Use of Euler-rotation angles for generating antenna patterns

Antenna patterns are most naturally expressed in spherical coordinates, since the pattern angles represent directions in space. All patterns can be represented by circles on the coordinate sphere. Most patterns are either polar great-circle patterns or equatorial patterns, referring to the coordinate sphere, where all look directions lie in a plane. In some cases, however, the surface containing all of the look directions is a cone, with the origin at the antenna, and the axis along one of the Cartesian-coordinate axes. In general, all patterns are conical, known in the trade as “conics”, and the axes of the cones can be in any direction. Conical patterns are useful in cases where the beam is electronically scanned, and the sidelobes in the plane perpendicular to the plane of the scan therefore fall on a cone, instead of in a plane. Additional examples are patterns of constant range or constant Doppler shift, for airborne-radar antennas, which are all conical patterns. Calculating antenna patterns, in such cases, presents a problem, since the antenna-pattern-coordinate system will generally not be the same as the antenna- or the antenna-radome-coordinate system. The two systems may not even be "square" with each other. But by using the proper Euler angles, the desired antenna-pattern look direction can be easily converted to a coordinate system better suited to the calculations