Showing papers on "Regular polygon published in 1972"

Minimum-Perimeter Polygons of Digitized Silhouettes

Abstract: The minimum-perimeter polygon of a silhouette has been shown to be a means for recognizing convex silhouettes and for smoothing the effects of digitization in silhouettes. We describe a new method of computing the minimum-perimeter polygon (MPP) of any digitized silhouette satisfying certain constraints of connectedness and smoothness, and establish the underlying theory. Such a digitized silhouette is called a ``regular complex,'' in accordance with the usage in piecewise linear topology. The method makes use of the concept of a stretched string constrained to lie in the cellular boundary of the digitized silhouette. We show that, by properly marking the virtual as well as the real vertices of an MPP, the MPP can serve as a precise representation of any regular complex, and that this representation is often an economical one.

Inequalitites on the probability content of convex regions for elliptically contoured distributions

Abstract: This work was supported in part by the National Science Foundation Grant No. 17172 at Stanford University, Grants No. 11021, 9593, and 21074 at the University of Minnesota and Grant No. 25911 at the University of Chicago; by the Army. Navy, Air Force and NASA under a contract (N00014-67-4-0097-0014, Task Number NR-042-242) administered by the Office of Naval Research at Yale University, by the Army under a contract (DA-ARO-D31-124-70-G-102) at the University of Minnesota. Reproduction in whole or in part is permitted for any purpose of the United States Government.

Computer graphics clipping system for polygons

Abstract: A system is disclosed for clipping three-dimensional polygons for use in a computer-graphics display. The system removes from each polygon all parts that lie outside an arbitrary, planefaced, convex polyhedron, e.g. a truncated pyramid defining a viewing volume. The polygon is defined by data representing a group of vertices and is clipped separately in its entirety against each clipping plane (of the polyhedron). In a multiplestage structure as disclosed, each stage clips the polygon against a single plane and requires storage for only two vertex values. A time-sharing embodiment of the system is also disclosed. The disclosed system also incorporates the use of a perspective transformation matrix which provides for arbitrary field-of-view angles and depth-of-field distances while utilizing simple, fixed clipping planes.

Averages for polygons formed by random lines in euclidean and hyperbolic planes

Abstract: We consider a countable number of independent random uniform lines in the hyperbolic plane (in the sense of the theory of geometrical probability) which divide the plane into an infinite number of convex polygonal regions. The main purpose of the paper is to compute the mean number of sides, the mean perimeter, the mean area and the second order moments of these quantities of such polygonal regions. For the Euclidean plane the problem has been considered by several authors, mainly Miles [4]-[9] who has taken it as the starting point of a series of papers which are the basis of the so-called stochastic geometry. GEOMETRICAL PROBABILITY; RANDOM LINES; RANDOM POLYGONS; PLANE OF CONSTANT CURVATURE; HYPERBOLIC PLANE; CONVEX DOMAINS; POISSON LINE PROCESS

Some theorems on convex polygons

Abstract: In this paper diagonals of various orders in a (strict) convex polygon P n are considered. The sums of lengths of diagonals of the same order are studied. A relationship between the number of consecutive diagonals which do not intersect a given maximal diagonal and lie on one side of it and the order of the smallest diagonal among them is established. Finally a new proof of a conjecture of P. Erdos, considered already in [1], is given.

Fraunhofer Diffraction at Apertures in the Form of Regular Polygons. I.

Abstract: In this second contribution the Fraunhofer diffraction patterns from apertures of the form of an equilateral triangle, a square, a regular pentagon, a hexagon and an octagon are compared with the calculated maps of the intensity distribution. For each shape of the aperture the first few maxima and minima of the intensity are tabulated and the formulae for the wave function are given characterizing the diffraction in the directions of the slowest and the steepest decrease of intensity.

