共查询到10条相似文献,搜索用时 156 毫秒
1.
Summary.
Motion by (weighted) mean curvature is a geometric evolution law for
surfaces, representing steepest descent with respect to (an)isotropic
surface energy. It has been proposed that this motion could
be computed by solving the analogous evolution law using a
``crystalline' approximation to the surface energy. We present the
first convergence analysis for a numerical scheme of this type. Our
treatment is restricted to one dimensional surfaces (curves in the
plane) which are graphs. In this context, the scheme amounts to a new
algorithm for solving quasilinear parabolic equations in one space
dimension.
Received January 28, 1993 相似文献
2.
Eckhard M. S. Hitzer 《Advances in Applied Clifford Algebras》2003,13(1):11-45
This paper introduces the new interactive Java sketching software KamiWaAi, recently developed at the University of Fukui.
Its graphical user interface enables the user without any knowledge of both mathematics or computer science, to do full three
dimensional “drawings” on the screen. The resulting constructions can be reshaped interactively by dragging its points over
the screen. The programming approach is new. KamiWaAi implements geometric objects like points, lines, circles, spheres, etc.
directly as software objects (Java classes) of the same name. These software objects are geometric entities mathematically
defined and manipulated in a conformal geometric algebra, combining the five dimensions of origin, three space and infinity.
Simple geometric products in this algebra represent geometric unions, intersections, arbitrary rotations and translations,
projections, distance, etc. To ease the coordinate free and matrix free implementation of this fundamental geometric product,
a new algebraic three level approach is presented. Finally details about the Java classes of the new GeometricAlgebra software
package and their associated methods are given. KamiWaAi is available for free internet download. 相似文献
3.
In this paper, we propose a new 2D segmentation model including geometric constraints, namely interpolation conditions, to detect objects in a given image. We propose to apply the deformable models to an explicit function using the level set approach (Osher and Sethian [24]); so, we avoid the classical problem of parameterization of both segmentation representation and interpolation conditions. Furthermore, we allow this representation to have topological changes. A problem of energy minimization on a closed subspace of a Hilbert space is defined and introducing Lagrange multipliers enables us to formulate the corresponding variational problem with interpolation conditions. Thus the explicit function evolves, while minimizing the energy and it stops evolving when the desired outlines of the object to detect are reached. The stopping term, as in the classical deformable models, is related to the gradient of the image. Numerical results are given.
AMS subject classification 74G65, 46-xx, 92C55 相似文献
4.
It iswell-known that the positionvector function is themost basic geometric object for a surface immersed in the three dimensional Euclidean space $\mathbb{E}^3 $ . In 2001, B.-Y. Chen defined constant ratio hypersurfaces in Euclidean n-spaces. Independently, in 2010, by using another approach in dimension 3, the second author classified constant slope surfaces. In this paper, we extend this concept in order to study surfaces with the property that the tangential component of the position vector is a principal direction on the surface. 相似文献
5.
Theodoros Vlachos 《manuscripta mathematica》2008,126(2):201-230
We deal with minimal surfaces in a sphere and investigate certain invariants of geometric significance, the Hopf differentials,
which are defined in terms of the complex structure and the higher fundamental forms. We discuss the holomorphicity of Hopf
differentials and provide a geometric interpretation for it in terms of the higher curvature ellipses. This motivates the
study of a class of minimal surfaces, which we call exceptional. We show that exceptional minimal surfaces are related to
Lawson’s conjecture regarding the Ricci condition. Indeed, we prove that, under certain conditions, compact minimal surfaces
in spheres which satisfy the Ricci condition are exceptional. Thus, under these conditions, the proof of Lawson’s conjecture
is reduced to its confirmation for exceptional minimal surfaces. In fact, we provide an affirmative answer to Lawson’s conjecture
for exceptional minimal surfaces in odd dimensional spheres or in S
4m
. 相似文献
6.
