In 3D graphics programming, triangles form the surfaces of solid figures, starting with the simplest of all three-dimensional figures, the triangular pyramid, or tetrahedron. But that square can be divided into two triangles, each of which is flat, although not necessarily on the same plane. A square in 3D space isn’t guaranteed to be flat because the fourth point might not be in the same plane as the other three. Indeed, one way to define a plane in 3D space is with three non-collinear points, and that’s a triangle. On the other hand, any other type of polygon can be decomposed into a collection of triangles.Įven in three dimensions, a triangle is always flat.
It’s nothing more than three points connected by three lines, and if you try to make it any simpler, it collapses into a single dimension. The triangle is the most basic two-dimensional figure. By construction there is no overlapping.Volume 29 Number 3 DirectX Factor : Triangles and Tessellation The remainder are connected to a unique vertex of Δ and belong to two new triangles which have a common edge.
Each of these lie in three new triangles which intersect at that vertex. Those which are vertices of the triangles attached to sides of the original triangle, which are connected to 2 vertices of Δ. This results in a total of 3 + (2 a – 3) + (2 b - 3) + (2 c - 3) = 2( a + b + c) - 6 new triangles. A more systematic way of performing the tiling is first to add a tile to each side (the reflection of the triangle in that edge) and then fill in the gaps at each vertex. The construction of P 2 can be understood more clearly by noting that some triangles or tiles are added twice, the three which have a side in common with the original triangle. Even in this degenerate case when an angle of π arises, the two collinear edge are still considered as distinct for the purposes of the construction. In the case when an angle of Δ equals π/3, a vertex of P 2 will have an interior angle of π, but this does not affect the convexity of P 2. It is made up of triangles which only intersect in edges or vertices, forms a convex polygon with all angles less than or equal to π and each side being the edge of a reflected triangle. The union of these new triangles together with the original triangle form a connected shape P 2. The triangles do not overlap except at the edges, half of them have their orientation reversed and they fit together to tile a neighborhood of the point. At each of the three vertices the triangle can be successively reflected through edges emanating from the vertices to produce 2 m copies of the triangle where the angle at the vertex is π/ m.
The original triangle Δ gives a convex polygon P 1 with 3 vertices. The construction of a tessellation will first be carried out for the case when a, b and c are greater than 2.
#CONSTRUCTION OF TRIANGLE TESSELLATION PLUS#
In the sphere there are three Möbius triangles plus one one-parameter family in the plane there are three Möbius triangles, while in hyperbolic space there is a three-parameter family of Möbius triangles, and no exceptional objects.Ī fundamental domain triangle ( p q r), with vertex angles π⁄ p, π⁄ q, and π⁄ r, can exist in different spaces depending on the value of the sum of the reciprocals of these integers:ġ p + 1 q + 1 r > 1 : Sphere 1 p + 1 q + 1 r = 1 : Euclidean plane 1 p + 1 q + 1 r 1 When these are whole numbers, the triangle is called a Möbius triangle, and corresponds to a non-overlapping tiling, and the symmetry group is called a triangle group. The value n⁄ d means the vertex angle is d⁄ n of the half-circle. Each Schwarz triangle on a sphere defines a finite group, while on the Euclidean or hyperbolic plane they define an infinite group.Ī Schwarz triangle is represented by three rational numbers ( p q r), each representing the angle at a vertex. These can be defined more generally as tessellations of the sphere, the Euclidean plane, or the hyperbolic plane. In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere ( spherical tiling), possibly overlapping, through reflections in its edges. Spherical triangle that can be used to tile a sphere
0 kommentar(er)
