarities to quantum mechanics, where only observable quantities are admitted in physical theory. The author believes that this paper not only provides a new perspective on protein engineering, but also promotes further collaboration between biology and other disciplines.
文摘This paper proposes a novel application of cohomology to protein structure analysis. Since proteins interact each other by forming transient protein complexes, their shape (e.g., shape complementarity) plays an important role in their functions. In our mathematical toy models, proteins are represented as a loop of triangles (2D model) or tetrahedra (3D model), where their interactions are defined as fusion of loops. The purpose of this paper is to describe the conditions for loop fusion using the language of cohomology. In particular, this paper uses cohomology to describe the conditions for “allosteric regulation”, which has been attracted attention in safer drug discovery. I hope that this paper will provide a new perspective on the mechanism of allosteric regulation. Advantages of the model include its topological nature. That is, we can deform the shape of loops by deforming the shape of triangles (or tetrahedra) as long as their folded structures are preserved. Another advantage is the simplicity of the “allosteric regulation” mechanism of the model. Furthermore, the effect of the “post-translational modification” can be understood as a resolution of singularities of a flow of triangles (or tetrahedra). No prior knowledge of either protein science, exterior calculus, or cohomology theory is required. The author hopes that this paper will facilitate the interaction between mathematics and protein science.
基金partially supported by the Sectorial Operational Program Human Resources Development(SOP HRD)financed from the European Social Fund and by the Romanian Government under the Project number POSDRU/89/1.5/S/59323.
文摘Along with the construction of non-Lorentz-invariant effective field theories, recent studies which are based on geometric models of Finsler space-time become more and more popular. In this respect, the Finslerian approach to the problem of Lorentz symmetry violation is characterized by the fact that the violation of Lorentz symmetry is not accompanied by a violation of relativistic symmetry. That means, in particular, that preservation of relativistic symmetry can be considered as a rigorous criterion of the viability for any non-Lorentz-invariant effective field theory. Although this paper has a review character, it contains (with few exceptions) only those results on Finsler extensions of relativity theory, that were obtained by the authors.
文摘A charming feature of symplectic geometry is that it is at the crossroad of many other mathematical disciplines. In this article we review the basic notions with examples of symplectic structures and show the connections of symplectic geometry with the various branches of differential geometry using important theorems.
文摘In this work, we discuss the topological transformation of quantum dynamics by showing the wave dynamics of a quantum particle on different types of topological structures in various dimensions from the fundamental polygons of the corresponding universal covering spaces. This is not the view from different perspectives of an observer who simply uses different coordinate systems to describe the same physical phenomenon but rather possible geometric and topological structures that quantum particles are endowed with when they are identified with differentiable manifolds that are embedded or immersed in Euclidean spaces of higher dimension. We present our discussions in the form of Bohr model in one, two and three dimensions using linear wave equations. In one dimension, the fundamental polygon is an interval and the universal covering space is the straight line and in this case the standing wave on a finite string is transformed into the standing wave on a circle which can be applied into the Bohr model of the hydrogen atom. In two dimensions, the fundamental polygon is a square and the universal covering space is the plane and in this case, the standing wave on the square is transformed into the standing wave on different surfaces that can be formed by gluing opposite sides of the square, which include a 2-sphere, a 2-torus, a Klein bottle and a projective plane. In three dimensions, the fundamental polygon is a cube and the universal covering space is the three-dimensional Euclidean space. It is shown that a 3-torus and the manifold K?× S1?defined as the product of a Klein bottle and a circle can be constructed by gluing opposite faces of a cube. Therefore, in three-dimensions, the standing wave on a cube is transformed into the standing wave on a 3-torus or on the manifold K?× S1. We also suggest that the mathematical degeneracy may play an important role in quantum dynamics and be associated with the concept of wavefunction collapse in quantum mechanics.
文摘A detailed geometric analysis of spherical triboelectric nanogenerators is presented.In comparison with earlier works on spherical triboelectric generators,the general case where the moving dielectric rolls on the inside surface of the larger sphere of the TENG is discussed in terms of maximum energy harvesting.An optimization analysis of geometrical parameters allows various cases of electrode geometry,either in the form of a spherical circle,spherical ellipse,spherical rectangle,or spherical isosceles trapezium,to be solved.The analytical insight and computational effective models provided by differential geometry make the mathematical model superior compared to standard three-dimensional(3D)numerical methods.
