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 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[edit]

Example asymmmetric and symmetric lenses
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.

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[edit]

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).

Applications[edit]

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[edit]

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

References[edit]

  • 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. 

External links[edit]