engleski

Special Sextics with Quadruple Line

We define a transformation $i^{;n+2};_\Psi : \mathbb P^3\rightarrow \mathbb P^3$ where corresponding points lie on the rays of the 1st order and $n$th class congruences $\mathcal C_n^1$ and are conjugate with respect to a quadric $\Psi$. It is shown that this inversion transforms a plane into the surface of the order $n+2$ which contains $n$-ple straight line. In 3-dimensional Euclidean space we shown that $i^{;n+2};_\Psi$, where $\Psi$ is any sphere with a center $P$, transforms the plane at infinity into the pedal surface of congruence $C^1_n$ with respect to a pole $P$. For special congruence $C^1_4$ (directing lines are Viviani's curve and a straight line which cut it into two points, where one of them is the double point of Viviani's curve) we derived the pedal surfaces which are the 6th order surfaces (sextics) $\mathcal P^6_4$ with a quadruple straight line. For this class we investigate the singularities: the condition for the existence of quintuple point and the type of its tangent cone, the number and type of pinch points on the quadruple line and the conditions for the existence of double points out of the quadruple line.

congruence of lines; inversion; pedal surface of congruence; quintuple point; quadruple straight line

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