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The cone-paraboloid intercept

Readers might at times have wondered at some of the reflector shapes to be seen at airports and satellite communication centres around the world. Typically, such shapes occur when a cone of energy emanates from a focal point to illuminate an offset portion of a paraboloidal reflector (a parabola of revolution). The purpose of this note is to show how such shapes can be determined mathematically using the tools of trigonometry and geometry established at school level. The cross-section of the energy cone is usually circular but in some instances it is elliptic and it is this case that is the focus of the present article.