Line segment

In geometry, a line segment is a part of a line, bounded by two distinct end points, contains every point on the line between its endpoints. A closed line segment includes both endpoints. Examples of line segments include the sides of a square. More when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge if they are adjacent vertices, or otherwise a diagonal; when the end points both lie on a curve such as a circle, a line segment is called a chord. If V is a vector space over R or C, L is a subset of V L is a line segment if L can be parameterized as L = for some vectors u, v ∈ V, in which case the vectors u and u + v are called the end points of L. Sometimes one needs to distinguish between "open" and "closed" line segments. One defines a closed line segment as above, an open line segment as a subset L that can be parametrized as L = for some vectors u, v ∈ V. Equivalently, a line segment is the convex hull of two points. Thus, the line segment can be expressed as a convex combination of the segment's two end points.

In geometry, it is sometimes defined that a point B is between two other points A and C, if the distance AB added to the distance BC is equal to the distance AC. Thus in R 2 the line segment with endpoints A = and C = is the following collection of points:. A line segment is a non-empty set. If V is a topological vector space a closed line segment is a closed set in V. However, an open line segment is an open set in V if and only if V is one-dimensional. More than above, the concept of a line segment can be defined in an ordered geometry. A pair of line segments can be any one of the following: intersecting, skew, or none of these; the last possibility is a way that line segments differ from lines: if two nonparallel lines are in the same Euclidean plane they must cross each other, but that need not be true of segments. In an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else be defined in terms of an isometry of a line.

Segments play an important role in other theories. For example, a set is convex if the segment that joins any two points of the set is contained in the set; this is important because it transforms some of the analysis of convex sets to the analysis of a line segment. The Segment Addition Postulate can be used to add congruent segment or segments with equal lengths and substitute other segments into another statement to make segments congruent. A line segment can be viewed as a degenerate case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints, the eccentricity goes to one. A standard definition of an ellipse is the set of points for which the sum of a point's distances to two foci is a constant. A complete orbit of this ellipse traverses the line segment twice; as a degenerate orbit this is a radial elliptic trajectory. In addition to appearing as the edges and diagonals of polygons and polyhedra, line segments appear in numerous other locations relative to other geometric shapes.

Some frequently considered segments in a triangle include the three altitudes, the three medians, the perpendicular bisectors of the sides, the internal angle bisectors. In each case there are various equalities relating these segment lengths to others as well as various inequalities. Other segment

Simplex

In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. A k-simplex is a k-dimensional polytope, the convex hull of its k + 1 vertices. More formally, suppose the k + 1 points u 0, …, u k ∈ R k are affinely independent, which means u 1 − u 0, …, u k − u 0 are linearly independent; the simplex determined by them is the set of points C =. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, a 4-simplex is a 5-cell. A single point may be considered a 0-simplex, a line segment may be considered a 1-simplex. A simplex may be defined as the smallest convex set containing the given vertices. A regular simplex is a simplex, a regular polytope. A regular n-simplex may be constructed from a regular -simplex by connecting a new vertex to all original vertices by the common edge length; the standard simplex or probability simplex is the simplex formed from the k + 1 standard unit vectors, or. In topology and combinatorics, it is common to “glue together” simplices to form a simplicial complex.

The associated combinatorial structure is called an abstract simplicial complex, in which context the word “simplex” means any finite set of vertices. The concept of a simplex was known to William Kingdon Clifford, who wrote about these shapes in 1886 but called them "prime confines". Henri Poincaré, writing about algebraic topology in 1900, called them "generalized tetrahedra". In 1902 Pieter Hendrik Schoute described the concept first with the Latin superlative simplicissimum and with the same Latin adjective in the normal form simplex; the regular simplex family is the first of three regular polytope families, labeled by Coxeter as αn, the other two being the cross-polytope family, labeled as βn, the hypercubes, labeled as γn. A fourth family, the infinite tessellation of hypercubes, he labeled as δn. A 0-simplex is a point. A 1-simplex is a line segment. A 2-simplex is a triangle. A 3-simplex is a tetrahedron. A 4-simplex is a 5-cell; the convex hull of any nonempty subset of the n + 1 points that define an n-simplex is called a face of the simplex.

Faces are simplices themselves. In particular, the convex hull of a subset of size m + 1 is an m-simplex, called an m-face of the n-simplex; the 0-faces are called the vertices, the 1-faces are called the edges, the -faces are called the facets, the sole n-face is the whole n-simplex itself. In general, the number of m-faces is equal to the binomial coefficient; the number of m-faces of an n-simplex may be found in column of row of Pascal's triangle. A simplex A is a coface of a simplex B if B is a face of A. Face and facet can have different meanings when describing types of simplices in a simplicial complex; the number of 1-faces of the n-simplex is the n-th triangle number, the number of 2-faces of the n-simplex is the th tetrahedron number, the number of 3-faces of the n-simplex is the th 5-cell number, so on. An - simplex can be constructed as a join of a point. An - simplex can be constructed as a join of an n-simplex; the two simplices are oriented to be normal from each other, with translation in a direction orthogonal to both of them.

A 1-simplex is the join of two points: ∨ = 2 ·. A general 2-simplex is the join of three points: ∨ ∨. An isosceles triangle is the join of a 1-simplex and a point: ∨. An equilateral triangle is 3· or. A general 3-simplex is the join of 4 points: ∨ ∨ ∨. A 3-simplex with mirror symmetry can be expressed as the join of an edge and two points: ∨ ∨. A 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point: 3.∨ or ∨. A regular tetrahedron is and so on. In some conventions, the empty set is defined to be a -simplex. Th