In the following we use the arc described by the two-dimensional super
quadric equation (taking its exponent upsilon to be any positive real
number) in the first quadrant only and revolve it about its major axis
to obtain a body of revolution family of geometric shapes called supe
rspheroids. For certain values of length and radius and assuming that
1 < upsilon < 2, we have determined new shapes that are appropriate fo
r high speed missile radomes. We have found that the superspheroid wit
h optimized exponent value upsilon = 1.381 can almost exactly reproduc
e the traditional Von Karman radome geometry. Incidence angle maps and
geometric properties have been determined for this superspheroidal fa
mily. We have used a ray tracing analysis to obtain boresight error in
duced by this family of shapes as a function of gimbal angle. The supe
rspheroids are mathematically simple, can approximate most of the trad
itional radome geometries quite well, and are exceptionally easy to ei
ther program or use analytically.