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A triangulation of the square that respects the gluings, like that shown below, also defines a triangulation of the torus. A two dimensional torus, represented as the gluing of a square via the map g, identifying its opposite sites; The projective plane admits a triangulation (see CW-complexes)
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This category concerns subdivisions of the plane or of regions of the plane into triangles, and related mathematical structures. For other unrelated meanings of "triangulation", see Triangulation (disambiguation).
Let + be the (+)-simplex. + is a combinatorial n-sphere with its triangulation as the boundary of the n+1-simplex. Given a triangulated piecewise linear (PL) n-manifold , and a co-dimension 0 subcomplex together with a simplicial isomorphism : ′ +, the Pachner move on N associated to C is the triangulated manifold () (+ ′).
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.
In geometry, a triangulation is a subdivision of a planar object into triangles, and by extension the subdivision of a higher-dimension geometric object into simplices. Triangulations of a three-dimensional volume would involve subdividing it into tetrahedra packed together.
In Spain there are 11,000 triangulation stations, concrete structures which typically consist of a cylinder 120 cm high and 30 cm in diameter over a concrete cubic base. They were erected by the Instituto Geográfico Nacional , usually painted in white, and can be marked with a metallic label with the warning: "The destruction of this sign is ...
The triangulation of a parametrically defined surface is simply achieved by triangulating the area of definition (see second figure, depicting the Monkey Saddle). However, the triangles may vary in shape and extension in object space, posing a potential drawback.