In
geometry, 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 (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
''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 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
Michel Marie Deza (27 April 1939. – 23 November 2016) was a Soviet and French mathematician, specializing in combinatorics, discrete geometry and graph theory. He was the retired director of research at the French National Centre for Sc ...
, 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.
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 bitangents 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