engleski

Generalized Rose-Surfaces

In [1] we studied circular surfaces CS($\alpha$, p)that are defined by a curve $\alpha$ and a congruence of circles C(p), where C(p) contains all circles passing through the points P(0, 0, +-p), p=$\sqrt{;q};$, $q\in\R$. Depending on the type of points P, C(p) is an elliptic, parabolic or hyperbolic congruence of circles. It was shown that the rose surfaces, treated in [2], are circular surfaces CS($\alpha$, p) where $\alpha$ is a rose (rhodonea) and C(p) is an elliptic or parabolic congurence. The rose lies in the plane z=p having the directing point P as the point of the highest multiplicity. If we extend $\alpha$ to all cyclic-harmonic curves with the polar equation, and include hyperbolic congruences C(p), numerous forms of new class of surfaces are obtained. This class we call generalized rose surfaces, study their algebraic properties and visulaize their shapes with the program Mathematica.

congruence of circles; circular surfaces; rose-surfaces

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