Introduction

This short paper is a companion to one delivered at Math/Science Education and Technology 2000 in San Diego, Feb. 5-8, 2000. It gives the details of a derivation for a quartic equation that is useful in computing anamorphic images for viewing with cylindrical mirrors. We start with a cylinder of radius 1 which has an axis coincident with the Z-axis. An ``eye'' is located above the XY plane and outside of the cylinder at the point E = (r,0,e), r>1, e>0. The eye gazes in the general direction of the axis of the cylinder and slightly downward at a ``painting'' which lies in an ``image plane'' centered at the point (f,0,e), 1<f<r, between the eye and the cylinder. The plane of the painting is to be perpendicular to the X axis. Given a point F = F(u,v) = (f,u,v+e) on the painting, we trace a ray from the eye through F until it hits the cylinder at a point R. At the point R the ray is reflected and follows a new path until it hits the XY plane at a point P = P(u,v) = (a,b,0) = (a(u,v), b(u,v), 0). We now paint the point P with the same color as the point F had in the painting. This is done for every point in the original painting and a distorted image is constructed in the XY plane. If the original painting is thrown away, the eye will still see the same colors at the same spots in the image plane as they are reflected back to the eye from the points of the distorted image. We wish to derive mathematical relationships between the points E, F, R, and P.