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For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).
This has always been the situation since numerical simulation and computer graphics were invented, because as computer hardware and simple equation-solving software have improved, people have been drawn to larger and more complex geometric models in a drive for greater fidelity, scientific insight, and artistic expression.
A mesh is a representation of a larger geometric domain by smaller discrete cells. Meshes are commonly used to compute solutions of partial differential equations and render computer graphics, and to analyze geographical and cartographic data.
In an analogous way, one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for f ′(x + h / 2 ) and f ′(x − h / 2 ) and applying a central difference formula for the derivative of f ′ at x, we obtain the central difference approximation of the second derivative of f:
Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It is used in the study of computer graphics and topological combinatorics. Topics in this area include: Discrete Laplace operator; Discrete ...
A modern rendering of the Utah teapot, an iconic model in 3D computer graphics created by Martin Newell in 1975. Computer graphics is a sub-field of computer science which studies methods for digitally synthesizing and manipulating visual content. Although the term often refers to the study of three-dimensional computer graphics, it also ...
Scene rendered with RRV [1] (simple implementation of radiosity renderer based on OpenGL) 79th iteration The Cornell box, rendered with and without radiosity by BMRT. In 3D computer graphics, radiosity is an application of the finite element method to solving the rendering equation for scenes with surfaces that reflect light diffusely.
In computer-aided engineering and finite element analysis, an object may be represented by a surface mesh of node points connected by triangles or quadrilaterals (polygon mesh). More accurate, but also far more CPU-intensive, results can be obtained by using a solid mesh. The process of creating a mesh is called tessellation. Once tessellated ...