Bitangent

This article is about the geometric concept. For the basis vector often called the "bitangent" in computer graphics, seeFrenet–Serret formulas.

TheTrott curve(black) has 28 real bitangents (red). This image shows 7 of them; the others are symmetric with respect to 90° rotations through the origin and reflections through the two blue axes.

Ingeometry,abitangentto acurveCis alineLthat touchesCin two distinct pointsPandQand that has the same direction asCat these points. That is,Lis atangent lineatPand atQ.

Bitangents of algebraic curves

In general, analgebraic curvewill have infinitely manysecant lines,but only finitely many bitangents.

Bézout's theoremimplies that analgebraic plane curvewith a bitangent must have degree at least 4. The case of the 28bitangents of a quarticwas a celebrated piece of geometry of the nineteenth century, a relationship being shown to the 27 lines on thecubic surface.

Bitangents of polygons

The four bitangents of two disjointconvex polygonsmay be found efficiently by an algorithm based onbinary searchin 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 maintainingconvex hullsdynamically(Overmars & van Leeuwen 1981). Pocchiola and Vegter (1996a,1996b) 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 onpseudotriangulation.

Bitangents may be used to speed up thevisibility graphapproach to solving theEuclidean shortest pathproblem: the shortest path among a collection of polygonal obstacles may only enter or leave the boundary of an obstacle along one of its bitangents, so the shortest path can be found by applyingDijkstra's algorithmto asubgraphof the visibility graph formed by the visibility edges that lie on bitangent lines (Rohnert 1986).

Related concepts

A bitangent differs from asecant linein that a secant line may cross the curve at the two points it intersects it. One can also consider bitangents that are not lines; for instance, thesymmetry setof a curve is the locus of centers of circles that are tangent to the curve in two points.

Bitangents to pairs of circlesfigure prominently inJakob Steiner's 1826 construction of theMalfatti circles,in thebelt problemof calculating the length of a belt connecting two pulleys, inCasey's theoremcharacterizing sets of four circles with a common tangent circle, and inMonge's theoremon the collinearity of intersection points of certain bitangents.

References

• Overmars, M. H.;van Leeuwen, J.(1981), "Maintenance of configurations in the plane",Journal of Computer and System Sciences,23(2): 166–204,doi:10.1016/0022-0000(81)90012-X,hdl:1874/15899.
• Pocchiola, Michel; Vegter, Gert (1996a),"The visibility complex",International Journal of Computational Geometry and Applications,6(3): 297–308,doi:10.1142/S0218195996000204,Preliminary versionin Ninth ACMSymposium on Computational Geometry(1993) 328–337]., archived fromthe originalon 2006-12-03,retrieved2007-04-12.
• Pocchiola, Michel; Vegter, Gert (1996b), "Topologically sweeping visibility complexes via pseudotriangulations",Discrete and Computational Geometry,16(4): 419–453,doi:10.1007/BF02712876.
• Rohnert, H. (1986), "Shortest paths in the plane with convex polygonal obstacles",Information Processing Letters,23(2): 71–76,doi:10.1016/0020-0190(86)90045-1.