On the rectifiability of a twisted cubic

Tsuruichi Hayashi
1919 Bulletin of the American Mathematical Society  
x 2 = bt 2 , x$ = et 3 , abc 4= 0, the condition that it be a helix is precisely the condition that it be algebraically rectifiable. Since her proof is an application of the common differential geometry, the coordinates are rectangular. The parametrical equations of the most general twisted cubics in rectangular coordinates x, y, z, are where F, /i, f 2 , ƒ3 are polynomials of degree 3 in the parameter t. By increasing t by a constant, and taking the axes along the tangent, the principal
more » ... and the binormal at the point on the curve where t is infinite, these equations are reduced to m _ apt 2 + ait + a 2 _ bjt + b 2 c 2 {1) X~ f + dt + e ' y~~t * + dt + e> Z~t * + dt+e> as Mr. W. H. Salmon has done in treating the twisted cubics of constant torsion.* But I will consider here the algebraic rectifiability of a less general type of twisted cubics whose equations are x = ait z + a^t 2 + arf + a±, y = b,f + b 2 t 2 + ht + 64, This type is even more general than that treated by Dr. Curtis, and contains the cubics which have the plane at infinity as their osculating plane. By changing the value of t by a constant, transferring the origin, and taking the axes of coordinates along the tangent, the principal normal and the binormal at the point on the
doi:10.1090/s0002-9904-1919-03256-x fatcat:cq6idihoxjgsbg4tk6y7youmx4