Lens (geometry)

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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 biconvex (convex-convex) shape comprising two circular arcs, joined at their endpoints. A similar concave-convex shape is called a lune.

Example asymmmetric and symmetric lenses

If the arcs 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.

The Vesica piscis is the intersection of two disks with the same radius, with the center of each disk on the perimeter of the other.

A lens can be seen as two circular segments, attached along their common chord. The area inside a symmetric lens can be defined by the radius R and arc lengths θ in radians:

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

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.


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  • 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.