Steiner Point (other)
A Steiner point (named after Jakob Steiner) may refer to: *Steiner point (computational geometry), a point added in solving a geometric optimization problem to make its solution better *Steiner point (triangle), a certain point on the circumcircle of a given triangle *One of 20 points associated with a given set of six points on a conic; see See also *Steiner tree problem, an algorithmic problem of finding extra Steiner points to add to a point set to reduce the cost of connecting the points **The median of three vertices in a median graph, the solution to the Steiner tree problem for those three vertices **The Fermat point In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest ...

Jakob Steiner
Jakob Steiner (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry. Life Steiner was born in the village of Utzenstorf, Canton of Bern. At 18, he became a pupil of Heinrich Pestalozzi and afterwards studied at Heidelberg. Then, he went to Berlin, earning a livelihood there, as in Heidelberg, by tutoring. Here he became acquainted with A. L. Crelle, who, encouraged by his ability and by that of Niels Henrik Abel, then also staying at Berlin, founded his famous ''Journal'' (1826). After Steiner's publication (1832) of his ''Systematische Entwickelungen'' he received, through Carl Gustav Jacob Jacobi, who was then professor at Königsberg University, and earned an honorary degree there; and through the influence of Jacobi and of the brothers Alexander and Wilhelm von Humboldt a new chair of geometry was founded for him at Berlin (1834). This he occupied until his death in Bern on 1 April 1863. He was described by Thomas Hirst as follows: ...
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Steiner Point (computational Geometry)
In computational geometry, a Steiner point is a point that is not part of the input to a geometric optimization problem but is added during the solution of the problem, to create a better solution than would be possible from the original points alone. The name of these points comes from the Steiner tree problem, named after Jakob Steiner, in which the goal is to connect the input points by a network of minimum total length. If the input points alone are used as endpoints of the network edges, then the shortest network is their minimum spanning tree. However, shorter networks can often be obtained by adding Steiner points, and using both the new points and the input points as edge endpoints. Another problem that uses Steiner points is Steiner triangulation. The goal is to partition an input (such as a point set or polygon) into triangles, meeting edge-to-edge. Both input points and Steiner points may be used as triangle vertices. See also *Delaunay refinement In mesh generation ...
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Steiner Point (triangle)
In triangle geometry, the Steiner point is a particular point associated with a triangle. It is a triangle center and it is designated as the center X(99) in Clark Kimberling's Encyclopedia of Triangle Centers. Jakob Steiner (1796–1863), Swiss mathematician, described this point in 1826. The point was given Steiner's name by Joseph Neuberg in 1886. Definition The Steiner point is defined as follows. (This is not the way in which Steiner defined it.) :Let be any given triangle. Let be the circumcenter and be the symmedian point of triangle . The circle with as diameter is the Brocard circle of triangle . The line through perpendicular to the line intersects the Brocard circle at another point . The line through perpendicular to the line intersects the Brocard circle at another point . The line through perpendicular to the line intersects the Brocard circle at another point . (The triangle is the Brocard triangle of triangle .) Let be the line through parallel t ...
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Steiner Tree Problem
In combinatorial mathematics, the Steiner tree problem, or minimum Steiner tree problem, named after Jakob Steiner, is an umbrella term for a class of problems in combinatorial optimization. While Steiner tree problems may be formulated in a number of settings, they all require an optimal interconnect for a given set of objects and a predefined objective function. One well-known variant, which is often used synonymously with the term Steiner tree problem, is the Steiner tree problem in graphs. Given an undirected graph with non-negative edge weights and a subset of vertices, usually referred to as terminals, the Steiner tree problem in graphs requires a tree of minimum weight that contains all terminals (but may include additional vertices). Further well-known variants are the ''Euclidean Steiner tree problem'' and the '' rectilinear minimum Steiner tree problem''. The Steiner tree problem in graphs can be seen as a generalization of two other famous combinatorial optimization ...
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Median Graph
In graph theory, a division of mathematics, a median graph is an undirected graph in which every three vertices ''a'', ''b'', and ''c'' have a unique ''median'': a vertex ''m''(''a'',''b'',''c'') that belongs to shortest paths between each pair of ''a'', ''b'', and ''c''. The concept of median graphs has long been studied, for instance by or (more explicitly) by , but the first paper to call them "median graphs" appears to be . As Chung, Graham, and Saks write, "median graphs arise naturally in the study of ordered sets and discrete distributive lattices, and have an extensive literature".. In phylogenetics, the Buneman graph representing all maximum parsimony Phylogenetic tree, evolutionary trees is a median graph. Median graphs also arise in social choice theory: if a set of alternatives has the structure of a median graph, it is possible to derive in an unambiguous way a majority preference among them. Additional surveys of median graphs are given by , , and . Examples ...
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