An application of Groebner bases to planarity of intersection of surfaces

TL;DR

This paper applies Groebner bases to analyze the planarity of intersections between algebraic surfaces, focusing on specific conoid sections with cubic egg curves, to determine conditions for conic plane sections.

Contribution

It introduces a novel application of Groebner bases to assess intersection planarity of algebraic surfaces, especially for conoids with cubic egg directrices.

Findings

Groebner bases effectively determine intersection planarity.

Conditions for conic sections of specific conoids are identified.

The method provides a systematic approach for algebraic surface analysis.

Abstract

In this paper we use Groebner bases theory in order to determine planarity of intersections of two algebraic surfaces in ${\bf R}^3$. We specially considered plane sections of certain type of conoid which has a cubic egg curve as one of the directrices. The paper investigates a possibility of conic plane sections of this type of conoid.