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
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 ( θ − sin θ ) . {\displaystyle 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 ( d 2 + r 2 − R 2 2 d r ) + R 2 cos − 1 ( d 2 + R 2 − r 2 2 d R ) − 2 Δ {\displaystyle 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
Δ = 1 4 ( − d + r + R ) ( d − r + R ) ( d + r − R ) ( d + r + R ) {\displaystyle \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. The two circles overlap if d < r + R {\displaystyle d<r+R} . For sufficiently large d {\displaystyle d} , the coordinate x {\displaystyle x} of the lens centre lies between the coordinates of the two circle centers:
For small d {\displaystyle d} the coordinate x {\displaystyle x} of the lens centre lies outside the line that connects the circle centres:
By eliminating y from the circle equations x 2 + y 2 = r 2 {\displaystyle x^{2}+y^{2}=r^{2}} and ( x − d ) 2 + y 2 = R 2 {\displaystyle (x-d)^{2}+y^{2}=R^{2}} the abscissa of the intersecting rims is
x = ( d 2 + r 2 − R 2 ) / ( 2 d ) {\displaystyle x=(d^{2}+r^{2}-R^{2})/(2d)} . The sign of x, i.e., d 2 {\displaystyle d^{2}} being larger or smaller than R 2 − r 2 {\displaystyle R^{2}-r^{2}} , distinguishes the two cases shown in the images. The ordinate of the intersection is
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