In geometry, the sagitta (sometimes abbreviated as sag, plural is sagittae) of a circular arc is the distance from the midpoint of the arc to the midpoint of its chord. It is used extensively in architecture when calculating the arc necessary to span a certain height and distance and also in optics where it is used to find the depth of a spherical mirror or lens. The name comes directly from Latin sagitta, meaning an "arrow" (with the circular arc representing a bow).
Formulas In the following equations, s {\displaystyle s} denotes the sagitta (the depth or height of the arc), r {\displaystyle r} equals the radius of the circle, and l {\displaystyle l} the length of the chord spanning the base of the arc. As 1 2 l {\displaystyle {\tfrac {1}{2}}l} and r − s {\displaystyle r-s} are two sides of a right triangle with r {\displaystyle r} as the hypotenuse, the Pythagorean theorem gives us
r 2 = ( 1 2 l ) 2 + ( r − s ) 2 . {\displaystyle r^{2}=\left({\tfrac {1}{2}}l\right)^{2}+\left(r-s\right)^{2}\,.}
This may be rearranged to give any one of s {\displaystyle s} , l {\displaystyle l} , or r {\displaystyle r} in terms of the other two:
s = r − r 2 − 1 4 l 2 , l = 2 2 r s − s 2 , r = s 2 + 1 4 l 2 2 s = s 2 + l 2 8 s . {\displaystyle {\begin{aligned}s&=r-{\sqrt {r^{2}-{\tfrac {1}{4}}l^{2}}}\,,\\[10mu]l&=2{\sqrt {2rs-s^{2}}}\,,\\[5px]r&={\frac {s^{2}+{\tfrac {1}{4}}l^{2}}{2s}}={\frac {s}{2}}+{\frac {l^{2}}{8s}}\,.\end{aligned}}}
The sagitta may also be calculated from the versine function, for an arc that spans an angle of Δ = 2θ, and coincides with the versine for unit circles. Generally, for a known value of θ, any of s, l, and r, can be computed from one of the others:
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