# Lens (geometry)

In 2-dimensional geometry, a lens is a convex set bounded by two circular arcs 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 of two circular disks. It can also be formed as the union of two circular segments (regions between the chord of a circle and the circle itself), joined along a common chord.

## Special cases

If the two arcs of a lens have equal radii, it is called a symmetric lens, otherwise is an asymmetric lens.

The vesica piscis is one form of a symmetrical 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

The area inside a symmetric lens can be expressed in terms of the radii R and arc lengths θ in radians:

${\displaystyle A=R^{2}\left(\theta -\sin \theta \right).}$

The area of an asymetric lens formed from circles of radii R and r with distance d between their centers is[1]

${\displaystyle A=r^{2}\cos ^{-1}\left({\frac {d^{2}+r^{2}-R^{2}}{2dr}}\right)+R^{2}\cos ^{-1}\left({\frac {d^{2}+R^{2}-r^{2}}{2dR}}\right)-2\Delta }$

where

${\displaystyle \Delta ={\frac {1}{4}}{\sqrt {(-d+r+R)(d-r+R)(d+r-R)(d+r+R)}}}$

is the area of a triangle with sides d, r, and R.

## Applications

A lens with a different shape forms part of the answer to Mrs. Miniver's problem, which asks how to bisect the area of a disk by an arc of another circle with given radius. One of the two areas into which the disk is bisected is a lens.

Lenses are used to define beta skeletons, 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.

• Lune, a related non-convex shape formed by two circular arcs, one bowing outwards and the other inwards

## References

1. Weisstein, Eric W. "Lens". MathWorld.
2. Weisstein, Eric W. "Lemon". Wolfram MathWorld. Retrieved 2019-11-04.
• Pedoe, D. (1995). "Circles: A Mathematical View, rev. ed". Washington, DC: Math. Assoc. Amer.
• Plummer, H. (1960). An Introductory Treatise of Dynamical Astronomy. York: Dover.
• Watson, G. N. (1966). A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press.