Concepts inComputing elevation maxima by searching the gauss sphere
Sphere
A sphere (from Greek ¿¿¿¿¿¿ ¿ sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle, which is in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point. This distance r is known as the "radius" of the sphere. The maximum straight distance through the sphere is known as the "diameter" of the sphere.
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Maxima and minima
In mathematics, the maximum and minimum of a function, known collectively as extrema, are the largest and smallest value that the function takes at a point either within a given neighborhood (local or relative extremum) or on the function domain in its entirety (global or absolute extremum). More generally, the maximum and minimum of a set (as defined in set theory) are the greatest and least element in the set.
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Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, ¿1 and ¿2, of the given point. It is an intrinsic measure of curvature, i.e. , its value depends only on how distances are measured on the surface, not on the way it is isometrically embedded in space. This result is the content of Gauss's Theorema egregium. Symbolically, the Gaussian curvature ¿ is defined as . where and are the principal curvatures.
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Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply.
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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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Manifold
In mathematics, a manifold is a mathematical object that on a small enough scale resembles Euclidean space. For example, seen from far away, the surface of the planet Earth is not flat and Euclidean, but on a smaller scale, one may describe each region via a geographic map, a projection of the surface onto the Euclidean plane. A precise mathematical definition of a manifold is given below. Lines and circles (but not figure eights) are one-dimensional manifolds (1-manifolds).
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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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