Original scientific paper

https://doi.org/10.31896/k.25.3

A Rarity in Geometry: a Septic Curve

Boris Odehnal orcid.org/0000-0002-7265-5132 ; University of Applied Arts Vienna, Vienna, Austria

Abstract

We study the locus C of all points in the plane whose pedal points on the six sides of a complete quadrangle lie on a conic. In the Euclidean plane, it turns out that C is an algebraic curve of degree 7 and genus 5 and not of degree 12 as it could be expected. Septic curves occur rather seldom in geometry which motivates a detailed study of this particular curve. We look at its singularities, focal points, and those points on C whose pedal conics degenerate. Then, we show that the septic curve occurs as the locus curve for a more general question. Further, we describe those cases where C degenerates or is of degree less than 7 depending on the shape of the initial quadrilateral.

Keywords

quadrilateral; complete quadrangle; pedal point; conic; six conconic points; septic curve; Simson line; Miquel point

Hrčak ID:

269179

URI

https://hrcak.srce.hr/269179

Publication date:

27.12.2021.

Article data in other languages: croatian

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