The boundary of a shape and its classification (original) (raw)

The algebra of shapes

1994

This thesis investigates a new representation scheme for geometric modeling, based on an algebraic model for shapes and formalized using a boundary representation. The algebraic model is mathematically uniform for shapes of all kinds and provides a natural and intuitive framework for mixed-dimensional shapes. The corresponding maximal element representation is essential to the concept of shape emergence. The representation scheme particularly supports computational design as a generative process of search or exploration. This thesis begins with a treatment of the algebraic and geometric properties of shapes and gives a formal and complete definition of the maximal element representation for n-dimensional shapes in a k-dimensional space (n le\lele k). Efficient algorithms are presented for the algebraic operations of sum, product, difference and symmetric difference on shapes of plane and volume segments. An exploration of related research in shape grammars, computational design and co...

Algorithms for classifying and constructing the boundary of a shape

J. of Design Research, 2006

This paper continues with the subject matter that we introduced previously (Krishnamurti and . Here, we describe algorithms for classifying the boundary of a shape with respect to another, coequal, shape and for constructing the description of a shape given parts of the boundary that make up the shape. Specifically, algorithms for classification and construction of shapes in U 23 (plane shapes) and in U 33 (volume shapes) are described in this paper. These procedures form a unified basis for shape arithmetic. Architecture and CAAD at ETH Zurich. His research interests include computational issues of description, modelling and representation for design in the areas of information exchange, collaboration, shape recognition and generation, geometric modelling and visualisation.

Algebras and grammars for shapes and their boundaries

Environment and Planning B: Planning and Design, 2001

A special kind of shape algebra is defined for concurrent computations with shapes and their boundaries, together with related shape grammars. Some new types of shape grammar are defined and classified.

Arithmetic operations among shapes using shape numbers

Pattern Recognition, 1981

In this work, a notation is given called the Discrete Geometry of Shapes, which describes the forms or shapes of flat regions limited by simply connected curves. A procedure is given that deduces from every region a unique number (its shape number) independent of translation and rotation, and optionally, of size and origin.All the integer numbers contain all the universe of discrete shapes (of course with different precision). In this universe there are shapes such as straight lines, circumferences, ellipses, parabolas, trigonometric functions, graphics of time, absorption waves, etc.The Discrete Geometry of Shapes is one-dimensional. It does not use the definition of equation and function to define shapes in a rectangular co-ordinate plane. With this notation it is possible to generate shapes with any characteristics by generating numerical sequences; also it is possible to do arithmetic operations among shapes. For example, the addition of a square and a circle, the average of a triangle and a circle, the square root of a pentagon, the numerical relations between given shapes, etc.Section V of this work describes the third dimension in the Discrete Geometry of Shapes for surfaces and volumes by means of a vector of shape numbers. It is possible to add surfaces, to divide volumes, to obtain the square root of a volume, etc.The main objective of this notation is the simplification of some mathematical and geometrical processes in this analysis of shapes and surfaces.

Mathematical morphological operations of boundary-represented geometric objects

Journal of Mathematical Imaging and Vision, 1996

The resemblance between the integer number system with multiplication and division and the system of convex objects with Minkowski addition and decomposition is really striking. The resemblance also indicates a computational technique which unifies the two Minkowski operations as a single operation. To view multiplication and division as a single operation, it became necessary to extend the integer number system to the rational number system. The unification of the two Minkowski operations also requires that the ordinary convex object domain must be appended by a notion of inverse objects or negative objects. More interestingly, the concept of negative objects permits further unification. A nonconvex object may be viewed as a mixture of ordinary convex object and negative object, and thereby, makes it possible to adopt exactly the same computational technique for convex as well as nonconvex objects. The unified technique, we show, can be easily understood and implemented if the input polygons and polyhedra are represented by their slope diagram representations.

A uniform characterization of augmented shapes

Computer-Aided Design, 2018

Shapes are considered as finite arrangements of spatial elements from among points, line and plane segments, circles and ellipses, (circular) arcs, quadratic Bezier curves, of limited but nonzero measure, in 2D and 3D. Augmented shapes are defined as shapes augmented with attributes, e.g., labels, weights, colors, enumerative values, and (parametric) descriptions. Different attribute types specify different behaviors under operations of sum, product and difference and a part relationship. We review different shape attribute propositions from the shape grammar literature and characterize them uniformly. This uniform characterization of augmented shapes is intended to assist in formalizing new shape attribute propositions that may have been visually conceived. Highlights:  A uniform characterization of the behavior of different attribute types is presented under operations of sum product and difference, and a part relationship.  A uniform description of the behavior of augmented shapes, i.e., shapes with attributes, is presented under the same operations and relationship.

A combinatorial description of shape theory

arXiv (Cornell University), 2022

We give a combinatorial description of shape theory using finite topological T 0-spaces (finite partially ordered sets). This description may lead to a sort of computational shape theory. Then we introduce the notion of core for inverse sequences of finite spaces and prove some properties.

A logic-based framework for shape representation

Computer-Aided Design, 1996

Shapes represent a very important way with which we perceive and reason about the world. In this article we develop a logic-based framework to represent graphical shapes in two dimensions. Based on the concept of halfplanes, this framework allows us to represent regions as predicates in logic. This representation is applied to demonstrate shape concepts associated with topology and emergence.