engleski

Singular Points on Surfaces P_4^6

The pedal surfaces $\mathcal P_4^6$ with respect to any pole $P$ and one special 1st order 4th class congruence $\mathcal C_4^1$ are 6th order surfaces with a quadruple line. The highest singularity which these surfaces can possess is a quintuple point. The quintuple points on $\mathcal P_4^6$ are classified according to the type of their 5th order tangent cone $-$ six types are obtained. Points on the quadruple line of $\mathcal P_4^6$ are quadriplanar. We distinguish nine types of these points and six of them are the types of pinch-points. Except the singular points on a quadruple line surface $\mathcal P_4^6$ has at least one real double point iff pole $P$ lies on one 5th degree ruled surface (see Fig.~1) and exactly two real double points iff it lies on one parabola.

quintuple point; quadruple point; tangent cone

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