The neighborhood of a sextactic point on a plane curve

Lane, Ernest P.

doi:10.1215/s0012-7094-35-00119-3

Duke Math. J.

1935

DOI: 10.1215/s0012-7094-35-00119-3

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The neighborhood of a sextactic point on a plane curve

Ernest P. Lane¹

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1941

1952

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“…The correspondence between A and A' is a projectivity with unique double point H(e, 0,/), namely, the Halphen point discovered by Lane [8]. When it is taken for (1, 0, 0), we have/ = 0.…”

mentioning

confidence: 98%

“…In §1 a covariant triangle of reference and a covariant unit point are constructed by means of the neighborhood of order six at a point of an ordinary plane curve. This leads us in §2 to reconstruct Lane's [8] canonical expansion of a plane curve at its sextactic point.…”

mentioning

confidence: 99%

“…where ££i is the tangent to C at P. We take £2 on the osculating conic C2 of C at £ and £i£2 as the tangent to the conic C¡ at P2, so that (1) reduces to (2) y = ax2 + dx« + ex« + fx1 + (8).…”

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confidence: 99%

“…The polar line of A (a, 0, 1) with respect to C2, (7) x2 = 2aax\ intersects C2 at (0,0, 1) and (8) A'(l, 2aa, 1/2«)…”

mentioning

confidence: 99%

“…At P a six-point cubic of C can be determined to be tangent to the lines (7) and .4.4', (9) -2o«*1 + x2/2 + 2aa2z» = 0, at (8) and (a, 0, 1) respectively; its equation is found to be (y -2aax)2(-2aax + y/2 + 2aa2) + (y -2aax)(-4da*y2/a) -2y(-2aax + y/2 + 2aa2)2 = 0.…”

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confidence: 99%

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A new foundation of the projective differential theory of curves in five-dimensional space

Chang¹

1946

Trans. Amer. Math. Soc.

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Introduction.The purpose of the present paper is to develop a purely geometric theory of the projective differential geometry of curves in a space of five dimensions, the methods hitherto adopted by various authors(2) being more or less analytic and artificial.The projective differential theory of plane curves is one of the cornerstones upon which our theory of curves in five dimensions rests. The problem of finding a covariant figure to associate projectively with the neighborhood of order five at an ordinary point of a plane curve is an interesting one as has been pointed out by . We have succeeded in solving this problem in an elementary way. Then using neighborhoods of order six we have obtained, besides a covariant triangle, three covariant points /,• (i = 1, 2, 3) any one of which can be selected as unit point.In five-dimensional space the osculating plane p at an ordinary point P oí a curve V intersects the developable hypersurface of T in a plane curve C, of which P is either an ordinary point or a &-ic (k = 6, 7, 8) point [2]. In any case a covariant unit point / and a covariant triangle {PPiP2} can be determined in p, PPi being the tangent to T at P.When a plane p, passing through PPi and lying in the osculating threespace of r at P, rotates about PPi, the Bompiani osculant On [l], associated with the point of inflexion P of the projection produced by projecting V on the plane p from the point aP+ßPi, constitutes a generator of a covariant quadric Q3. Since P, Pi, P2 are three vertices of a quadrilateral on Q3, we may take the fourth vertex for P3. In a similar way we define two other vertices Pi and P6 of a covariant pyramid {PPiPiP3PiPf,} and two other covariant quadrics Q4 and Q$ which pass through the quadrilaterals PPiP^PJ* and

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“…The correspondence between A and A' is a projectivity with unique double point H(e, 0,/), namely, the Halphen point discovered by Lane [8]. When it is taken for (1, 0, 0), we have/ = 0.…”

mentioning

confidence: 98%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…The polar line of A (a, 0, 1) with respect to C2, (7) x2 = 2aax\ intersects C2 at (0,0, 1) and (8) A'(l, 2aa, 1/2«)…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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A new foundation of the projective differential theory of curves in five-dimensional space

Chang¹

1946

Trans. Amer. Math. Soc.

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Über die Invarianten von Linienelementen und die projektive Geometrie der Kurvengewebe

Bortolotti¹

1941

Monatsh. f. Mathematik und Physik

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1. Allgemeines. Die Beriihrungsinvarianten yon Smith und Nehmke. In diesem Vortrage mbehte ieh mieh mit den einfaehsten projektiven Differentialinvarianten besehiiftigen~ die zu Kurvenelementen versehiedener Ordnung in der Ebene oder im Raum gehSren.Die Grundaufgabe liilgt sieh folgendermagen formulieren: Oegeben sind eine oder mehrere Kurven und auf jeder ein Punkt; zu bestimmen sind diejenigen geometrisehen und numerisehen Elemente, die mit den Umgebungen yon gewissen gegebenen Ordnungen der obigen Punkte projektiv verkntipft werden kiSnnen. Diese geometrisehen und numerisehen Elemente h~ngen also yon den Kurvenseharen ab, die die gegebenen Kurven in den gegebenen Pankten yon den gegebenen Ordnungen beriihren.Ein erster Beitrag zu diesem Thema findet sieh in einer Arbeit yon It. J. S. Smith (1869) und sp~,tter, unabhitngig davon, in Untersuehungen yon R. Mehmke (1891) und E. W51ffing (1893). Es handelt sieh dabei um die Entdeekung, dal~ das Verhiiltnis der Kriimmungen zweier ebenen Kurven in einem Bertihrungspunkt nieht nut eine metrisehe~ sondern sogar eine projektive Invariante der beiden Kurven ist. Naeh Mehmke gilt entspreehendes aueh ttir das gerh~iltnis der Gaugsehen Kriimmungen zweier Fl~iehenl).Noeh etwas frtiher finden sieh bei E. Beltrami (1865)zwei bemerkenswerte Beispiele soleher Invarianten~ allerdings im Rahmen 1) Vgl. 2, 5, 6, 7 und aueh (vgl. 9, 29)

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Affine und projektive differentialgeometrie der singulären ebenen kurvenelemente

Ancochea¹

1952

Abh.Math.Semin.Univ.Hambg.

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The neighborhood of a sextactic point on a plane curve

Cited by 4 publications

References 0 publications

A new foundation of the projective differential theory of curves in five-dimensional space

A new foundation of the projective differential theory of curves in five-dimensional space

Über die Invarianten von Linienelementen und die projektive Geometrie der Kurvengewebe

Affine und projektive differentialgeometrie der singulären ebenen kurvenelemente

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