Given a right triangle and two inscribed squares, we show that the reciprocals of the hypotenuse and the sides of the squares satisfy an interesting Pythagorean equality. This gives new ways to obtain rational (integer) right triangles from a given one. 1. Harmonic and Symphonic Squares Consider an arbitrary triangle with altitude α corresponding to base β (see Figure 1a). Assuming that the bas...

If equilateral triangles are erected outwardly on the sides of any given triangle, then the circumcenters of the three erected triangles form an equilateral triangle. This statement, known as Napoleon’s theorem, and the configuration involved, usually called the Torricelli configuration of the initial triangle, were generalized to d-dimensional simplices (d ≥ 3) in [12]. It is obvious that for ...

This paper presents a new method for computing whether or not two triangles in three dimensions intersect. The code is very efficient and requires minimum arithmetic precision. Indeed, all branching decisions are carried out by evaluating the signs of degree three polynomials. In addition, an efficient test is proposed for the two-dimensional case. Key-words: geometric predicates, low degree pr...

We construct a Teichmüller curve uniformized by the Fuchsian triangle group ∆(m, n, ∞) for every m < n ≤ ∞. Our construction includes the Teichmüller curves constructed by Veech and Ward as special cases. The construction essentially relies on properties of hypergeometric differential operators. For small m, we find bil-liard tables that generate these Teichmüller curves. We interprete some of ...

This paper presents a fast algorithm for generating triangle strips from triangulated meshes, providing a compact representation suitable for transmission and rendering of the models. A data structure that allows efficient triangle strip generation is also described. The method is based on simple heuristics, significantly reducing the number of vertices used to describe the triangulated models....

We give a comprehensive theoretical treatment for calculating the viewshed of a given point, present an analytical solution to the viewshed problem and a new algorithm for finding the viewshed on a triangulated terrain. We implement our algorithm on a real terrain. Some algorithms make use of the horizon information of the terrain to calculate viewshed. The vertices of the horizon of the terrai...

Given a right triangle and two inscribed squares, we show that the reciprocals of the hypotenuse and the sides of the squares satisfy an interesting Pythagorean equality. This gives new ways to obtain rational (integer) right triangles from a given one. 1. Harmonic and Symphonic Squares Consider an arbitrary triangle with altitude α corresponding to base β (see Figure 1a). Assuming that the bas...

Because of their improved numerical properties, quadrilateral meshes have become a popular representation for finite elements computations and computer animation. In this paper we address the problem of optimally representing quadrilateral meshes as generalized triangle strips (with one swap bit per triangle). This is important because 3D rendering hardware is optimized for rendering triangle m...

With the extensive applications of lithium ion batteries, many batteries fire and explosion accidents were reported. Base on the combustion triangle theory, the combustion triangle contributions of lithium ion battery were analysed. By using C80 micro calorimeter, the thermal behavior studies on the materials show that the flammable electrolyte, oxygen generated by charged cathode and anode dec...

Let Pf (N ) be the set of finite nonempty subsets of N and for F,G ∈ Pf (N ) write F < G when maxF < minG. Let X = {(F,G) : F,G ∈ Pf (N ) and F < G}. A triangle in X is a set of the form {(F ∪ H,G), (F,G), (F,H ∪ G)} where F < H < G. Motivated by a question of Erdős, Nešetŕıl, and Rödl regarding three term arithmetic progressions, we show that any finite subset Y of X contains a relatively larg...