Concepts inShape sensitivity of constructive representations
Constructive proof
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object with certain properties by creating or providing a method for creating such an object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem) which proves the validity of a proposition without considering an example.
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Constructive solid geometry
Constructive solid geometry (CSG) is a technique used in solid modeling. Constructive solid geometry allows a modeler to create a complex surface or object by using Boolean operators to combine objects. Often CSG presents a model or surface that appears visually complex, but is actually little more than cleverly combined or decombined objects. In 3D computer graphics and CAD CSG is often used in procedural modeling.
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Solid
Solid is one of the three classical states of matter. It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a gas does. The atoms in a solid are tightly bound to each other, either in a regular geometric lattice or irregularly.
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Boundary representation
In solid modeling and computer-aided design, boundary representation¿often abbreviated as B-rep or BREP¿is a method for representing shapes using the limits. A solid is represented as a collection of connected surface elements, the boundary between solid and non-solid.
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Shape optimization
Shape optimization is part of the field of optimal control theory. The typical problem is to find the shape which is optimal in that it minimizes a certain cost functional while satisfying given constraints. In many cases, the functional being solved depends on the solution of a given partial differential equation defined on the variable domain. Topology optimization is, in addition, concerned with the number of connected components/boundaries belonging to the domain.
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Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system.
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Domain of a function
In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined. That is, the function provides an "output" or value for each member of the domain. For instance, the domain of cosine is the set of all real numbers, while the domain of the square root consists only of numbers greater than or equal to 0 (ignoring complex numbers in both cases).
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