| dbo:abstract
|
- In projective geometry, the Laguerre–Forsyth invariant is a cubic differential that is an invariant of a projective plane curve. It is named for Edmond Laguerre and Andrew Forsyth, the latter of whom analyzed the invariant in an influential book on ordinary differential equations. Suppose that is a three-times continuously differentiable immersion of the projective line into the projective plane, with homogeneous coordinates given by then associated to p is the third-order ordinary differential equation Generically, this equation can be put into the form where are rational functions of the components of p and its derivatives. After a change of variables of the form , this equation can be further reduced to an equation without first or second derivative terms The invariant is the Laguerre–Forsyth invariant. A key property of P is that the cubic differential P(dt)3 is invariant under the automorphism group of the projective line. More precisely, it is invariant under , , and . The invariant P vanishes identically if (and only if) the curve is a conic section. Points where P vanishes are called the sextactic points of the curve. It is a theorem of Herglotz and Radon that every closed strictly convex curve has at least six sextactic points. This result has been extended to a variety of optimal minima for simple closed (but not necessarily convex) curves by , depending on the curve's homotopy class in the projective plane. (en)
|
| dbo:wikiPageExternalLink
| |
| dbo:wikiPageID
| |
| dbo:wikiPageLength
|
- 2635 (xsd:nonNegativeInteger)
|
| dbo:wikiPageRevisionID
| |
| dbo:wikiPageWikiLink
| |
| dbp:wikiPageUsesTemplate
| |
| dcterms:subject
| |
| rdfs:comment
|
- In projective geometry, the Laguerre–Forsyth invariant is a cubic differential that is an invariant of a projective plane curve. It is named for Edmond Laguerre and Andrew Forsyth, the latter of whom analyzed the invariant in an influential book on ordinary differential equations. Suppose that is a three-times continuously differentiable immersion of the projective line into the projective plane, with homogeneous coordinates given by then associated to p is the third-order ordinary differential equation Generically, this equation can be put into the form (en)
|
| rdfs:label
|
- Laguerre–Forsyth invariant (en)
|
| owl:sameAs
| |
| prov:wasDerivedFrom
| |
| foaf:isPrimaryTopicOf
| |
| is dbo:wikiPageRedirects
of | |
| is dbo:wikiPageWikiLink
of | |
| is foaf:primaryTopic
of | |
