The `clamshell problem´-interpolation of shaped reflectors and other smooth surfaces

An interpolation method for shaped reflector antennas and similar smooth surfaces is developed using cubic spline interpolants in a parametric representation. Ordinary one-dimensional splines are combined in tensor products to give a two-dimensional interpolant for each cartesian component of points of the surface. The two independent variables are the polar coordinates of rays traced from an aperture disk. This ray tracing maps a grid of lines from the aperture to the surface being interpolated and gives a general-purpose method for interpolating smooth surfaces. The polar coordinates are partitioned into uniform intervals, which simplifies the calculations. The interpolant is differentiated to provide partial derivatives of the surface coordinates, and these derivatives are combined to give surface normals and Jacobians. The bicubic spline is also integrated to give a general-purpose two-dimensional integration routine. The parametric form makes it easy to find a variety of cross sections, boundaries, inflection points, and other characteristics of the surface