Open AccessA Rarity in Geometry
Author(s) -
Boris Odehnal
Publication year - 2021
Publication title -
kog/kog
Language(s) - English
Resource type - Journals
eISSN - 1846-4068
pISSN - 1331-1611
DOI - 10.31896/k.25.3
Subject(s) - conic section , mathematics , plane curve , locus (genetics) , gravitational singularity , geometry , quadrilateral , algebraic curve , euclidean geometry , algebraic geometry , degree (music) , quadrangle , degenerate energy levels , combinatorics , pure mathematics , mathematical analysis , physics , geology , thermodynamics , biochemistry , chemistry , quantum mechanics , geomorphology , finite element method , acoustics , gene
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