Counting tritangent planes of space curves
type: Item
instance of: scholarly article
title: Counting tritangent planes of space curves
zbMATH Open document ID: 0567.53001
DOI: 10.1016/0040-9383(85)90041-2
author: Terence Gaffney
author: Clint McCrory
author: Tom Banchoff
published in: Topology
publication date: 1985-01-01T00:00:00Z
review text: In this paper a formula is derived for the parity of the number of tritangent planes of a generic closed curve C in \(R^ 3\). More precisely the number T(C) of planes which are tangent at exactly three points is shown to have the same parity as the sum of the indices of the so-called ''stalls'' \((=points\) with vanishing torsion). This index is half of the number of other intersection points of C with the osculating plane through a stall. The proof makes essentially use of the dual surface of C defined by the space of all tangent planes of C. In the same issue \textit{T. Ozawa} [Topology 24, 1-14 (1985; Zbl 0525.53002)] independently obtained a formula for the integer number of tritangencies itself (not just modulo 2). This depends on finding a suitable sign for the indices of the stalls. Ozawa also has a formula regarding the different possibilities how the three points of tangency can coincide.
Mathematics Subject Classification ID: 53A04
zbMATH DE Number: 3905352
zbMATH Keywords: tritangent planes
zbMATH Keywords: closed curve
zbMATH Keywords: points with vanishing torsion
zbMATH Keywords: stall
reviewed by: Wolfgang Kühnel
MaRDI profile type: Publication
full work available at URL: 0040-9383(85)90041-2
OpenAlex ID: W2088299508
description: scientific article; zbMATH DE number 3905352