engleski

Visualiztions of Rose-Surfaces

Roses or rhodonea curves $R(n, d)$ can be expressed by polar equations $r(\varphi)=\cos\frac{; ; ; m}; ; ; {; ; ; n}; ; ; \, \varphi$ or $r(\varphi)=\sin\frac{; ; ; m}; ; ; {; ; ; n}; ; ; \, \varphi$, where $\frac{; ; ; n}; ; ; {; ; ; d}; ; ; $ 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 $R(n, d)$ be a rose in the plane $z=0$. A rose-surface $\mathcal R(n, d, p)$ 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 $R(n, d)$ and the plane $\zeta$. We derive the parametric and implicit equations of $\mathcal R(n, d, p)$, visualized their shapes with the program Mathematica and investigate some of their properties such as its order and the number and kind of their singular lines and points.

algebraic surfaces; higher singularities; roses; rose-surfaces

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