engleski

Rose-Surfaces

We consider roses or rhondonea curves R(m, n) which can be expressed by polar equations $r(\varphi)=\cos\frac{;m};{;n};\, \varphi$ or $r(\varphi)=\sin\frac{;m};{;n};\, \varphi$, where $\frac{;m};{;n};$ is a rational number in the simplest form. For such curves we construct surfaces in the following way: Let P(0, 0, p) be any point on the axis z and let R(m, n) be a rose in the plane z=0. A rose-surface R(m, n, p) is the system of circles which lie in the planes $\zeta$ through the axis z and have diameters $\overline{;PR_i};$, where $R_i\neq O$ are the intersection points of the rose $R(m, n)$ and the plane $\zeta$.}; We derive the parametric and implicit equations of R(m, n, p), visualized their shapes with the program Mathematica and investigate some of their properties such as the number and the kind of their singular lines and points.

roses; singularities of algebraic surfaces; Mathematica

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