Benedict Geihe Martin Lenz Martin Rumpf Rüdiger Schultz 《Mathematical Programming》2013,141(1-2):383-403
Shape optimization of the fine scale geometry of elastic objects is investigated under stochastic loading. Thus, the object geometry is described via parametrized geometric details placed on a regular lattice. Here, in a two dimensional set up we focus on ellipsoidal holes as the fine scale geometric details described by the semiaxes and their orientation. Optimization of a deterministic cost functional as well as stochastic loading with risk neutral and risk averse stochastic cost functionals are discussed. Under the assumption of linear elasticity and quadratic objective functions the computational cost scales linearly in the number of basis loads spanning the possibly large set of all realizations of the stochastic loading. The resulting shape optimization algorithm consists of a finite dimensional, constraint optimization scheme where the cost functional and its gradient are evaluated applying a boundary element method on the fine scale geometry. Various numerical results show the spatial variation of the geometric domain structures and the appearance of strongly anisotropic patterns. 相似文献
7.
The objective of the work presented in this paper is an attempt at solving and transforming of the known from the classical mechanics three dimensional – single mass mathematical and mechanical vibration models in a higher order dimensional space with any virtual sectional curvature – positive or negative, constant or variable. The object of the investigation is a class of three dimensional surfaces. The aims of the work presented in this paper are to illustrate the performance of the common algorithm in three dimensional linear motion transformation, that means to transform 3D space in a higher order dimensional space and a comparison is derived on the behavior of the common algorithm depending on the surface properties. A characterization of the Riemannian Manifolds is performed by means of curvature operators in the three dimensional solution. The computer codes Mathematica and MATLAB are used in the numerical simulation. The system motion is investigated in a 3-D qualitative aspect in time and frequency domain. The application can be in topology when geodesists make snap shots of the surface profile, then the curved lines can be analyzed and transformed in the desired space dimension. Any kind of a trajectory of motion can be transformed successfully in a higher order dimensional space and vice verse by means of applying of the common algorithm. 相似文献
8.
The aim of this paper is to present and discuss an approach based on the integration of the boundary element method (BEM) with β-spline geometric modeling of the different surfaces involved in the external bone remodeling phenomena. The purpose of combining these two techniques is to avoid the jagged edges shapes and thus, to increase the convergence speed of the bone remodeling function. In this study, the external bone remodeling model proposed by Fridez et al. [P. Fridez, L. Rakotomanana, A. Terrier, P.F. Leyvraz, Three dimensional model of bone external adaptation, Comput. Methods Biomech. Biomed. Eng. 2 (1998) 189–196] is used. This model shows the change of the external bone surface remodeling at a boundary point, as a function of the stimulus variable Ψ. This variable is related to the stress tensor and the normal vector to that point. The β-spline surfaces were used because they are simple and reliable to smooth the contour by using the less possible number of geometric constraints. Some numerical examples are presented and discussed in order to show the versatility of the proposed approach. 相似文献
9.
10.
Wolfgang Carl 《Foundations of Computational Mathematics》2016,16(5):1115-1150
This paper studies a Laplace operator on semi-discrete surfaces. A semi-discrete surface is represented by a mapping into three-dimensional Euclidean space possessing one discrete variable and one continuous variable. It can be seen as a limit case of a quadrilateral mesh, or as a semi-discretization of a smooth surface. Laplace operators on both smooth and discrete surfaces have been an object of interest for a long time, also from the viewpoint of applications. There are a wealth of geometric objects available immediately once a Laplacian is defined, e.g., the mean curvature normal. We define our semi-discrete Laplace operator to be the limit of a discrete Laplacian on a quadrilateral mesh, which converges to the semi-discrete surface. The main result of this paper is that this limit exists under very mild regularity assumptions. Moreover, we show that the semi-discrete Laplace operator inherits several important properties from its discrete counterpart, like symmetry, positive semi-definiteness, and linear precision. We also prove consistency of the semi-discrete Laplacian, meaning that it converges pointwise to the Laplace–Beltrami operator, when the semi-discrete surface converges to a smooth one. This result particularly implies consistency of the corresponding discrete scheme. 相似文献