Computation of Self-Intersections of Offsets of Be´zier Surface Patches

1997 ◽  
Vol 119 (2) ◽  
pp. 275-283 ◽  
Author(s):  
Takashi Maekawa ◽  
Wonjoon Cho ◽  
Nicholas M. Patrikalakis
Keyword(s):  
Differential Geometry ◽  
Distance Function ◽  
Polynomial Equations ◽  
Intersection Curve ◽  
Parameter Domain ◽  
Surface Patches ◽  
Intersection Curves

Self-intersection of offsets of regular Be´zier surface patches due to local differential geometry and global distance function properties is investigated. The problem of computing starting points for tracing self-intersection curves of offsets is formulated in terms of a system of nonlinear polynomial equations and solved robustly by the interval projected polyhedron algorithm. Trivial solutions are excluded by evaluating the normal bounding pyramids of the surface subpatches mapped from the parameter boxes computed by the polynomial solver with a coarse tolerance. A technique to detect and trace self-intersection curve loops in the parameter domain is also discussed. The method has been successfully tested in tracing complex self-intersection curves of offsets of Be´zier surface patches. Examples illustrate the principal features and robustness characteristics of the method.

scholarly journals Differential geometry of non-transversal intersection curves of three implicit hypersurfaces in Euclidean 4-space

2016 ◽  
Vol 308 ◽  
pp. 20-38 ◽  
Author(s):  
O. Aléssio ◽  
M. Düldül ◽  
B. Uyar Düldül ◽  
Nassar H. Abdel-All ◽  
Sayed Abdel-Naeim Badr
Keyword(s):  
Differential Geometry ◽  
Transversal Intersection ◽  
Intersection Curves

scholarly journals Singular Points of Curves

2018 ◽  
Vol 25 (6) ◽  
pp. 692-710
Author(s):  
Artem D. Uvarov
Keyword(s):  
Iterative Process ◽  
Geometric Modeling ◽  
Global Minimum ◽  
Iterative Algorithms ◽  
Singular Points ◽  
Intersection Curve ◽  
Software Environment ◽  
Advantages And Disadvantages ◽  
Intersection Curves ◽  
Complex Cases

In this paper, we consider the key problem of geometric modeling, connected with the construction of the intersection curves of surfaces. Methods for constructing the intersection curves in complex cases are found: by touching and passing through singular points of surfaces. In the first part of the paper, the problem of determining the tangent line of two surfaces given in parametric form is considered. Several approaches to the solution of the problem are analyzed. The advantages and disadvantages of these approaches are revealed. The iterative algorithms for finding a point on the line of tangency are described. The second part of the paper is devoted to methods for overcoming the difficulties encountered in solving a problem for singular points of intersection curves, in which a regular iterative process is violated. Depending on the type of problem, the author dwells on two methods. The first of them suggests finding singular points of curves without using iterative methods, which reduces the running time of the algorithm of plotting the intersection curve. The second method, considered in the final part of the article, is a numerical method. In this part, the author introduces a function that achieves a global minimum only at singular points of the intersection curves and solves the problem of minimizing this function. The application of this method is very effective in some particular cases, which impose restrictions on the surfaces and their arrangement. In conclusion, this method is considered in the case when the function has such a relief, that in the neighborhood of the minimum point the level surfaces are strongly elongated ellipsoids. All the images given in this article are the result of the work of algorithms on methods proposed by the author. Images are built in the author’s software environment.

