engleski

Special nth Order Surfaces with (n-2)-ple Line

In this paper, in Euclidean space $\mathbb E^3$, we treat the pedal surfaces of special line congruences $\mathcal C^1_{;2k};$ which are of the 1st order and the $2k$th class. We derive the parametric and implicit equations of these surfaces which enable Mathematica visualizations and proving some properties such as their order is $2k+2$, they possess one $2k$-ple straight line and pass through the absolute conic of $E^3$. The properties of their singularities, which do not lie on $2k$-ple line, and of the pinch points on the $2k$-ple line, are also shown.

congruence of lines; inversion; pedal surfaces of congruence; multiple line; multiple point; pinch point

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