{{distinguish|text=the supporting line of a [[line segment]]}} [[File:Reuleaux supporting lines.svg|thumb|Parallel supporting lines of a [[Reuleaux triangle]]]] [[File:00-02 Tau - Square - Reuleaux Wikipedia.gif|thumb|Animation of parallel supporting lines around a Reuleaux triangle.]]
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''.<ref name=Busemann>"The geometry of geodesics", Herbert Busemann, [https://books.google.com/books?id=t22O0XBtyJsC&dq=%22supporting+line%22&pg=PA154 p. 158]</ref> In other words, ''C'' lies completely in one of the two [[closed set|closed]] [[half-plane]]s defined by ''L'' and has at least one point on ''L''.
==Properties== 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.
==Generalizations== 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.<ref name=deza>"Encyclopedia of Distances", by [[Michel M. Deza]], [[Elena Deza]], [https://books.google.com/books?id=LXEezzccwcoC&dq=%22Grenander+distance%22&pg=PA178 p. 179]</ref>
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== If two bounded connected planar shapes have disjoint [[convex hull]]s that are separated by a positive distance, then they necessarily have exactly four common lines of support, the [[bitangent]]s of the two convex hulls. Two of these lines of support separate the two shapes, and are called '''critical support lines'''.<ref name=deza/> 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 (mathematics)|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== {{reflist}}
[[Category:Line (geometry)]]