Interpolation by polyhedral functions (original) (raw)
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On relatively equilateral polygons inscribed in a convex body
Publicationes Mathematicae Debrecen
Let C ⊂ E 2 be a convex body. The C-length of a segment is the ratio of its length to the half of the length of a longest parallel chord of C. By a relatively equilateral polygon inscribed in C we mean an inscribed convex polygon all of whose sides are of equal C-length. We prove that for every boundary point x of C and every integer k ≥ 3 there exists a relatively equilateral k-gon with vertex x inscribed in C. We discuss the C-length of sides of relatively equilateral k-gons inscribed in C and we reformulate this question in terms of packing C by k homothetical copies which touch the boundary of C. Let C be a convex body in Euclidean n-space E n. If pq is a longest chord of C in a direction l, we say that points p and q are opposite and we call pq a diametral chord of C in direction l. By the C-distance dist C (a, b) of a and b we mean the ratio of the Euclidean distance |ab| of a and b to the half of the Euclidean distance of end-points of a diametral chord of C parallel to ab (comp. [7]). We use here the term relative distance if there is no doubt about C. By the C-length of the segment ab we mean dist C (a, b). If C ⊂ E 2 , we define a C-equilateral k-gon as a convex k-gon all of whose sides have equal C-lengths. We also use the name relatively equilateral k-gon when C is fixed. Section 1 is of an auxiliary nature. It presents properties of the Cdistance, and especially properties of the C-distance of boundary points
On Polygons Enclosing Point Sets II
Graphs and Combinatorics, 2009
Let R and B be disjoint point sets such that R ∪ B is in general position. We say that B encloses by R if there is a simple polygon P with vertex set B such that all the elements in R belong to the interior of P. In this paper we prove that if the vertices of the convex hull of R ∪ B belong to B, and |R| ≤ |Conv (B)| − 1 then B encloses R. The bound is tight. This improves on results of a previous paper in which it was proved that if |R| ≤ 56|Conv (B)| then B encloses R. To obtain our result we prove the next result which is interesting on its own right: Let P be a convex polygon with n vertices p 1 ,. .. , p n and S a set of m points contained in the interior of P , m ≤ n − 1. Then there is a convex decomposition {P 1 ,. .. , P n } of P such that all points from S lie on the boundaries of P 1 ,. .. , P n , and each P i contains a whole edge of P on its boundary.
On polygons enclosing point sets
2001
Let R and B be point sets such that R ∪ B is in general position. We say that B is enclosed by R if there is a simple polygon P with vertex set R such that all the elements in B belong to the interior of P. In this paper we prove that if the vertices of the convex hull of R ∪ B belong to B, and |R| ≤ |Conv (R)| − 1 then B encloses R. The bound is tight. This improves on results of a previous paper in which it was proved that if |R| ≤ 56|Conv (B)| then B encloses R. To obtain our result we prove the next result which is interesting on its own right: Let P be a convex polygon with n vertices p 1 ,. .. , p n and S a set of m points contained in the interior of P , m ≤ n − 1. Then there is a convex decomposition {P 1 ,. .. , P n } of P such that all points from S lie on the boundaries of P 1 ,. .. , P n , and each P i contains a whole edge of P on its boundary.
Resolution of regular polygons: the coefficient ν
Resolution of regular polygons - the coefficient Ni, 2019
In the study of regular polygons, the greatest attention has always been given to resolutive formulas that take into account either measures, at least two plus the number of sides, or the width of any of the available angles, or to the trigonometric aspect of the polygon intended as the sum of right-angled triangles congruent to each other. In dealing with this work, attention was instead given for the first time to a particular relationship, that between the area constant φ and the so-called "fixed number" f, which came to light during the study of different geometry problems abstract related to the possible possibility of being able to obtain all the direct and indirect measurements, and from there find all the possible relationships between the polygon in question and all the other geometrical figures derived from it, starting from a single datum: the Area. Of this new coefficient and of its immediate applicability to the calculation, despite the numerous researches carried out subsequently, there is no apparent trace to date. And this ratio, which has been called "coefficient ν" $Ni), however, immediately simplifies most of the necessary calculations and has the advantage of being immediately recognizable-being equal to the number of sides of the polygon divided by two-and of allow to work on a single piece of data available, not only the area as it seemed at the beginning, but also the apothem, the side or the radius, without even using the tables of fixed numbers and area constants, from which also directly derives. An Excel file is attached to this work. In this spreadsheet all the formulas analyzed here are applied, in addition to being calculated, as it was in the original intentions, all the possible relationships between the polygon on which one works and all those derivable: the circumferences inscribed, circumscribed and equivalent, the respective rays and the relative right-angled triangles constructed on the basis of the second proposition of Archimedes. The file calculates the relationship between all these figures and their elements as well as between said elements among them, on the basis of criteria of belonging to similar groups.
