In geometry, a supporting lineL of a curveC in the plane is a line that contains a point of C, but does not separate any two points of C.[1] In other words, C lies completely in one of the two closedhalf-planes defined by L and has at least one point on L.
Parallel supporting lines of a Reuleaux triangleAnimation of parallel supporting lines around a Reuleaux triangle.
There can be many supporting lines for a curve at a given point. When a tangent exists at a given point, then it is the unique supporting line at this point, if it does not separate the curve.
Generalizationsedit
The notion of supporting line is also discussed for planar shapes. In this case a supporting line may be defined as a line which has common points with the boundary of the shape, but not with its interior.[2]
The notion of a supporting line to a planar curve or convex shape can be generalized to n dimension as a supporting hyperplane.
Critical support linesedit
If two bounded connected planar shapes have disjoint convex hulls that are separated by a positive distance, then they necessarily have exactly four common lines of support, the bitangents of the two convex hulls. Two of these lines of support separate the two shapes, and are called critical support lines.[2] Without the assumption of convexity, there may be more or fewer than four lines of support, even if the shapes themselves are disjoint. For instance, if one shape is an annulus that contains the other, then there are no common lines of support, while if each of two shapes consists of a pair of small disks at opposite corners of a square then there may be as many as 16 common lines of support.
Referencesedit
^"The geometry of geodesics", Herbert Busemann, p. 158
supporting, line, geometry, supporting, line, curve, plane, line, that, contains, point, does, separate, points, other, words, lies, completely, closed, half, planes, defined, least, point, parallel, supporting, lines, reuleaux, triangle, animation, parallel, . In geometry a supporting line L of a curve C in the plane is a line that contains a point of C but does not separate any two points of C 1 In other words C lies completely in one of the two closed half planes defined by L and has at least one point on L Parallel supporting lines of a Reuleaux triangle Animation of parallel supporting lines around a Reuleaux triangle Contents 1 Properties 2 Generalizations 3 Critical support lines 4 ReferencesProperties editThere can be many supporting lines for a curve at a given point When a tangent exists at a given point then it is the unique supporting line at this point if it does not separate the curve Generalizations editThe notion of supporting line is also discussed for planar shapes In this case a supporting line may be defined as a line which has common points with the boundary of the shape but not with its interior 2 The notion of a supporting line to a planar curve or convex shape can be generalized to n dimension as a supporting hyperplane Critical support lines editIf two bounded connected planar shapes have disjoint convex hulls that are separated by a positive distance then they necessarily have exactly four common lines of support the bitangents of the two convex hulls Two of these lines of support separate the two shapes and are called critical support lines 2 Without the assumption of convexity there may be more or fewer than four lines of support even if the shapes themselves are disjoint For instance if one shape is an annulus that contains the other then there are no common lines of support while if each of two shapes consists of a pair of small disks at opposite corners of a square then there may be as many as 16 common lines of support References edit The geometry of geodesics Herbert Busemann p 158 a b Encyclopedia of Distances by Michel M Deza Elena Deza p 179 Retrieved from https en wikipedia org w index php title Supporting line amp oldid 1192332380, wikipedia, wiki, book, books, library,
