geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a bitangent to 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 ...

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

at and at .

Bitangents of algebraic curves

In general, an

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...

will have infinitely many

secant line Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to: * a secant line, in geometry * the secant variety, in algebraic geometry * secant (trigonometry) (Latin: secans), the multiplicative inverse (or recipr ...

s, but only finitely many bitangents.

Bézout's theorem Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the deg ...

implies that an

algebraic plane curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...

with a bitangent must have degree at least 4. The case of the 28

bitangents of a quartic In the theory of algebraic plane curves, a general quartic plane curve has 28 bitangent lines, lines that are tangent to the curve in two places. These lines exist in the complex projective plane, but it is possible to define quartic curves for wh ...

was a celebrated piece of geometry of the nineteenth century, a relationship being shown to the 27 lines on the

cubic surface In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather th ...

Bitangents of polygons

The four bitangents of two disjoint

convex polygon In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ...

s may be found efficiently by an algorithm based on

binary search In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search algorithm that finds the position of a target value within a sorted array. Binary search compares the target value to the m ...

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 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 dynamically . 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 In Euclidean plane geometry, a pseudotriangle (''pseudo-triangle'') is the simply connected subset of the plane that lies between any three mutually tangent convex sets. A pseudotriangulation (''pseudo-triangulations'') is a partition of a region ...

. Bitangents may be used to speed up the

visibility graph In computational geometry and robot motion planning, a visibility graph is a graph of intervisible locations, typically for a set of points and obstacles in the Euclidean plane. Each node in the graph represents a point location, and each edge repr ...

approach to solving the

Euclidean shortest path The Euclidean shortest path problem is a problem in computational geometry: given a set of polyhedral obstacles in a Euclidean space, and two points, find the shortest path between the points that does not intersect any of the obstacles. Two d ...

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 applying

Dijkstra's algorithm Dijkstra's algorithm ( ) is an algorithm for finding the shortest paths between nodes in a graph, which may represent, for example, road networks. It was conceived by computer scientist Edsger W. Dijkstra in 1956 and published three years ...

to a subgraph of the visibility graph formed by the visibility edges that lie on bitangent lines .

Related concepts

A bitangent differs from a

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, the

symmetry set In geometry, the symmetry set is a method for representing the local symmetries of a curve, and can be used as a method for representing the shape of objects by finding the topological skeleton. The medial axis The medial axis of an object is ...

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 in

Jakob Steiner Jakob Steiner (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry. Life Steiner was born in the village of Utzenstorf, Canton of Bern. At 18, he became a pupil of Heinrich Pestalozzi and afterwards st ...

's 1826 construction of the

Malfatti circles In geometry, the Malfatti circles are three circles inside a given triangle such that each circle is tangent to the other two and to two sides of the triangle. They are named after Gian Francesco Malfatti, who made early studies of the problem o ...

, in the

belt problem The belt problem is a mathematics problem which requires finding the length of a crossed belt that connects two circular pulleys with radius ''r''1 and ''r''2 whose centers are separated by a distance ''P''. The solution of the belt problem requ ...

of calculating the length of a belt connecting two pulleys, in

Casey's theorem In mathematics, Casey's theorem, also known as the generalized Ptolemy's theorem, is a theorem in Euclidean geometry named after the Irish mathematician John Casey. Formulation of the theorem Let \,O be a circle of radius \,R. Let \,O_1, O_2 ...

characterizing sets of four circles with a common tangent circle, and in

Monge's theorem In geometry, Monge's theorem, named after Gaspard Monge, states that for any three circles in a plane, none of which is completely inside one of the others, the intersection points of each of the three pairs of external tangent lines are collinea ...

on the collinearity of intersection points of certain bitangents.

References

*. *. *. *{{citation , last = Rohnert , first = H. , doi = 10.1016/0020-0190(86)90045-1 , issue = 2 , journal =

Information Processing Letters ''Information Processing Letters'' is a peer reviewed scientific journal in the field of computer science, published by Elsevier Elsevier () is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its ...

, pages = 71–76 , title = Shortest paths in the plane with convex polygonal obstacles , volume = 23 , year = 1986. Differential geometry Algebraic curves