Minimum Distortion

As we experienced with spherical and parabloid mirrors, when the mirror is of a quadratic function, there is a great distortion in the image.  In a study done by Andrew Hicks, University of Pennsylvania, it was found that when the distance between the object and the image plane is of a linear function, the image captured by a mirror in both perspective and orthographic projections has no distortion. 

In the picture below, you're looking at an orthographic projection.  From the Law of Reflection, we know, the angle of incidence = the angle of reflection = ϴ. From the paper we were given the
tan(ϴ )=F'(x) and we used the image plane, tangent line from the point on the mirror, and the parallel light ray to uderstand where the equation came from.  In the picture, the angle between the normal line and tangent line is 90 degrees.  To get the smaller angle for our traingle and solve for the unknown angle formed by the interesection of the image plane and tangent line, we identified the angle as (90-ϴ ).  The angle formed from the interesection of the parallel line an the image plane was know to be 90 degrees.  With this information from the picture, we go that ϴ was equal to our unknown angle, and the tan(ϴ) was equal to F'(x) because it represented the raise and run from the image plance to the point on the mirror containing F(x). Our was as follows





ORTHOGRAPHIC PROJECTION GRAPH

 With the infromation from the picture, we used the double angle identity for  the tan function to get

We were then able to use Maple to put in the function and get  an equation for F'(x).  In the new form of an ordinary differentiable equation, we solve for F(x) using it's derivative F'(x).  The equation for F(x) was the surface of our mirror with minimal distortion and we then graphed it in Maple to get a visual idea of what our discovered mirror surface looked like.  THE END

FUNCTION OF A MIRROR SURFACE IN ORTHOGRAPHIC PROJECTION 



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