Ingeometry, abitangent to acurveC is alineL that touchesC in two distinct pointsP andQ and that has the same direction asC at these points. That is,L is atangent line atP and atQ.
In general, analgebraic curve will have infinitely manysecant lines, but only finitely many bitangents.
Bézout's theorem implies that analgebraic plane curve with a bitangent must have degree at least 4. The case of the 28bitangents of a quartic was a celebrated piece of geometry of the nineteenth century, a relationship being shown to the 27 lines on thecubic surface.
The four bitangents of two disjointconvex polygons may be found efficiently by an algorithm based onbinary 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 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 graph approach to solving theEuclidean shortest path problem: 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 algorithm to asubgraph of the visibility graph formed by the visibility edges that lie on bitangent lines (Rohnert 1986).
A bitangent differs from asecant line in 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 set of a curve is the locus of centers of circles that are tangent to the curve in two points.
Bitangents to pairs of circles figure prominently inJakob Steiner's 1826 construction of theMalfatti circles, in thebelt problem of calculating the length of a belt connecting two pulleys, inCasey's theorem characterizing sets of four circles with a common tangent circle, and inMonge's theorem on the collinearity of intersection points of certain bitangents.