Homologous Deformations of Tiltable Telescopes

The deformation of a structure is defined to be homologous if a given geometrical relation holds for a given number of structural points, before, during, and after the deformation. Applied to radio telescopes, the task is to find a structure where the gravitational deformations of the surface, when tilted, transform one paraboloid of revolution into another one, yielding an exactly focussing mirror for any angle of tilt. A mathematical method is presented for obtaining homology solutions, in terms of the bar areas of all members, for any telescope structure with given geometrical shape. The method is linearized and iterative; it uses the notations of linear algebra. Finally, a two-gradient method is suggested for obtaining that homology solution which fulfils all external load conditions with a minimum amount of total weight.