Lens (geometry)

For other uses, see Lens (optics).

A lens contained between two circular arcs of radius R, and centers at O₁ and O₂

In 2-dimensional geometry, a lens is a convex region 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.

Types

Example of two asymmetric lenses (left and right) and one symmetric lens (in the middle)

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

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

Asymmetric

The area of an asymmetric lens formed from circles of radii R and r with distance d between their centers is

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

where

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

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.

See also

• Lune, a related non-convex shape formed by two circular arcs, one bowing outwards and the other inwards
• Lemon, created by a lens rotated around an axis through its tips.

A lemon.

References

• 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.