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Journal for Geometry and Graphics 10 (2006), No. 1, 015--021
Copyright Heldermann Verlag 2006

A Transformation Based on the Cubic Parabola y = x3

Eugeniusz Korczak
ul. sw. Rocha 6B m. 5, 61-142 Poznan, Poland


A particular geometric transformation is investigated, the $\Lambda$-transformation. It is defined on the set $T$ of tangent lines of the cubic parabola $C^3: y = x^3$ in the Euclidean plane $R^2$. Let $t$ be any line from the set $T$. The point $X\in t$ is called the image of a certain point $M\in t$ under the $\Lambda$-transformation, if the condition $(PQMX) = \lambda$ ($\lambda\in R$ and $\lambda \neq 0,1$) holds, where $(PQMX)$ is the cross-ratio of the four points; $P$ is the point of contact, and $Q$ is the remaining point of intersection between the tangent line $t$ and the basic curve $C^3$. Varying the line $t$ in the set $T$ and the point $M$ along the line $t$ we obtain a transformation of the plane $R^2$ into $R^2$. The image of any straight line $p \in R^2$ is discussed too.

Keywords: Lambda-transformation, quadratic transformation.

MSC: 51N15; 51N35

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