Through every point in space not on the twisted cubic one and only one secant to the cubic can be drawn.
§ 104. The fact that all the secants through a point on the cubic form a quadric cone shows that the centres of the projective pencils generating the cubic are not distinguished from any other points on the cubic. If we take any two points S, S′ on the cubic, and draw the secants through each of them, we obtain two quadric cones, which have the line SS′ in common, and which intersect besides along the cubic. If we make these two pencils having S and S′ as centres projective by taking four rays on the one cone as corresponding to the four rays on the other which meet the first on the cubic, the correspondence is determined. These two pencils will generate a cubic, and the two cones of secants having S and S′ as centres will be identical with the above cones, for each has five rays in common with one of the first, viz. the line SS′ and the four lines determined for the correspondence; therefore these two cones intersect in the original cubic. This gives the theorem:—
On a twisted cubic any two points may be taken as centres of projective pencils which generate the cubic, corresponding planes being those which meet on the same secant.
Of the two projective pencils at S and S′ we may keep the first fixed, and move the centre of the other along the curve. The pencils will hereby remain projective, and a plane α in S will be cut by its corresponding plane α′ always in the same secant a. Whilst S′ moves along the curve the plane α′ will turn about a, describing an axial pencil.
Authorities.—In this article we have given a purely geometrical theory of conics, cones of the second order, quadric surfaces, &c. In doing so we have followed, to a great extent, Reye’s Geometrie der Lage, and to this excellent work those readers are referred who wish for a more exhaustive treatment of the subject. Other works especially valuable as showing the development of the subject are: Monge, Géométrie descriptive: Carnot, Géométrie de position (1803), containing a theory of transversals; Poncelet’s great work Traité des propriétés projectives des figures (1822); Möbins, Barycentrischer Calcul (1826); Steiner, Abhängigkeit geometrischer Gestalten (1832), containing the first full discussion of the projective relations between rows, pencils, &c.; Von Staudt, Geometrie der Lage (1847) and Beiträge zur Geometrie der Lage (1856-1860), in which a system of geometry is built up from the beginning without any reference to number, so that ultimately a number itself gets a geometrical definition, and in which imaginary elements are systematically introduced into pure geometry; Chasles, Aperçu historique (1837), in which the author gives a brilliant account of the progress of modern geometrical methods, pointing out the advantages of the different purely geometrical methods as compared with the analytical ones, but without taking as much account of the German as of the French authors; Id., Rapport sur les progrès de la géométrie (1870), a continuation of the Aperçu; Id., Traité de géométrie supérieure (1852); Cremona, Introduzione ad una teoria geometrica delle curve piane (1862) and its continuation Preliminari di una teoria geometrica delle superficie (German translations by Curtze). As more elementary books, we mention: Cremona, Elements of Projective Geometry, translated from the Italian by C. Leudesdorf (2nd ed., 1894); J.W. Russell, Pure Geometry (2nd ed., 1905).
(O. H.)
III. Descriptive Geometry
This branch of geometry is concerned with the methods for representing solids and other figures in three dimensions by drawings in one plane. The most important method is that which was invented by Monge towards the end of the 18th century. It is based on parallel projections to a plane by rays perpendicular to the plane. Such a projection is called orthographic (see [Projection], § 18). If the plane is horizontal the projection is called the plan of the figure, and if the plane is vertical the elevation. In Monge’s method a figure is represented by its plan and elevation. It is therefore often called drawing in plan and elevation, and sometimes simply orthographic projection.
§ 1. We suppose then that we have two planes, one horizontal, the other vertical, and these we call the planes of plan and of elevation respectively, or the horizontal and the vertical plane, and denote them by the letters π1 and π2. Their line of intersection is called the axis, and will be denoted by xy.