Sagitta (geometry)


In geometry, the sagitta of a circular arc is the distance from the center of the arc to the center of its base. 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.

Formulae

In the following equations, denotes the sagitta, equals the radius of the circle, and the length of the chord spanning the base of the arc. As and are two sides of a right triangle with as the hypotenuse, the Pythagorean theorem gives us
This may be rearranged to give any of the other three:
The sagitta may also be calculated from the versine function, for an arc that spans an angle of, and coincides with the versine for unit circles

Approximation

When the sagitta is small in comparison to the radius, it may be approximated by the formula
Alternatively, if the sagitta is small and the sagitta, radius, and chord length are known, they may be used to estimate the arc length by the formula
where is the length of the arc; this formula was known to the Chinese mathematician Shen Kuo, and a more accurate formula also involving the sagitta was developed two centuries later by Guo Shoujing.

Applications

Architects, engineers, and contractors use these equations to create "flattened" arcs that are used in curved walls, arched ceilings, bridges, and numerous other applications.
The sagitta also has uses in physics where it is used, along with chord length, to calculate the radius of curvature of an accelerated particle. This is used especially in bubble chamber experiments where it is used to determine the momenta of decay particles.