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 defined by the radius R and arc lengths θ in radians:
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
- 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.