Concepts inShortest paths on polyhedral surfaces and terrains
Polyhedron
In elementary geometry a polyhedron (plural polyhedra or polyhedrons) is a geometric solid in three dimensions with flat faces and straight edges. The word polyhedron comes from the Classical Greek ¿¿¿¿¿¿¿¿¿, as poly- (stem of ¿¿¿¿¿, "many") + -hedron (form of ¿¿¿¿, "base", "seat", or "face"). A polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions.
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Shortest path problem
In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. An example is finding the quickest way to get from one location to another on a road map; in this case, the vertices represent locations and the edges represent segments of road and are weighted by the time needed to travel that segment.
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Path (graph theory)
In graph theory, a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence. A path may be infinite, but a finite path always has a first vertex, called its start vertex, and a last vertex, called its end vertex. Both of them are called terminal vertices of the path. The other vertices in the path are internal vertices. A cycle is a path such that the start vertex and end vertex are the same.
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Polynomial-time approximation scheme
In computer science, a polynomial-time approximation scheme (PTAS) is a type of approximation algorithm for optimization problems (most often, NP-hard optimization problems). A PTAS is an algorithm which takes an instance of an optimization problem and a parameter ¿ > 0 and, in polynomial time, produces a solution that is within a factor 1 + ¿ of being optimal (or 1 - ¿ for maximization problems).
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Vertex (geometry)
In geometry, a vertex (plural vertices) is a special kind of point that describes the corners or intersections of geometric shapes.
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Face (geometry)
In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube. The suffix -hedron is derived from the Greek word ¿¿¿¿ (hedra) which means "face". Sometimes, in the case of a pyramid, the term face is understood to exclude the base. The (two-dimensional) polygons that bound higher-dimensional polytopes are also commonly called faces.
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Metric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric. A topological space whose topology can be described by a metric is called metrizable.
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