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This data might be useful to classify topological spaces up to homeomorphism but only given that the characteristics are also topological invariants, meaning, they do not depend on the chosen triangulation. For the data listed here, this is the case. [4] For details and the link to singular homology, see topological invariance.
The four datasets composing Anscombe's quartet. All four sets have identical statistical parameters, but the graphs show them to be considerably different. Anscombe's quartet comprises four datasets that have nearly identical simple descriptive statistics, yet have very different distributions and appear very different when graphed.
Each edge or triangle of the Delaunay triangulation may be associated with a characteristic radius: the radius of the smallest empty circle containing the edge or triangle. For each real number α, the α-complex of the given set of points is the simplicial complex formed by the set of edges and triangles whose radii are at most 1/α.
Many methods have been invented to extract a low-dimensional structure from the data set, such as principal component analysis and multidimensional scaling. [27] However, it is important to note that the problem itself is ill-posed, since many different topological features can be found in the same data set.
The TIN model was developed in the early 1970s as a simple way to build a surface from a set of irregularly spaced points. The first triangulated irregular network program for GIS was written by W. Randolph Franklin, under the direction of David Douglas and Thomas Peucker (Poiker), at Canada's Simon Fraser University, in 1973. [2]
In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy category.
Head and cerebral structures (hidden) extracted from 150 MRI slices using marching cubes (about 150,000 triangles). Marching cubes is a computer graphics algorithm, published in the 1987 SIGGRAPH proceedings by Lorensen and Cline, [1] for extracting a polygonal mesh of an isosurface from a three-dimensional discrete scalar field (the elements of which are sometimes called voxels).
Triangulation today is used for many purposes, including surveying, navigation, metrology, astrometry, binocular vision, model rocketry and, in the military, the gun direction, the trajectory and distribution of fire power of weapons. The use of triangles to estimate distances dates to antiquity.