A Generalization of Finsler Geometry

1962 ◽  
Vol 14 ◽  
pp. 87-112 ◽  
Author(s):  
J. R. Vanstone
Keyword(s):  
Differential Geometry ◽  
General Theory ◽  
Distance Function ◽  
Riemannian Geometry ◽  
Finsler Geometry ◽  
Riemannian Metric ◽  
Elliptic Partial Differential Equations ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Metric Function

Modern differential geometry may be said to date from Riemann's famous lecture of 1854 (9), in which a distance function of the form F(xi, dxi) = (γij(x)dxidxj½ was proposed. The applications of the consequent geometry were many and varied. Examples are Synge's geometrization of mechanics (15), Riesz’ approach to linear elliptic partial differential equations (10), and the well-known general theory of relativity of Einstein.Meanwhile the results of Caratheodory (4) in the calculus of variations led Finsler in 1918 to introduce a generalization of the Riemannian metric function (6). The geometry which arose was more fully developed by Berwald (2) and Synge (14) about 1925 and later by Cartan (5), Busemann, and Rund. It was then possible to extend the applications of Riemannian geometry.

scholarly journals Twisted Quadrics and Algebraic Submanifolds in $\mathbb {R}^{n}$

2020 ◽  
Vol 23 (4) ◽  
Author(s):  
Gaetano Fiore ◽  
Davide Franco ◽  
Thomas Weber
Keyword(s):  
Differential Geometry ◽  
Level Sets ◽  
Vector Fields ◽  
General Procedure ◽  
Killing Vector ◽  
Polynomial Equations ◽  
De Sitter ◽  
Killing Vector Fields ◽  
Gauss Theorem ◽  
Levi Civita Connection

AbstractWe propose a general procedure to construct noncommutative deformations of an algebraic submanifold M of $\mathbb {R}^{n}$ ℝ n , specializing the procedure [G. Fiore, T. Weber, Twisted submanifolds of$\mathbb {R}^{n}$ ℝ n , arXiv:2003.03854] valid for smooth submanifolds. We use the framework of twisted differential geometry of Aschieri et al. (Class. Quantum Grav. 23, 1883–1911, 2006), whereby the commutative pointwise product is replaced by the ⋆-product determined by a Drinfel’d twist. We actually simultaneously construct noncommutative deformations of all the algebraic submanifolds Mc that are level sets of the fa(x), where fa(x) = 0 are the polynomial equations solved by the points of M, employing twists based on the Lie algebra Ξt of vector fields that are tangent to all the Mc. The twisted Cartan calculus is automatically equivariant under twisted Ξt. If we endow $\mathbb {R}^{n}$ ℝ n with a metric, then twisting and projecting to normal or tangent components commute, projecting the Levi-Civita connection to the twisted M is consistent, and in particular a twisted Gauss theorem holds, provided the twist is based on Killing vector fields. Twisted algebraic quadrics can be characterized in terms of generators and ⋆-polynomial relations. We explicitly work out deformations based on abelian or Jordanian twists of all quadrics in $\mathbb {R}^{3}$ ℝ 3 except ellipsoids, in particular twisted cylinders embedded in twisted Euclidean $\mathbb {R}^{3}$ ℝ 3 and twisted hyperboloids embedded in twisted Minkowski $\mathbb {R}^{3}$ ℝ 3 [the latter are twisted (anti-)de Sitter spaces dS2, AdS2].

scholarly journals Differential geometry of intersection curve of two surfaces

2009 ◽  
Vol 50 ◽  
Author(s):  
Kazimieras Navickis
Keyword(s):  
Differential Geometry ◽  
Euclidean Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Intersection Curve ◽  
Curvature And Torsion

In this this article the differential geometry of intersection curve of two surfaces in the three dimensional euclidean space is considered.In case, curvature and torsion formulas for such curve are defined.

Curvature Connection of Surface Patches around a Common Vertex

2013 ◽  
Vol 353-356 ◽  
pp. 3585-3588
Author(s):  
Qing Xian Meng ◽  
Hui Hui Meng
Keyword(s):  
Differential Geometry ◽  
New Method ◽  
Common Vertex ◽  
Sufficient Condition ◽  
System Of Equations ◽  
Surface Patches

Based on the properties of curvature connection and theory of differential geometry, a sufficient condition of curvature connection between two adjacent surfaces is obtained, and then a new method of curvature connection of surface patches around a common vertex is put forward by using these conditions and consistency. As a result, the obtained surface patches have the lowest degrees, whose degrees are five, and the relevant system of equations can be solved easily.

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