Triangulation of a surface means * a ''net'' of triangles, which covers a given surface partly or totally, ''or'' * the ''procedure'' of generating the points and triangles of such a net of triangles.

Approaches

This article describes the generation of a net of triangles. In literature there are contributions which deal with the optimization of a given net. Surface triangulations are important for * visualizing surfaces and * the application of

finite element method Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat tran ...

s. 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. This can be minimized through adaptive methods that consider step width while triangulating the parameter area. To triangulate an ''

implicit surface In mathematics, an implicit surface is a Surface (geometry), surface in Euclidean space defined by an equation : F(x,y,z)=0. An ''implicit surface'' is the set of Zero of a function, zeros of a Function of several real variables, function of ...

'' (defined by one or more equations) is more difficult. There exist essentially two methods. * One method divides the 3D region of consideration into cubes and determines the intersections of the surface with the edges of the cubes in order to get polygons on the surface, which thereafter have to be triangulated (''cutting cube method''). The expenditure for managing the data is great. * The second and simpler concept is the ''marching method''.S. Akkouche & E Galin: ''Adaptive Implicit Surface Polygonization Using Marching Triangles'', COMPUTER GRAPHICS forum (2001), Vol. 20, pp. 67–80 The triangulation starts with a triangulated hexagon at a starting point. This hexagon is then surrounded by new triangles, following given rules, until the surface of consideration is triangulated. If the surface consists of several components, the algorithm has to be started several times using suitable starting points. The cutting cube algorithm determines, at the same time, all components of the surface within the surrounding starting cube depending on prescribed limit parameters. An advantage of the marching method is the possibility to prescribe boundaries (see picture). Polygonizing a surface means to generate a

polygon mesh In 3D computer graphics and solid modeling, a polygon mesh is a collection of , s and s that defines the shape of a polyhedron, polyhedral object's surface. It simplifies Rendering (computer graphics), rendering, as in a wire-frame model. The fac ...

. The triangulation of a surface should not be confused with the triangulation of a

discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit * Discrete group, ...

prescribed plane ''set of points''. See Delaunay triangulation. File:Triang-cyl-sph4.svg, Triangulation: cylinder, surface File:Triang-cyl-sph4-pov.png, Triangulation: cylinder, surface {{math, 1=''x''{{sup, 4 + ''y''{{sup, 4 + ''z''{{sup, 4 = 1, POV-Ray image File:Torus-triang.svg, Torus: triangulated by the marching method File:Torus-cutting-cube.svg, Torus: polygonized by the cutting cube method

References

External links

*Tasso Karkanis & A. James Stewart: ''Curvature-Dependent Triangulation of Implicit Surfaces'

Software

Surface reconstruction tutorial
an

in the Point Cloud Library Surfaces Computer-aided design Finite element method Triangle geometry

Approaches

See also

References

External links

Software