18. Twisted Curves.—In conclusion a little may be said as to curves of double curvature, otherwise twisted curves or curves in space. The analytical theory by Cartesian co-ordinates was first considered by Alexis Claude Clairaut, Recherches sur les courbes à double courbure (Paris, 1731). Such a curve may be considered as described by a point, moving in a line which at the same time rotates about the point in a plane which at the same time rotates about the line; the point is a point, the line a tangent, and the plane an osculating plane, of the curve; moreover the line is a generating line, and the plane a tangent plane, of a developable surface or torse, having the curve for its edge of regression. Analogous to the order and class of a plane curve we have the order, rank and class of the system (assumed to be a geometrical one), viz. if an arbitrary plane contains m points, an arbitrary line meets r lines, and an arbitrary point lies in n planes, of the system, then m, r, n are the order, rank and class respectively. The system has singularities, and there exist between m, r, n and the numbers of the several singularities equations analogous to Plücker’s equations for a plane curve.
It is a leading point in the theory that a curve in space cannot in general be represented by means of two equations U = 0, V = 0; the two equations represent surfaces, intersecting in a curve; but there are curves which are not the complete intersection of any two surfaces; thus we have the cubic in space, or skew cubic, which is the residual intersection of two quadric surfaces which have a line in common; the equations U = 0, V = 0 of the two quadric surfaces represent the cubic curve, not by itself, but together with the line.
Authorities.—In addition to the copious authorities mentioned in the text above, see Gabriel Cramer, Introduction à l’analyse des lignes courbes algébriques (Geneva, 1750). Bibliographical articles are given in the Ency. der math. Wiss. Bd. iii. 2, 3 (Leipzig, 1902-1906); H. C. F. von Mangoldt, “Anwendung der Differential- und Integralrechnung auf Kurven und Flächen,” Bd. iii. 3 (1902); F. R. v. Lilienthal, “Die auf einer Fläche gezogenen Kurven,” Bd. iii. 3 (1902); G. W. Scheffers, “Besondere transcendente Kurven,” Bd. iii. 3 (1903); H. G. Zeuthen, “Abzahlende Methoden,” Bd. iii. 2 (1906); L. Berzolari, “Allgemeine Theorie der höheren ebenen algebraischen Kurven,” Bd. iii. 2 (1906). Also A. Brill and M. Noether, “Die Entwicklung der Theorie der algebraischen Funktionen in älterer und neuerer Zeit” (Jahresb. der deutschen math. ver., 1894); E. Kötter, “Die Entwickelung der synthetischen Geometrie” (Jahresb. der deutschen math. ver., 1898-1901); E. Pascal, Repertorio di matematiche superiori, ii. “Geometrìa” (Milan, 1900); H. Wieleitner, Bibliographie der höheren algebraischen Kurven für den Zeitabschnitt von 1890-1894 (Leipzig, 1905).
Text-books:—G. Salmon, A Treatise on the Higher Plane Curves (Dublin, 1852, 3rd ed., 1879); translated into German by O. W. Fiedler, Analytische Geometrie der höheren ebenen Kurven (Leipzig, 2te Aufl., 1882); L. Cremona, Introduzione ad una teoria geometrica delle curve piane (Bologna, 1861); J. H. K. Durège, Die ebenen Kurven dritter Ordnung (Leipzig, 1871); R. F. A. Clebsch and C. L. F. Lindemann, Vorlesungen über Geometrie, Band i. and i2 (Leipzig, 1875-1876); H. Schroeter, Die Theorie der ebenen Kurven dritter Ordnung (Leipzig, 1888); H. Andoyer, Leçons sur la théorie des formes et la géométrie analytique supérieure (Paris, 1900); Wieleitner, Theorie der ebenen algebraischen Kurven höherer Ordnung (Leipzig, 1905).
(A. Ca.; E. B. El.)
[1] In solid geometry infinity is a plane—its intersection with any given plane being the right line which is the infinity of this given plane.
CURVILINEAR, in architecture, that which is formed by curved or flowing lines; the roofs over the domes and vaults of the Byzantine churches were generally curvilinear. The term is also given to the flowing tracery of the Decorated and the Flamboyant styles.