View source for Lens (geometry)

{{short description|Convex plane region bounded by two circular arcs}}
[[Image:Geometric lens.gif|thumb|A lens contained between two circular arcs of radius {{mvar|R}}, and centers at {{math|''O''{{sub|1}}}} and {{math|''O''{{sub|2}}}}]]

In 2-dimensional [[Geometry|geometry]], a '''lens''' is a [[Convex set|convex]] region bounded by two [[Circular arc|circular arc]]s joined to each other at their endpoints. In order for this shape to be convex, both arcs must bow outwards (convex-convex). This shape can be formed as the [[Intersection|intersection]] of two [[Disk (mathematics)|circular disks]]. It can also be formed as the union of two [[Circular segment|circular segment]]s (regions between the [[Chord (geometry)|chord]] of a circle and the circle itself), joined along a common chord.

==Types==
[[File:geometric_lens_examples.png|thumb|Example of two asymmetric lenses (left and right) and one symmetric lens (in the middle)]]
[[File:Vesica_piscis_circles.svg|thumb|right|upright=1|The [[Vesica piscis|Vesica piscis]] is the intersection of two disks with the same radius, R, and with the distance between centers also equal to R.]]
If the two arcs of a lens have equal radius, it is called a '''symmetric lens''', otherwise is an '''asymmetric lens'''.

The [[Vesica piscis|vesica piscis]] is one form of a symmetric lens, formed by arcs of two circles whose centers each lie on the opposite arc. The arcs meet at angles of 120° at their endpoints.

==Area==
;Symmetric
The [[Area|area]] of a symmetric lens can be expressed in terms of the radius ''R'' and arc lengths ''&theta;'' in radians:
:<math>A =  R^2\left(\theta - \sin \theta \right).</math>

;Asymmetric
The area of an asymmetric lens formed from circles of radii ''R'' and ''r'' with distance ''d'' between their centers is<ref>{{MathWorld|Lens|Lens}}</ref>

:<math>A=r^2 \cos^{-1} \left(\frac{d^2+r^2-R^2}{2dr}\right) +R^2\cos^{-1}\left( \frac{d^2+R^2-r^2}{2dR}\right) -2\Delta</math>

where 

:<math>\Delta = \frac{1}{4} \sqrt{(-d+r+R)(d-r+R)(d+r-R)(d+r+R)}</math>

is the [[Triangle#Using Heron's formula|area of a triangle]] with sides ''d'', ''r'', and ''R''.

The two circles overlap if <math>d<r+R</math>. For sufficiently large <math>d</math>, the coordinate <math>x</math> of the lens centre lies between the coordinates of the two circle centers:
 
[[Image:Two overlapping circles with large distance.svg|300px|A lens contained between two circular arcs of radii ''R'' and ''r'' at distance of ''d'']]

For small <math>d</math> the coordinate <math>x</math> of the lens centre lies outside the line that connects the circle centres:

[[Image:Two overlapping circles with small distance.svg|300px|A lens contained between two circular arcs of radii ''R'' and ''r'' at distance of ''d'']]

By eliminating ''y'' from  the circle equations <math>x^2+y^2=r^2</math> and <math>(x-d)^2+y^2=R^2</math> the [[Abscissa and ordinate|abscissa]] of the intersecting rims is 

:<math>x=(d^2+r^2-R^2)/(2d)</math>.

The sign of ''x'', i.e., <math>d^2</math> being larger or smaller than <math>R^2-r^2</math>, distinguishes the two cases shown in the images.

The [[Abscissa and ordinate|ordinate]] of the intersection is

:<math>y=\sqrt{r^2-x^2} = \frac{\sqrt{[(R-d)^2-r^2][r^2-(R+d)^2]}}{2d}</math>.

Negative values under the square root indicate that the rims of the two circles do not touch
because the circles are too far apart or one circle lies entirely within the other.

The value under the square root is a biquadratic polynomial of ''d''. The four roots of this polynomial are  associated with ''y=0'' and with the four values of ''d'' where the two circles have only one point in common.

The angles in the blue triangle of sides ''d'', ''r'' and ''R'' are
:<math> \sin a_r  = y/r;\quad \sin a_R = y/R</math>
where ''y'' is the ordinate of the intersection. The branch of the arcsin with <math>a_r>\pi/2</math> is to be taken if <math>d^2<R^2-r^2</math>.

The [[Triangle#Computing the area of a triangle|area]] of the triangle is <math>\Delta = \frac12 yd</math>.

The area of the asymmetric lens is <math>A=a_r r^2+a_R R^2-yd</math>, where the two angles are measured in radians.
[This is an application of the Inclusion-exclusion principle: the two circular sectors centered at (0,0) and (d,0) with central
angles <math>2a_r</math> and <math>2a_R</math> have areas <math>2a_r r^2</math> and <math>2a_R R^2</math>. Their union covers the triangle, the flipped triangle with corner at (x,-y),  and twice the lens area.]

==Applications==
A lens with a different shape forms the answer to [[Mrs. Miniver's problem]], on finding a lens with half the area of the union of the two circles.

Lenses are used to define [[Beta skeleton|beta skeleton]]s, geometric graphs defined on a set of points by connecting pairs of points by an edge whenever a lens determined by the two points is empty.

==See also==
*Circle–circle intersection
*[[Lune (geometry)|Lune]], a related non-convex shape formed by two circular arcs, one bowing outwards and the other inwards
*[[Lemon (geometry)|Lemon]], created by a lens rotated around an axis through its tips.<ref name=mathworld>{{cite web|url=http://mathworld.wolfram.com/Lemon.html|title=Lemon|website=Wolfram MathWorld|author=Weisstein, Eric W.|access-date=2019-11-04}}</ref>
[[File:Lemon (geometry).png|thumb|A [[Lemon (geometry)|lemon]].]]

==References==
{{reflist}}

* {{cite journal|
author=Pedoe, D.|
year=1995|
title=Circles: A Mathematical View, rev. ed.|
journal=Washington, DC: Math. Assoc. Amer.|mr=1349339}}
* {{cite book|
author=Plummer, H.|
year=1960|
title=An Introductory Treatise of Dynamical Astronomy| url=https://archive.org/details/introductorytrea0000plum| url-access=registration|
location=York | publisher=Dover|bibcode=1960aitd.book.....P}}
* {{cite book|
author=Watson, G. N.|
year=1966|
title=A Treatise on the Theory of Bessel Functions, 2nd ed.|
location=Cambridge, England | publisher=Cambridge University Press|mr=1349110}}
* {{cite web|first1=M. P. |last1=Fewell|title=Area of common overlap of three circles|
year=2006|url=https://apps.dtic.mil/sti/citations/ADA463920
|archive-url=https://web.archive.org/web/20220303121106/https://apps.dtic.mil/sti/citations/ADA463920
|url-status=live
|archive-date=March 3, 2022
|institution=Defence Science and Technology Organisation}}
*{{cite journal|first1=Federico|last1=Librion|first2=Marco|last2=Levorato|first3=Michele|last3=Zorzi
|title=An algorithmic solution for computing circle intersection areas and its application to wireless communications
|journal=Wirel. Commun. Mobile Comput.|year=2012|volume=14|number=18|pages=1672–1690|doi=10.1002/wcm.2305|s2cid=2828261 |arxiv=1204.3569}}


[[Category:Convex geometry]]

{{Sourceattribution|Lens (geometry)}}