# Supporting line

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Supporting_line
> Markdown URL: https://mediated.wiki/source/Supporting_line.md
> Source: https://en.wikipedia.org/wiki/Supporting_line
> Source revision: 1343081144
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

Not to be confused with the supporting line of a [line segment](/source/Line_segment).

Parallel supporting lines of a [Reuleaux triangle](/source/Reuleaux_triangle)

Animation of parallel supporting lines around a Reuleaux triangle.

In [geometry](/source/Geometry), a **supporting line** *L* of a [curve](/source/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](/source/Closed_set) [half-planes](/source/Half-plane) 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](/source/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.[2]

The notion of a supporting line to a planar curve or convex shape can be generalized to n dimension as a [supporting hyperplane](/source/Supporting_hyperplane).

## Critical support lines

If two bounded connected planar shapes have disjoint [convex hulls](/source/Convex_hull) that are separated by a positive distance, then they necessarily have exactly four common lines of support, the [bitangents](/source/Bitangent) 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](/source/Annulus_(mathematics)) 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

1. **[^](#cite_ref-Busemann_1-0)** "The geometry of geodesics", Herbert Busemann, [p. 158](https://books.google.com/books?id=t22O0XBtyJsC&dq=%22supporting+line%22&pg=PA154)

1. ^ [***a***](#cite_ref-deza_2-0) [***b***](#cite_ref-deza_2-1) "Encyclopedia of Distances", by [Michel M. Deza](/source/Michel_M._Deza), [Elena Deza](/source/Elena_Deza), [p. 179](https://books.google.com/books?id=LXEezzccwcoC&dq=%22Grenander+distance%22&pg=PA178)

---
Adapted from the Wikipedia article [Supporting line](https://en.wikipedia.org/wiki/Supporting_line) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Supporting_line?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.