Summary: The reconstruction of surfaces in R 3 by reflection.
by William K. Allard
1. Introduction. Among my fondest memories from childhood are the times when my entire elementary
school went on an excursion to an amusement park right before summer vacation. This park had a wonderful
fun house, in which, among such things as tilted and moving floors, were curvy mirrors. When you stood in
front of these mirrors you would appear distorted in peculiar ways. In this paper we will describe how to
construct the surface of such a curved mirror from knowledge of how a known surface reflects off it.
I was motivated to do this work by more practical considerations, however. In the field of corneal
topography, a device called a keratometer is used to map a patient's cornea. In these devices, the reflection
of a pattern on a known surface is captured. By what are often proprietary methods the data is processed
to obtain an approximate mapping of the the cornea. It appears that these methods approximately solve a
family of ordinary differential equations whose exact solutions only approximately correspond to the corneal
data desired. Moreover, providing initial conditions for these equations is problematical. See [1,5,6]. The
results turn out to be unsatisfactory in some respects . I believe the reconstruction method provided below
will overcome these difficulties. Evidence that this is indeed the case is given in the Ph.D. thesis of James S.
Rolf  who simulated numerically my reconstruction method and obtained very satisfactory results using
parameters that I believe are like those one encounters in the field.
For the remainder of this Introduction the reader should refer to Figure One. Suppose Y is a smooth
embedded 2dimensional submanifold of R 3 ¸ f0g with the property that the mapping
U : Y ! S 2