Tiling Polygons with Lattice Triangles

Given a simple polygon with rational coordinates having one vertex at the origin and an adjacent vertex on the x-axis, we look at the problem of the location of the vertices for a tiling of the polygon using lattice triangles (i.e., triangles which are congruent to a triangle with the coordinates of the vertices being integer). We show that the coordinates of the vertices in any tiling are rationals with the possible denominators odd numbers dependent on the cotangents of the angles in the triangles.     

Configurations of inscribed equilateral triangles

This paper explores equilateral triangles XYZ with vertices on sidelines of a given triangle ABC such that one side of XYZ is parallel to the corresponding side of ABC. There are six such triangles. They have many interesting properties which we investigate using trilinear coordinates. Our results improve and add to the earlier results of Blas Herrera Gómez about these configurations. We obtain new characterizations of several central points of the triangles and identify interesting pairs of triangles that are homologic (or perspective) and orthologic. The recognition of the Darboux cubic of a triangle is also accomplished in these configurations. Triples of circles intersecting in a point and six points on a conic also appear.      

Asymptotic properties of some triangulations of the sphere

In this paper we analyse a method for triangulating the sphere originally proposed by Baumgardner and Frederickson in 1985. The method is essentially a refinement procedure for arbitrary spherical triangles that fit into a hemisphere. Refinement is carried out by dividing each triangle into four by introducing the midpoints of the edges as new vertices and connecting them in the usual ‘red’ way. We show that this process can be described by a sequence of piecewise smooth mappings from a reference triangle onto the spherical triangle. We then prove that the whole sequence of mappings is uniformly bi-Lipschitz and converges uniformly to a non-smooth parameterization of the spherical triangle, recovering the Baumgardner and Frederickson spherical barycentric coordinates. We also prove that the sequence of triangulations is quasi-uniform, that is, areas of triangles and lengths of the edges are roughly the same at each refinement level. Some numerical experiments confirm the theoretical results.     

An algebraic approach to finding the Fermat–Torricelli point

Using a calculus and an algebraic approach, the Cartesian coordinates of the Fermat–Torricelli point are deduced for triangles with no internal angle greater than 120°. Although in theory, the deduction of these coordinates could be made ‘by hand’, in practice it is very laborious to obtain them without the aid of mathematical computer software, but with human guidance, since there are mathematical artifices not yet incorporated into the software. It is also shown that these coordinates can be conveniently expressed in terms of the side lengths and the area of the triangle. These coordinates are contrasted with the coordinates of a similar point: one whose sum of the squares of the distances to the vertices of an arbitrary triangle is a minimum.     

Triangles III: Complex triangle functions

Summary This paper is the third in a series of three examining Euclidean triangle geometry via complex cross ratios. In the first two papers, we looked at triangle shapes and triangle coordinates. In this paper, we look at the triangle coordinates of the special points of a triangle, and show that they are functions of its shape. We then show how these functions can be used to prove theorems about triangles, and to gain some insight into what makes a special point of a triangle a centre.     

Adaptive cubature over a collection of triangles using the d-transformation

We describe an automatic routine to integrate a function over a collection of triangles where on each triangle the function is well behaved, or has singularities of certain types at one or more vertices or edges. The underlying algorithm is globally adaptive and incorporates the d-transformation for extrapolation. Results from performance profile testing indicate that the routine is superior to other published routines when the singularities are located along edges of the triangles.     

The hexaparagon

A hexagon with each pair of opposite sides parallel to a side of a triangle will be called a hexaparagon for that triangle. One way to construct a hexaparagon for a given triangle ABC is to use as vertices the centroids P, Q, R, S, T, and U of the six non-overlapping sub-triangles formed by the three medians of triangle ABC. The perimeter of this hexaparagon is half the perimeter of triangle ABC. The ratio of the areas of triangle ABC to this hexaparagon is 36 to 13 and the lengths of the parallel sides are in the ratio 6 to 2 to 1. The vertices of this hexaparagon lie on an ellipse and, with a second type of hexaparagon introduced later, hexaparagons tile the plane.     

