# 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:

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

where

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.

## See also

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

## References

- Weisstein, Eric W. "Lens".
*MathWorld*. - 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.