In all of the discussion to follow, we use K to describe the conic sections.
Rewriting Eq. (3.5.2) in terms of the magnification m by substituting Eq. (2.3.3)
into Eq. (3.5.2) gives
K=-^^^.
(3.5.4)
(m
-\f
Transforming Eq. (3.5.3) to get the equation for the surface of revolution gives
r^ - 2i?z + (1 + K)z^ = 0, (3.5.5)
where r^ =zx^
-{-yP-.
At this point it is instructive to calculate
Ri^,
the local radius of curvature at a
point (r, z) on the mirror surface. The relation for radius of curvature is
where z' = dz/dr, z!' = d^z/dr^. Solving Eq. (3.5.5) for z and carrying out the
calculation gives
Ri,
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