Computer representation of surfacesIn technical applications of 3D computer graphics (CAx) such as computer-aided design and computer-aided manufacturing, surfaces are one way of representing objects. The other ways are wireframe (lines and curves) and solids. Point clouds are also sometimes used as temporary ways to represent an object, with the goal of using the points to create one or more of the three permanent representations. If one considers a local parametrization of a surface: then the curves obtained by varying u while keeping v fixed are coordinate lines, sometimes called the u flow lines.
DimensionIn physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it - for example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on it - for example, both a latitude and longitude are required to locate a point on the surface of a sphere.
Geometry processingGeometry processing, or mesh processing, is an area of research that uses concepts from applied mathematics, computer science and engineering to design efficient algorithms for the acquisition, reconstruction, analysis, manipulation, simulation and transmission of complex 3D models. As the name implies, many of the concepts, data structures, and algorithms are directly analogous to signal processing and .
Surface (topology)In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solid figures; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.
Hidden-surface determinationIn 3D computer graphics, hidden-surface determination (also known as shown-surface determination, hidden-surface removal (HSR), occlusion culling (OC) or visible-surface determination (VSD)) is the process of identifying what surfaces and parts of surfaces can be seen from a particular viewing angle. A hidden-surface determination algorithm is a solution to the visibility problem, which was one of the first major problems in the field of 3D computer graphics .
Surface areaThe surface area (symbol A) of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with flat polygonal faces), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces.
Geometric primitiveIn vector computer graphics, CAD systems, and geographic information systems, geometric primitive (or prim) is the simplest (i.e. 'atomic' or irreducible) geometric shape that the system can handle (draw, store). Sometimes the subroutines that draw the corresponding objects are called "geometric primitives" as well. The most "primitive" primitives are point and straight line segment, which were all that early vector graphics systems had. In constructive solid geometry, primitives are simple geometric shapes such as a cube, cylinder, sphere, cone, pyramid, torus.
Implicit surfaceIn mathematics, an implicit surface is a surface in Euclidean space defined by an equation An implicit surface is the set of zeros of a function of three variables. Implicit means that the equation is not solved for x or y or z. The graph of a function is usually described by an equation and is called an explicit representation. The third essential description of a surface is the parametric one: where the x-, y- and z-coordinates of surface points are represented by three functions depending on common parameters .
Projective representationIn the field of representation theory in mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear group where GL(V) is the general linear group of invertible linear transformations of V over F, and F∗ is the normal subgroup consisting of nonzero scalar multiples of the identity transformation (see Scalar transformation). In more concrete terms, a projective representation of is a collection of operators satisfying the homomorphism property up to a constant: for some constant .
Parametric surfaceA parametric surface is a surface in the Euclidean space which is defined by a parametric equation with two parameters . Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization.
