The maximal representation of a shape
نویسنده
چکیده
The maximal representation of a shape is defined and algorithms for shape arithmetic are developed. Introduction A shape is any arrangement of spatial elements from among points, or lines, planes, volumes, or higher dimensional hyperplanes of limited but nonzero measure. According to Stiny (1991), a shape is in algebra U, or simply in U, whenever it is made up of elements from U. Thus, a shape is in U0 if it consists of points; in Ux if it consists of lines; and, in general, in Un if it consists of n-dimensional hyperplanes. Un, n > 0, is constructed by taking the closure under union and the Euclidean transformations of any appropriate set of n-dimensional hyperplanes. A shape may consist of more than one type of spatial element, in which case its algebra is given by the Cartesian product of the algebras of its spatial element types. Thus, a shape that is made up from points, lines, and planes is in U0x UX x U2. When dealing with shapes made up of points, the points can be distinguished by attaching nongeometric labels; in this case, a shape consisting of labeled points is in V0 where V0 is the closure under union and the Euclidean transformations of a set of labels each associated with a point. Thus, a labeled shape made up of labeled points, lines, and planes is in V0xU1xjJ2. In general, a shape s is a tuple of shapes (s, s,..., s,...), where s is a shape in algebra U, k > 0. With the exception of sets of points, each shape s can be specified in indeterminately many ways as a collection of ^-dimensional spatial elements, where U is identical to Un. The algebras U 1 and IP', i # y, may be identical. For instance, we may choose to describe the plan, elevation, and section of a building collectively as a shape in V0 x Ul x U2 x V0 x Ux x U2 x V0 x Ul x U2 . The spatial elements in shape s in algebra U, k > 0, may be combined with other spatial elements in the same shape s to form larger spatial elements. A spatial element in a shape that cannot be so combined is called a maximal spatial element. Thus, every shape s in U, k > 0, can be uniquely and minimally represented by its set of maximal spatial elements; if the dimensionality of U equals n, s is uniquely represented by its maximal set of w-planes. The partitioning of a shape into its distinct sets of maximal n-planes is termed the maximal representation of the shape. ^ B y Euclidean transformation is meant the isometric transformations augmented with scale. W The set Ux is the least set closed under union and Euclidean transformation of a line; U2 is the least set closed under union and Euclidean transformation of all triangles, in fact, the set of all right-angled triangles will suffice. Equally, U2 is the least set closed under union and affine transformation of any triangle. U3 is likewise given by the least set closed under union and Euclidean transformation of the smallest set of all tetrahedra that includes all possible angle measures; in general, Un is given by the least set closed under union and Euclidean transformation of all ^-dimensional simplices defined in « + l linearly independent points.
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