This paper describes calculation methods to determine an optical reflected position on a mirror ball. Optical measurement is useful to develop a smart sensor for measuring small displacement, since it has a lot of merits to be free from electromagnetic noise, temperature, in addition to its high response and much information of transmission. Our idea is to apply the measurement to a direction-of-action sensor utilizing optical reflection on a mirror ball in a spherical vessel. For this purpose, geometry of the optical reflection is analyzed so that an appropriate calculation procedure of the reflected position can be obtained.
It is shown that the reflected position is given by solving six degree linear equation. To get the solution, a method of numerical iteration is proposed. Also, three approximation methods are introduced to get the position uniquely. These are called 1) concentric tangential method, 2) two-step angular bisection method, and 3) simultaneous angular bisection method. Calculated results are compared with the most likely data by numerical iteration to show that the concentric tangential method gives the position with minimum error in the three methods.