Polynomial of Degree Four Interpolation on Triangles

A method for constructing the $C^1$ piecewise polynomial surface of degree four on triangles is presented in this paper. On every triangle, only nine interpolation conditions, which are function values and first partial derivatives at the vertices of the triangle, are needed for constructing the surface.     

Triangle Contact Representations and Duality

A?contact representation by triangles of a graph is a set of triangles in the plane such that two triangles intersect on at most one point, each triangle represents a vertex of the graph and two triangles intersects if and only if their corresponding vertices are adjacent. De Fraysseix, Ossona de Mendez and Rosenstiehl proved that every planar graph admits a contact representation by triangles. We strengthen this in terms of a simultaneous contact representation by triangles of a planar map and of its dual. A?primal?Cdual contact representation by triangles of a planar map is a contact representation by triangles of the primal and a contact representation by triangles of the dual such that for every edge uv, bordering faces f and g, the intersection between the triangles corresponding to u and v is the same point as the intersection between the triangles corresponding to f and g. We prove that every 3-connected planar map admits a primal?Cdual contact representation by triangles. Moreover, the interiors of the triangles form a tiling of the triangle corresponding to the outer face and each contact point is a corner of exactly three triangles. Then we show that these representations are in one-to-one correspondence with generalized Schnyder woods defined by Felsner for 3-connected planar maps.     

Triangles I: Shapes

Summary This paper is the first in a series of three examining Euclidean triangle geometry via complex cross ratios. In this paper we show that every triangle can be characterized up to similarity by a single complex number, called its shape. We then use shapes and two basic theorems about shapes to prove theorems about similar triangles. The remaining papers in this series will examine complex triangle coordinates and complex triangle functions.     

Area functions on affine planes

Oriented area functions are functions defined on the set of ordered triangles of an affine plane which are antisymmetric under odd permutations of the vertices and which behave additively when triangles are cut into two. We compare several elementary properties which such an area function may have (roughly speaking shear invariance, equality of area of the two triangles obtained by cutting a parallelogram along a diagonal, and equality of area of the two triangles obtained by cutting a triangle along a median). It turns out purely by arguments of elementary affine geometry (if cleverly arranged) that these properties are grosso modo equivalent, although one has to be careful about “pathological” situations. Furthermore, all oriented area functions satisfying these properties are explicitly determined. Finally they are compared with so-called geometric valuations.     

Drawing planar 3-trees with given face areas

We study straight-line drawings of planar graphs such that each interior face has a prescribed area. It was known that such drawings exist for all planar graphs with maximum degree 3. We show here that such drawings exist for all planar partial 3-trees, i.e., subgraphs of a triangulated planar graph obtained by repeatedly inserting a vertex in one triangle and connecting it to all vertices of the triangle. Moreover, vertices have rational coordinates if the face areas are rational, and we can bound the resolution. We also give some negative results for other graph classes.     

Triangles II: Complex triangle coordinates

Summary This paper is the second in a series of three examining Euclidean triangle geometry via complex cross ratios. In the first paper of the series, we examined triangle shapes. In this paper, we coordinatize the Euclidean plane using cross ratios, and use these triangle coordinates to prove theorems about triangles. We develop a complex version of Ceva's theorem, and apply it to proofs of several new theorems. The remaining paper of this series will deal with complex triangle functions.     

Each maximal planar graph with exactly two separating triangles is Hamiltonian

A classical result of Whitney states that each maximal planar graph without separating triangles is Hamiltonian, where a separating triangle is a triangle whose removal separates the graph. Chen [Any maximal planar graph with only one separating triangle is Hamiltonian J. Combin. Optim. 7 (2003) 79-86] proved that any maximal planar graph with only one separating triangle is still Hamiltonian. In this paper, it is shown that the conclusion of Whitney's Theorem still holds if there are exactly two separating triangles.     