Some Metric Properties and a Constructive Task of a Semi-Regular 2n-Sides Polygon
American Journal of Applied Mathematics, 2020
A simple polygon that either has equal all sides or all interior angles is called a semi-regular polygon. In terms of this definition, we can distinguish between two types of semi-regular polygons: equilateral polygons (that have equal all sides and different interior angles) and equiangular polygons (that have equal interior angles and different sides). To analyze the metric properties of semi-regular polygons, knowing only one basic element, e.g. the length of a side, as in regular polygons, is not enough. Therefore, in addition to the side of a semi-regular polygon, we use another characteristic element of it to analyze the metric features, and that is the angle δ=∠(a,b) between the side of a semi-regular polygon P N and the side b of its inscribed regular polygon P n. Some metric properties of a semi-regular equilateral 2n-sides polygon are analyzed in this paper with respect to these two characteristic elements. Some of the problems discussed in the paper are: convexity, calculation of surface area, dependence on the length of sides a and δ, calculation of the radius of the inscribed circle depending on the sides a and angles δ, and calculation of the surface area in which the radius of the inscribed circle is known, as well as the relationship between them. It has been shown that the formula for calculating the surface area of regular polygons results from the formula for the surface area of 2n-side semi-regular, equilateral polygons. Further, by using these results, it has been shown that the cross-sections of regular polygons inscribed to semi-regular equilateral polygons, the vertices of equiangular semi-regular polygons, as well as the sides of the regular polygons inscribed to it, intersect in the same manner at the vertices of the equilateral semi-regular polygon. It has further been shown that the sides of the equiangular semi-regular polygon refer to each other as the sines of the angles created by the sides of the inscribed polygons and the side of the semi-regular polygon.
International Journal of Computational Geometry & Applications, 2008
Interpolating a piecewise-linear triangulated surface between two polygons lying in parallel planes has attracted a lot of attention in the literature over the past three decades. This problem is the simplest variant of interpolation between parallel slices, which may contain multiple polygons with unrestricted geometries and/or topologies. Its solution has important applications to medical imaging, digitization of objects, geographical information systems, and more. Practically all currently-known methods for surface reconstruction from parallel slices assume a priori the existence of a non-self-intersecting triangulated surface defined over the vertices of the two polygons, which connects them. Gitlin et al. were the first to specify a nonmatable pair of polygons. In this paper we provide proof of the nonmatability of a “simpler” pair of polygons, which is less complex than the example given by Gitlin et al. Furthermore, we provide a family of polygon pairs with unbounded complexi...
Global geometry of polygons. I: The theorem of Fabricius-Bjerre
Proceedings of the American Mathematical Society, 1974
Deformation methods provide a direct proof of a polygonal analogue of a theorem proved by Fabricius-Bjerre and by Halpern relating the numbers of crossings, pairs of inflections, and lines of double tangency for smooth closed plane curves.
On Weighted Sums of Numbers of Convex Polygons in Point Sets
Discrete & Computational Geometry
Let S be a set of n points in general position in the plane, and let X_{k,\ell }(S)Xk,ℓ(S)bethenumberofconvexk−gonswithverticesinSthathaveexactlyX k , ℓ ( S ) be the number of convex k-gons with vertices in S that have exactlyXk,ℓ(S)bethenumberofconvexk−gonswithverticesinSthathaveexactly\ell ℓpointsofSintheirinterior.Weproveseveralequalitiesforthenumbersℓ points of S in their interior. We prove several equalities for the numbersℓpointsofSintheirinterior.WeproveseveralequalitiesforthenumbersX_{k,\ell }(S)$$ X k , ℓ ( S ) . This problem is related to the Erdős–Szekeres theorem. Some of the obtained equations also extend known equations for the numbers of empty convex polygons to polygons with interior points. Analogous results for higher dimension are shown as well.
A unifying approach for a class of problems in the computational geometry of polygons
The Visual Computer, 1985
A generalized problem is defined in terms of functions on sets and illustrated in terms of the computational geometry of simple planar polygons. Although its apparent time complexity is O (n2), the problem is shown to be solvable for several cases of interest (maximum and minimum distance, intersection detection and rerporting) in O (n log n), O (n), or O (log n) time, depending on the nature of a specialized "selection" function. (Some of the cases can also be solved by the Voronoi diagram method; but time complexity increases with that approach.) A new use of monotonicity and a new concept of" locality" in set mappings contribute significantly to the derivation of the results.