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The vertices of triangles are associated not only with spatial position but also with other values used to render the object correctly. Most attributes of a vertex represent vectors in the space to be rendered. These vectors are typically 1 (x), 2 (x, y), or 3 (x, y, z) dimensional and can include a fourth homogeneous coordinate (w).
The OBJ file format is a simple data-format that represents 3D geometry alone – namely, the position of each vertex, the UV position of each texture coordinate vertex, vertex normals, and the faces that make each polygon defined as a list of vertices, and texture vertices. Vertices are stored in a counter-clockwise order by default, making ...
One way is for the 3D modeller to unfold the triangle mesh at the seams, automatically laying out the triangles on a flat page. If the mesh is a UV sphere, for example, the modeller might transform it into an equirectangular projection. Once the model is unwrapped, the artist can paint a texture on each triangle individually, using the ...
With index arrays, a mesh is represented by two separate arrays, one array holding the vertices, and another holding sets of three indices into that array which define a triangle. The graphics system processes the vertices first and renders the triangles afterwards, using the index sets working on the transformed data.
A common type of lattice graph (known under different names, such as grid graph or square grid graph) is the graph whose vertices correspond to the points in the plane with integer coordinates, x-coordinates being in the range 1, ..., n, y-coordinates being in the range 1, ..., m, and two vertices being connected by an edge whenever the corresponding points are at distance 1.
A claw-free graph is a graph in which no induced subgraph is a claw; i.e., any subset of four vertices has other than only three edges connecting them in this pattern. Equivalently, a claw-free graph is a graph in which the neighborhood of any vertex is the complement of a triangle-free graph.
A planar symmetry group has a polygonal fundamental domain and can be represented by its group notation: the sequence of the reflection orders of the fundamental domain vertices. A fundamental domain triangle is denoted ( p q r ), where p , q , r are whole numbers > 1, i.e. ≥ 2; a fundamental domain right triangle is denoted ( p q 2).
The neighbourhood of a vertex v in a graph G is the subgraph of G induced by all vertices adjacent to v, i.e., the graph composed of the vertices adjacent to v and all edges connecting vertices adjacent to v. The neighbourhood is often denoted or (when the graph is unambiguous) . The same neighbourhood notation may also be used ...