Reading optics books there is sometimes a reference to the radius of curvature of a paraboloid or some other non spherical curve. Since the radius of the curve changes from center to edge how can this be? Finally I found that if the zone of 'radius of curvature' of a lens or mirror is unspecified they are generally referring to its 'radius of curvature' at its center.
If we plot the difference between a sphere and a parabola zone by zone we get a curve like the top curve in the family of curves above. If we specify that the reference sphere is not at the center but is now at the 70 percent zone we get a difference curve like the middle curve above. If the reference sphere has the 'radius of curvature' of the mirror edge we get the bottom curve.
When doing the knife edge test it turns out that the apparent landscape of a parabolic mirror looks much like the shadows that would be cast if a light were to be shining across these curves at a low angle.
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