Śrı̄ Yantra Geometry

1999

This paper presents an approach for solving geometric constraint problems using auxiliary constructions. This approach converts a description of constrained geometry into a sequential ruler-and-compass construction, thereby converting variational geometry constraints into parametric geometry. The solution strategy is based on constructing auxiliary geometry and new constraints by systematically applying transformations to parts of the original constraint problem. The transformation of constraints results in creation of new constraint chains, yielding new paths in the constraint graph. Use of auxiliary constraints can thus decompose strongly connected components in the original constraint graph and provide an additional method for breaking loops due to simultaneous reference. The approach is demonstrated in two dimensions. Fig 12. (a) A variational geometry problem with triple symmetry, (b) transcript of automated construction process.

Discrete and Computational Geometry, 2010

Let L be a set of n lines in R d , for d ≥ 3. A joint of L is a point incident to at least d lines of L, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum possible number of joints of L is Θ(n d/(d−1)). For d = 3, this is a considerable simplification of the orignal algebraic proof of Guth and Katz [9], and of the follow-up simpler proof of Elekes et al. [6]. Some extensions, e.g., to the case of joints of algebraic curves, are also presented. Let L be a set of n lines in R d , for d ≥ 3. A joint of L is a point incident to at least d lines of L, not all in a common hyperplane. A simple construction, using the axis-parallel lines in a k × k × • • • × k grid, for k = Θ(n 1/(d−1)), has dk d−1 = Θ(n) lines and k d = Θ(n d/(d−1)) joints. In this paper we prove that this is a general upper bound. That is: Theorem 1 The maximum possible number of joints in a set of n lines in R d is Θ(n d/(d−1)). Background. The problem of bounding the number of joints, for the 3-dimensional case, has been around for almost 20 years [3, 7, 11] (see also [2, Chapter 7.1, Problem 4]), and, until very recently, the best known upper bound, established by Sharir and Feldman [7], was O(n 1.6232). The proof techniques were rather complicated, involving a battery of tools from combinatorial geometry, including forbidden subgraphs in extremal graph theory, space decomposition techniques, and some basic results in the geometry of lines in space (e.g., Plücker coordinates). Wolff [12] observed a connection between the problem of counting joints to the Kakeya problem. Bennett et al. [1] exploited this connection and proved an upper bound on the number of so-called θ-transverse joints in R 3 , namely, joints incident to at least one triple

Journal of Intelligent & Robotic Systems, 1994

We superimpose weaving patterns on planar line arrangements, and we face the question when they can be realized by lines in 3-space. Using the combinatorial type of the given lines in the plane we derive a class of nonrealizable weavings.

Computer Applications in Engineering Education, 2011

Dynamic Geometry, also known as Interactive Geometry, refers to computer programs where accurate construction of (generally) planar drawings can be made. The key characteristic of this software is that, when dragging certain elements of the configuration, the geometric properties of the construction are preserved. In this paper, we describe an educational web-based application that complements standard dynamic geometry programs in a mathematically sound manner. We put the focus on computing the geometric locus of distinguished points in linkages and other geometric constrained configurations, since knowing the equations of such loci is a typical engineering task. The tool is located at

Journal of Combinatorial Theory, Series A, 1982

We show a purity assumption which seems to be implicit in the theory of the geometry of diagrams (developed by F. Buekenhout in: Geom. Dedicata 8 , 253-257; 296-298; J. Combin. Theory Ser. A 27, (1979), 121-151); characterizations of those structures on which this assumption holds are given (pure structures), and a suffrcent condition on a structure to be pure is also presented.

SIAM Journal on Discrete Mathematics, 2004

This paper investigates the local uniqueness of designs of m-circles (lines and circles) in the plane up to inversion under a set of angles of intersection as constraints. This local behavior is studied through the Jacobian of the angle measurements in a form analogous to the rigidity matrix for a framework of points with distance constraints. After showing directly that the complete set of angle constraints on v distinct m-circles gives a matrix of rank 3v − 6, we show that the Jacobian is column equivalent by a geometric correspondence to the rigidity matrix for a bar-andjoint framework in Euclidean 3-space. As a corollary, the complexity of the independence of angle constraints on generic plane circles is the complexity of the old unsolved combinatorial problem of generic rigidity in 3-space. This theory is not known to have a polynomial time algorithm for generic independence that offers a warning about the complexity of general systems of geometric constraints even in the plane.

2006

La habilidad del Razonamiento Geométrico es central a muchas aplicaciones de CAD/CAM/CAPP (Computer Aided Design, Manufacturing and Process Planning). Existe una demanda creciente de sistemas de Razonamiento Geométrico que evalúen la factibilidad de escenas virtuales, especificados por relaciones geométricas. Por lo tanto, el problema de Satisfacción de Restricciones Geométricas o de Factibilidad de Escena (GCS/SF) consta de un escenario básico conteniendo entidades geométricas, cuyo contexto es usado para proponer relaciones de restricción entre entidades aún indefinidas. Si la especificación de las restricciones es consistente, la respuesta al problema es uno del finito o infinito número de escenarios solución que satisfacen las restricciones propuestas. De otra forma, un diagnóstico de inconsistencia es esperado. Las tres principales estrategias usadas para este problema son: numérica, procedimental y matemática. Las soluciones numérica y procedimental resuelven solo parte del problema, y no son completas en el sentido de que una ausencia de respuesta no significa la ausencia de ella. La aproximación matemática previamente presentada por los autores describe el problema usando una serie de ecuaciones polinómicas. Las raíces comunes a este conjunto de polinomios caracterizan el espacio solución para el problema. Ese trabajo presenta el uso de técnicas con Bases de Groebner para verificar la consistencia de las restricciones. Ella también integra los subgrupos del grupo especial Euclídeo de desplazamientos SE(3) en la formulación del problema para explotar la estructura implicada por las relaciones geométricas. Aunque teóricamente sólidas, estas técnicas requieren grandes cantidades de recursos computacionales. Este trabajo propone técnicas de Dividir y Conquistar aplicadas a subproblemas GCS/SF locales para identificar conjuntos de entidades geométricas fuertemente restringidas entre sí. La identificación y pre-procesamiento de dichos conjuntos locales, generalmente reduce el esfuerzo requerido para resolver el problema completo. La identificación de dichos sub-problemas locales está relacionada con la identificación de ciclos cortos en el grafo de Restricciones Geométricas del problema GCS/SF. Su preprocesamiento usa las ya mencionadas técnicas de Geometría Algebraica y Grupos en los problemas locales que corresponden a dichos ciclos. Además de mejorar la eficiencia de la solución, las técnicas de Dividir y Conquistar capturan la esencia física del problema. Esto es ilustrado por medio de su aplicación al análisis de grados de libertad de mecanismos. MSC: 68U07

2007

Arrangement of lines is the subdivision of a plane by a finite set of lines. Arrangement is an important structure which can aid in solving many problems in theoretical computer science. Arrangement of lines also has important application in robot motion ...

International Journal of Computational Geometry & Applications, 1992

Recent solutions of three “art gallery” problems are reported. Three related still-open problems are discussed.

Geometric Constraints Solving use graph-based methods to decompose systems of geometric constraints. These methods have intrinsic and unavoidable limitations which are overcome by the witness method presented here.