Vertex radius measurement of an off-axis parabola with a three-ball spherometer

A spherometer is often used to precisely measure the radius of curvature of a spherical surface. It can also measure the vertex radius of a more complex surface such as an off-axis parabola (OAP). This paper provides a reliable algorithm to find the vertex radius of an OAP by solving a few equations based on the test geometry. This algorithm can also be easily expanded to any conic surface with high-order aspheric coefficients. The algorithm was verified by measuring an 8-inch diameter OAP and comparing the results with its known prescription. Results show good agreement. An example of measuring the vertex radius of a 4-m diameter OAP is also presented. In addition to this, a calculation was done to show that the coma and astigmatism are independent of the clocking of the spherometer on the optic.