Hamilton cycles in plane triangulations

We extend Whitney's Theorem that every plane triangulation without separating triangles is hamiltonian by allowing some separating triangles. More precisely, we define a decomposition of a plane triangulation G into 4‐connected ‘pieces,’ and show that if each piece shares a triangle with at most three other pieces then G is hamiltonian. We provide an example to show that our hypothesis that each piece shares a triangle with at most three other pieces' cannot be weakened to ‘four other pieces.’ As part of our proof, we also obtain new results on Tutte cycles through specified vertices in planar graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 138–150, 2002     

Minimum Covering for Hexagon Triple Systems

A hexagon triple is a graph consisting of three triangles of the form (a, x, b), (b, y, c), and (c,z,a), where a, b, c, x, y, z are distinct. The triangle (a, b, c) is called the inside triangle and the triangles (a, x, b), (b,y,c), and (c, z, a) are called outside triangles. A 3k-fold hexagon triple system of order n is a pair (X, H), where H is an edge-disjoint collection of hexagon triples which partitions the edge set of 3kK ⁿ with vertex set X. Note that the outside triangles form a 3k-fold triple system. If the 3k-fold hexagon triple system (X, H) has the additional property that the inside triangles form a k-fold triple system, then (X, H) is said to be perfect. A covering of 3kK ⁿ with hexagon triples is a triple (X, H, P) such that: 1.3kK ⁿ has vertex set X. 2.P is a subset of E(λ K ⁿ ) with vertex set X for some λ, and 3.H is an edge disjoint partition of E(3kK ⁿ )∪ P with hexagon triples. If P is as small as possible (X, H, P) is called a minimum covering of 3kK ⁿ with hexagon triples. If the inside triangles of the hexagon triples in H form a minimum covering of kK ⁿ with triangles, the covering is said to be perfect. A complete solution for the problem of constructing perfect 3k-fold hexagon triple system and perfect maximum packing of 3kK ⁿ with hexagon triples was given recently by the authors [2]. In this work, we give a complete solution of the problem of constructing perfect minimum covering of 3kK ⁿ with hexagon triples.     

On an algorithm for calculating diffraction integrals

An algorithm for calculating integrals of rapidly oscillating functions given on a smooth two-dimensional surface is proposed. The surface is approximated by a collection of flat triangles with the values of the integrand known at their vertices. These values are used as reference ones to extend the function to other points of a triangle. The integral of the extended function over the surface of a triangle is calculated exactly. The desired value of the full diffraction integral is determined as the sum of the integrals calculated over the surfaces of all triangles. The resulting formulas for integral calculation involve singularities (indeterminate forms). Much attention is given to representations of these formulas in such a way that the indeterminate forms are automatically evaluated. Numerical results are presented.     

The inverse Fourier transform of the filtering function and its integral powers

The convex coordinates of a point, P,in a closed triangular region are defined as weights which must be placed at the vertices of the region to shift the balance point of the weight distribution to P. The vertices of the triangle are interpreted as energy states of a physical system, and the convex coordinates become probabilities. The Maxwell‐Boltzmann distribution function is derived with the aid of convex coordinates.     

Coverage problems and visibility regions on topographic surfaces

The viewshed of a point on an irregular topographic surface is defined as the area visible from the point. The area visible from a set of points is the union of their viewsheds. We consider the problems of locating the minimum number of viewpoints to see the entire surface, and of locating a fixed number of viewpoints to maximize the area visible, and possible extensions. We discuss alternative methods of representing the surface in digital form, and adopt a TIN or triangulated irregular network as the most suitable data structure. The space is tesselated into a network of irregular triangles whose vertices have known elevations and whose edges join vertices which are Thiessen neighbours, and the surface is represented in each one by a plane. Visibility is approximated as a property of each triangle: a triangle is defined as visible from a point if all of its edges are fully visible. We present algorithms for determination of visibility, and thus reduce the problems to variants of the location set covering and maximal set covering problems. We examine the performance of a variety of heuristics.     

A generalization of the circumcircle theorem

The theorem relating the bisectors of the edges of a triangle and the corresponding circumscribing circle is established as a special case of a theorem for triangles with weighted vertices where the edges are partitioned with circular arcs in the proportions of the weights. The circular arcs are established as being uniquely determined by the weights and the triangle, and are given by three circles with collinear centres. These circles either intersect in zero, one or two real points, these latter points being the triple points.