Generalization of minimal surfaces and simulation of the shape of an orthotropic material construction

An axisymmetric reflector consists of parabolic spokes and metallic fabric attached to them. A piece of this fabric between the adjacent spokes is subject to the so-called ‘mattress effect’ (bending into the parabolic bowl). Simulation of the form of this petal is complicated by the fact that the stretching ratios for the fabric in two opposite orthogonal directions are not equal (the orthotropic property). The problem of modelling is solved with the help of the notion of a pseudominimal surface. It is a surface for which the ratio of main curvatures is constant (in other words, asymptotical lines intersect at a constant angle). The existence theorem for pseudo-minimal surfaces has been proven. The breadth of the class of such surfaces has been determined (it coincides with that of the class of minimal surfaces). An example of a pseudominimal surface of revolution has been constructed. A special composite surface is used for the geometric simulation of a deformed metallic fabric petal: a parabola located in the normal plane of the line $L$ is attached to each point of some line $L$. The vertex of the parabola is on the line $L$, the parabola crosses the adjacent hard edges of the reflector structure, and the ratio of curvatures of the line $L$ and attached parabola does not depend on the point of pasting. Thus, a new substantiation of the construction applied by the author earlier has been developed. The standard mean square deviation of reflector's deformed petal from the parent paraboloid has been calculated. For the complicated functions describing the simulation, polynomial approximations have been constructed and their reliability have been estimated.