All cubics are self-isotomic
type: Item
instance of: scholarly article
title: All cubics are self-isotomic
zbMATH Open document ID: 1365.51017
author: Roger C. Alperin
author: Dennis L. Johnson
published in: Journal of Geometry
publication date: 2017-04-20T00:00:00Z
review text: In [J. Geom. 105, No. 2, 243--271 (2014; Zbl 1301.51029)], the authors show that every planar cubic \(\mathcal J\) is self-isogonal with respect to some triangle on \(\mathcal J\). Here, the authors show that \(\mathcal J\) is also self-isotomic. Their proof uses the self-isogonal transform of \(\mathcal J\) but does not fit into the general framework to study Cremona transformations that fix \(\mathcal J\), as developed in above paper. This singular triangle of the self-isotomic transform can be constructed as follows: Denote by \(G\) a pole of the line at infinity \(L_\infty\) with respect to \(\mathcal J\) and by \(P\), \(Q\), \(R\) the intersection points of \(\mathcal J\) and \(L_\infty\). There exists an isogonal transform \(\mu\) with \(\mu(G) = G\), \(\mu(\mathcal J) = \mathcal J\) and a conic \(\mathcal K\) through \(\mu(P)\), \(\mu(Q)\), and \(\mu(R)\) such that \(G\), \(L_\infty\) are dual with respect to \(\mathcal K\). The residual intersection points \(A\), \(B\), \(C\) of \(\mathcal K\) with \(\mathcal J\) determine the singular triangle of an isotomic transform that fixes \(\mathcal J\). The authors do not explicitly mention this but it seems that their results are true in projective planes over algebraically closed fields as otherwise the pole \(G\) does not need to exist.
reviewed by: Hans-Peter Schröcker
Mathematics Subject Classification ID: 51N35
Mathematics Subject Classification ID: 14N05
zbMATH DE Number: 6706780
zbMATH Keywords: cubic curve
zbMATH Keywords: Cremona transformation
zbMATH Keywords: isotomic cubics
MaRDI profile type: Publication
full work available at URL: s00022-015-0307-x
OpenAlex ID: W2296928380
cites work: All cubics are self-isogonal
cites work: Topics in Elementary Geometry
DOI: 10.1007/S00022-015-0307-X
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description: scientific article; zbMATH DE number 6706780