engleski

Circular Curves in Euclidean Plane

In the real projective plane the Euclidean metric defines the Euclidean plane with the \textit{; ; absolute}; ; (\textit{; ; circular}; ; ) \textit{; ; points}; ; $(0, 1, i)$ and $(0, 1, -i)$. An algebraic curve passing through the absolute points is called \textit{; ; circular curve}; ; . If it contains absolute points as its $p-$fold points, the curve is called $p-$\textit{; ; circular}; ; . Every $p-$circular curve has the implicit equation in homogeneous coordinates of the following form: \begin{; ; equation}; ; \nonumber \sum_{; ; j=0}; ; ^{; ; p-1}; ; x_0^j (x_1^{; ; 2}; ; +x_2^{; ; 2}; ; )^{; ; p-j}; ; f_{; ; n-2p+j}; ; (x_1, x_2)+\sum_{; ; j=p}; ; ^{; ; n}; ; x_0^j g_{; ; n-j}; ; (x_1, x_2)=0, \end{; ; equation}; ; where $f_{; ; k}; ; $, $k=n-2p, ..., n-p-1$, and $g_{; ; k}; ; $, $k=0, ..., n-p$, are homogeneous algebraic polynomials of degree $k$. \\ Obviously $n$ must be at least $2p$. If $n=2p$, the curve is called \textit{; ; entirely circular}; ; . We present some properties of circular curves and visualize their forms with the program \textit{; ; Mathematica}; ; .

circular curves; Euclidean plane; absolute points

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