基金Peking University for providing financial support for the first author
文摘The authors study rotational hypersurfaces with constant Gauss-Kronecker curvature in R^(n).They solve the ODE associated with the generating curve of such hypersurface using integral expressions and obtain several geometric properties of such hypersurfaces.In particular,they discover a class of non-compact rotational hypersurfaces with constant and negative Gauss-Kronecker curvature and finite volume,which can be seen as the higher-dimensional generalization of the pseudo-sphere.
基金funded by the Swiss National Science Foundation under Grant 200020-162343
文摘Existing shape models with spherical topology are typically designed either in the discrete domain using interpolating polygon meshes or in the continuous domain using smooth but non-interpolating schemes such as subdivision or NURBS. Both polygon models and subdivision methods require a large number of parameters to model smooth surfaces.NURBS need fewer parameters but have a complicated rational expression and non-uniform shifts in their formulation. We present a new method to construct deformable closed surfaces, which includes exact spheres, by combining the best of two worlds: a smooth, interpolating model with a continuously varying tangent plane and well-defined curvature at every point on the surface. Our formulation is considerably simpler than NURBS and requires fewer parameters than polygon meshes. We demonstrate the generality of our method with applications including intuitive user-interactive shape modeling,continuous surface deformation, shape morphing,reconstruction of shapes from parameterized point clouds, and fast iterative shape optimization for image segmentation. Comparisons with discrete methods and non-interpolating approaches highlight the advantages of our framework.
基金the Program of Advance Research between France and Chinese(No.PRA SI 03-03)the Region Rhone-Alpes of France within the Project"MIRA Research 2003"the Project of Image Guided Surgery of Shanghai,China(No.045115001)
文摘Elliptical splats are used to represent and render the isosurface of volume data. The method consists of two steps. The first step is to extract points on the isosurface by looking up the case table. In the second step, properties of splats are computed based on local geometry. Rendering is achieved using surface splatting algorithm. The obtained results show that the extraction time of isosurfaces can be reduced by a factor of three. So our approach is more appropriate for interactive visualization of large medical data than the classical marching cubes (MC) technique.
基金The authors would like to thank the Norwegian University of Science and Technology for the financial support of this project.
文摘We present a coupled thermal-fluid model for Benard-Marangoni convection in a three-dimensional fluid layer.The governing equations are derived in detail for two reasons:first,we do not assume a flat free surface as commonly done;and second,we prepare for the use of flexible discretizations.The governing equations are discretized using spectral elements in space and an operator splitting approach in time.Since we are here primarily interested in steady state solutions,the focus is on the spatial discretization.The overall computational approach is very attractive to use for several reasons:(i)the solution can be expected to have a high degree of regularity,and rapid convergence can be expected;(ii)the spectral element decomposition automatically gives a convenient parameterization of the free surface that allows powerful results from differential geometry to easily be exploited;(iii)free surface deformation can readily be included;(iv)both normal and tangential stresses are conveniently accounted for through a single surface integral;(v)no differentiation of the surface tension is necessary in order to include thermocapillary effects(due to integration-byparts twice);(vi)the geometry representation of the free surface need only be C0 across element boundaries even though curvature effects are included.Three-dimensional simulation results are presented,including the free surface deflection due to buoyancy and thermocapillary effects.
基金supported by the National Natural Science Foundation of China(Grant No.12002370).
文摘This paper develops a novel approach and some main results on the varying-speed missile guided by pure proportional navigation(PPN)against a stationary target in the planar interception problem.The missile kinematic equation is established in the arc-length domain based on the differential geometry theory,which eliminates the influence of time-varying missile speed.Then,the closed-form solutions of line-of-sight(LOS)rate,leading angle,closing speed,and curvature command are derived in the arc-length domain.The performance of the varyingspeed missile is analyzed,including the maximum relative distance,maximum curvature command,accurate path-to-go,and curvature increment.Additionally,the capture region is obtained considering the missile maneuvering acceleration limit.These new theoretical results could be extended to improve the performance of existing guidance laws designed under the constant-speed assumption.
