Concepts inSurface matching using consistent pants decomposition
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R ¿ for example, the surface of a ball. On the other hand, there are surfaces, such as the Klein bottle, that cannot be embedded in three-dimensional Euclidean space without introducing singularities or self-intersections.
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Trousers
Trousers are an item of clothing worn from the waist to the ankles, covering both legs separately (rather than with cloth stretching across both as in skirts and dresses). The word trousers is used in the UK and Ireland, but some other English-speaking countries such as Canada, South Africa, and the United States can also refer to such items of clothing as pants. Additional synonyms include slacks, strides, kegs or kex, breeches, or breeks.
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Topology
Topology (from the Greek ¿¿¿¿¿, ¿place¿, and ¿¿¿¿¿, ¿study¿) is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation. Ideas that are now classified as topological were expressed as early as 1736.
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Curve
In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but which is not required to be straight. This entails that a line is a special case of curve, namely a curve with null curvature. Often curves in two-dimensional or three-dimensional (space curves) Euclidean space are of interest.
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Piecewise
In mathematics, a piecewise-defined function (also called a piecewise function) is a function whose definition changes depending on the value of the independent variable. Mathematically, a real-valued function f of a real variable x is a relationship whose definition is given differently on disjoint subsets of its domain (known as subdomains).
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Harmonic map
A (smooth) map ¿:M¿N between Riemannian manifolds M and N is called harmonic if it is a critical point of the Dirichlet energy functional This functional E will be defined precisely below¿one way of understanding it is to imagine that M is made of rubber and N made of marble (their shapes given by their respective metrics), and that the map ¿:M¿N prescribes how one "applies" the rubber onto the marble: E(¿) then represents the total amount of elastic potential energy resulting from tension in the rubber.
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Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous". Continuity of functions is one of the core concepts of topology, which is treated in full generality below.
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