geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a supporting line ''L'' of a

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

''C'' in the plane is a line that contains a point of ''C'', but does not separate any two points of ''C''."The geometry of geodesics", Herbert Busemann
p. 158
/ref> In other words, ''C'' lies completely in one of the two

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

half-planes 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 In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

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."Encyclopedia of Distances", by Michel M. Deza,

Elena Deza Elena Ivanovna Deza (russian: Елена Ивановна Деза, née Panteleeva; born 23 August 1961) is a French and Russian mathematician known for her books on metric spaces and figurate numbers. Education and career Deza was born on 23 A ...

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 In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...

s that are separated by a positive distance, then they necessarily have exactly four common lines of support, the

bitangent In geometry, a bitangent to a curve is a line that touches in two distinct points and and that has the same direction as at these points. That is, is a tangent line at and at . Bitangents of algebraic curves In general, an algebraic curv ...

s of the two convex hulls. Two of these lines of support separate the two shapes, and are called critical support lines. 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 Annulus (or anulus) or annular indicates a ring- or donut-shaped area or structure. It may refer to: Human anatomy * ''Anulus fibrosus disci intervertebralis'', spinal structure * Annulus of Zinn, a.k.a. annular tendon or ''anulus tendineus com ...

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 Geometry