A Rarity in Geometry | Zendy

Boris Odehnal | Zendy

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A Rarity in Geometry

Author(s) -

Boris Odehnal

Publication year - 2021

Publication title -

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

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