A quadrilateral-free arrangement of sixteen lines

Abstract: n lines in general position in the real projective plane determine (n2-n+2)/2 convex polygons. It has been observed ([1], [2]) that for almost all arrangements of n>3 lines that one or more of these polygons must be quadrilaterals. In fact, only three arrangements, one each for n=3, 6 and 10, have been reported in which there are no quadrilaterals. Because of this, Branko Griinbaum has recently conjectured [3] that for all n $3, 6 or 10 at least one quadrilateral will be formed by every possible arrangement of n lines in the projective plane. The figure shows that this conjecture is not true by displaying an

An index for set-valued maps in infinite-dimensional spaces

Abstract: Previous fixed point indexes defined for a set-valued map in an infinite-dimensional space have required the values of this map to be convex sets. The corresponding assumption of this paper is that the values be (co-)acyclic sets, i.e., that the reduced Alexander cohomology group of each of these sets be trivial in each dimension. Other assumptions are that the space is locally convex and that the map is compact and upper semicontinuous with no fixed points on the boundary of its domain. The index is defined, proved to be homotopy invariant, and proved to vanish in case there are no fixed points. The main methods used are finite-dimensional approximation and the Vietoris-Begle mapping theorem.

Averages for polygons formed by random

Abstract: We consider a countable number of independent random uniform lines in the hyperbolic plane (in the sense of the theory of geometrical probability) which divide the plane into an infinite number of convex polygonal regions. The main purpose of the paper is to compute the mean number of sides, the mean perimeter, the mean area and the second order moments of these quantities of such polygonal regions. For the Euclidean plane the problem has been considered by several authors, mainly Miles [4]-[9] who has taken it as the starting point of a series of papers which are the basis of the so-called stochastic geometry. GEOMETRICAL PROBABILITY; RANDOM LINES; RANDOM POLYGONS; PLANE OF CONSTANT CURVATURE; HYPERBOLIC PLANE; CONVEX DOMAINS; POISSON LINE PROCESS

The Limit Concept on the Geoboard.

Abstract: circle contains the interior of the inscribed polygon and is contained in the interior of the circumscribed polygon. The corre sponding measures of the interiors (areas) satisfy the following: measure of interior of inscribed polygon < measure of interior of circle < measure of interior of circum scribed polygon. However, the computa tional task in calculating the measures of the interiors of polygons is formidable if the traditional expression is used: A = ^ap, where a = apothem, = perimeter. In this calculation roots of expressions in volving roots are needed. Roots of expres sions involving roots are not reasonably accessible without the aid of calculating devices. We find that not only elementary school teachers but also secondary school teachers with instinctive good judgment avoid the formidable calculations of the traditional formula. It seems highly de sirable to salvage the limit concept em bodied irr the method of exhaustion by devising a new computational approach. Computations involving only counting procedures and operations with rational numbers are within the scope of even upper elementary-grade students. The nu merical-graphical experiences provided in this paper are expected to produce a strong intuitive feeling for the limit con cept. This sequence of activities not only contains the computational replacement for the traditional formula but also gen eralizes the use of the Archimedean method of exhaustion. Generalization is accom plished with a reliance on only the most rudimentary arithmetical procedures. Moreover, the new procedure no longer relies on inscribing regular polygons but uses the polygon best suited to the simple closed curve.

8 – Geometry of the Plane

Abstract: This chapter contains several examples illustrating existence which are not normally included in courses in the geometry of the plane. We discuss convex sets, tilings of the rectangle, tessellations of the plane, and equivalence classes of demi-dominoes. Convex sets are important in optimization theory, particularly in linear programming. Although the tiling of rectangles with noncongruent squares appears to be devoid of application, it is closely related to ideas in electrical engineering. Tessellations are interesting plane figures often mentioned in books on mathematical recreations. The idea of equivalence classes is a useful algebraic concept which has important implications in geometry.

Semi-regular plane polygons of integral type

Abstract: A polygon is called semi-regular if its interior angles are equal to one another. The paper deals mainly with semi-regular polygons whose sides have integral lengths.

Rate of convergence in the "circle formation" problem

Abstract: We consider transformations obtained from local homogeneous rules of motion of polygons in the plane for which regular polygons are stationary (the so-called "circle formation" problem). These transformations are studied for initial states that are close to regular polygons. A method of determination of the rate of convergence to regular polygons is presented; on this basis we obtain estimates of the highest rate of convergence. Figures: 1. Bibliography: 4 items.