# Bitangent

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Bitangent
> Markdown URL: https://mediated.wiki/source/Bitangent.md
> Source: https://en.wikipedia.org/wiki/Bitangent
> Source revision: 1213066445
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{short description|Line tangent to a curve at two locations}}
{{about|the geometric concept|the basis vector often called the "bitangent" in computer graphics|Frenet–Serret formulas}}
[[Image:Trott bitangents.png|right|frame|The [Trott curve](/source/Trott_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.]]

In [geometry](/source/geometry), a '''bitangent''' to a [curve](/source/curve) {{mvar|C}} is a [line](/source/Line_(geometry)) {{mvar|L}} that touches {{mvar|C}} in two distinct points {{mvar|P}} and {{mvar|Q}} and that has the same direction as {{mvar|C}} at these points. That is, {{mvar|L}} is a [tangent line](/source/tangent_line) at {{mvar|P}} and at {{mvar|Q}}.

==Bitangents of algebraic curves==
In general, an [algebraic curve](/source/algebraic_curve) will have infinitely many [secant line](/source/secant_line)s, but only finitely many bitangents.

[Bézout's theorem](/source/B%C3%A9zout's_theorem) implies that an [algebraic plane curve](/source/algebraic_plane_curve) with a bitangent must have degree at least 4. The case of the 28 [bitangents of a quartic](/source/bitangents_of_a_quartic) was a celebrated piece of geometry of the nineteenth century, a relationship being shown to the 27 lines on the [cubic surface](/source/cubic_surface).

==Bitangents of polygons==
The four bitangents of two disjoint [convex polygon](/source/convex_polygon)s may be found efficiently by an algorithm based on [binary search](/source/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 hull](/source/convex_hull)s [dynamically](/source/dynamic_convex_hull) {{harv|Overmars|van Leeuwen|1981}}. {{harvs|last1=Pocchiola|last2=Vegter|txt|year=1996a|year2=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 on [pseudotriangulation](/source/pseudotriangulation).

Bitangents may be used to speed up the [visibility graph](/source/visibility_graph) approach to solving the [Euclidean shortest path](/source/Euclidean_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 applying [Dijkstra's algorithm](/source/Dijkstra's_algorithm) to a [subgraph](/source/Glossary_of_graph_theory) of the visibility graph formed by the visibility edges that lie on bitangent lines {{harv|Rohnert|1986}}.

==Related concepts==
A bitangent differs from a [secant line](/source/secant_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, the [symmetry set](/source/symmetry_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](/source/Tangent_lines_to_two_circles) figure prominently in [Jakob Steiner](/source/Jakob_Steiner)'s 1826 construction of the [Malfatti circles](/source/Malfatti_circles), in the [belt problem](/source/belt_problem) of calculating the length of a belt connecting two pulleys, in [Casey's theorem](/source/Casey's_theorem) characterizing sets of four circles with a common tangent circle, and in [Monge's theorem](/source/Monge's_theorem) on the collinearity of intersection points of certain bitangents.

==References==
*{{citation
 | last1 = Overmars | first1 = M. H. | author1-link = Mark Overmars
 | last2 = van Leeuwen | first2 = J. | author2-link = Jan van Leeuwen
 | doi = 10.1016/0022-0000(81)90012-X
 | issue = 2
 | journal = [Journal of Computer and System Sciences](/source/Journal_of_Computer_and_System_Sciences)
 | pages = 166–204
 | title = Maintenance of configurations in the plane
 | volume = 23
 | year = 1981| hdl = 1874/15899 | hdl-access = free
 }}.
*{{citation
 |last1        = Pocchiola
 |first1       = Michel
 |last2        = Vegter
 |first2       = Gert
 |doi          = 10.1142/S0218195996000204
 |id           = [http://portal.acm.org/citation.cfm?id=160985.161159 Preliminary version] in Ninth ACM [Symposium on Computational Geometry](/source/Symposium_on_Computational_Geometry) (1993) 328–337].
 |issue        = 3
 |journal      = [International Journal of Computational Geometry and Applications](/source/International_Journal_of_Computational_Geometry_and_Applications)
 |pages        = 297–308
 |title        = The visibility complex
 |url          = https://www.di.ens.fr/~pocchiol/postscript/pv-vc-93.ps
 |volume       = 6
 |year         = 1996a
 |access-date  = 2007-04-12
 |archive-url  = https://web.archive.org/web/20061203144536/https://www.di.ens.fr/~pocchiol/postscript/pv-vc-93.ps
 |archive-date = 2006-12-03
 |url-status     = dead
}}.
*{{citation
 | last1 = Pocchiola | first1 = Michel
 | last2 = Vegter | first2 = Gert
 | journal = [Discrete and Computational Geometry](/source/Discrete_and_Computational_Geometry)
 | pages = 419–453
 | title = Topologically sweeping visibility complexes via pseudotriangulations
 | volume = 16
 | year = 1996b
 | doi = 10.1007/BF02712876
 | issue = 4| doi-access = free
 }}.
*{{citation
 | last = Rohnert | first = H.
 | doi = 10.1016/0020-0190(86)90045-1
 | issue = 2
 | journal = [Information Processing Letters](/source/Information_Processing_Letters)
 | pages = 71–76
 | title = Shortest paths in the plane with convex polygonal obstacles
 | volume = 23
 | year = 1986}}.

Category:Differential geometry
Category:Algebraic curves

---
Adapted from the Wikipedia article [Bitangent](https://en.wikipedia.org/wiki/Bitangent) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Bitangent?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.