Bitangent
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In mathematics, a bitangent to a curve C is a line L that touches C in two distinct points P and Q and that has the same direction to C at these points. That is, L is an tangent line at P and at Q. It differs from a secant line in that a secant line may cross the curve at the two points it intersects it. In general, an algebraic curve will have infinitely many secant lines, but only finitely many bitangents.
Bézout's theorem implies that a plane curve with a bitangent must have degree at least 4. The case of the 28 bitangents to a general plane quartic curve was a celebrated piece of geometry of the nineteenth century, a relationship being shown to the 27 lines on the cubic surface. Such bitangents are in general defined over the complex numbers, and are not real (see Salmon's Higher Plane Curves). For an example where all bitangents are real, see Trott curve.
The four bitangents of two disjoint convex polygons may be found efficiently by an algorithm based on binary search in which one maintains a binary search pointer into the lists of edges of each polygon and moves one of the pointers left or right at each steps depending on where the tangent lines to the edges at the two pointers cross each other. This bitangent calculation is a key subroutine in data structures for maintaining convex hulls dynamically (Overmars and van Leeuwen, 1981). Pocchiola and Vegter (1996a,b) describe an algorithm for efficiently listing all bitangent line segments that do not cross any of the other curves in a system of multiple disjoint convex curves, using a technique based on pseudotriangulation.
One can also consider bitangents that are not lines; for instance, the symmetry set of a curve is the locus of centers of circles that are tangent to the curve in two points.
[edit] References
- Overmars, M. H.; van Leeuwen, J. (1981). "Maintenance of configurations in the plane". J. Comput. Sys. Sci. 23 (2): 166–204. doi:.
- Pocchiola, Michel; Vegter, Gert (1996a). "The visibility complex". International Journal of Computational Geometry and Applications 6 (3): 297–308. doi:. Preliminary version in Ninth ACM Symp. Computational Geometry (1993) 328–337..
- Pocchiola, Michel; Vegter, Gert (1996b). "Topologically sweeping visibility complexes via pseudotriangulations". Discrete and Computational Geometry 16: 419–453.
