geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, 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

half-plane In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example ...

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

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
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 In geometry, a supporting hyperplane of a Set (mathematics), set S in Euclidean space \mathbb R^n is a hyperplane that has both of the following two properties: * S is entirely contained in one of the two closed set, closed Half-space (geometry), h ...

Critical support lines

If two bounded connected planar shapes have disjoint

convex hull In geometry, the convex hull, 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 cu ...

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 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