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A 3D torus prim created in Second Life, an example of a parametric shape. A Parametric shape is a standardized two-dimensional or three-dimensional shape defined by a minimal set of parameters, such as an ellipse defined by two points at its foci, or three points at its center, vertex, and co-vertex.
Computing the convex hull means that a non-ambiguous and efficient representation of the required convex shape is constructed. The complexity of the corresponding algorithms is usually estimated in terms of n , the number of input points, and sometimes also in terms of h , the number of points on the convex hull.
Solid modeling (or solid modelling) is a consistent set of principles for mathematical and computer modeling of three-dimensional shapes . Solid modeling is distinguished within the broader related areas of geometric modeling and computer graphics , such as 3D modeling , by its emphasis on physical fidelity. [ 1 ]
Geometric modeling is a branch of applied mathematics and computational geometry that studies methods and algorithms for the mathematical description of shapes.The shapes studied in geometric modeling are mostly two- or three-dimensional (solid figures), although many of its tools and principles can be applied to sets of any finite dimension.
Similar to the example above, hashing applies to higher-dimensional data. For three-dimensional data points, three points are also needed for the basis. The first two points define the x-axis, and the third point defines the y-axis (with the first point). The z-axis is perpendicular to the created axis using the right-hand rule.
Alpha shapes are closely related to alpha complexes, which are subcomplexes of the Delaunay triangulation of the point set. 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.
A convenient property of CSG shapes is that it is easy to classify arbitrary points as being either inside or outside the shape created by CSG. The point is simply classified against all the underlying primitives and the resulting boolean expression is evaluated. [6] This is a desirable quality for some applications such as ray tracing. [6]
Most commonly, it is the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called 3-manifolds. The term may also refer colloquially to a subset of space, a three-dimensional region (or 3D domain), [1] a solid figure.