Trilinear co-ordinates (see Geometry: Analytical) were first
used by E. E. Bobillier in the memoir Essai sur un nouveau mode
de recherche des propriétés de l’étendue (Gerg. t. xviii., 1827–1828).
It is convenient to use these rather than Cartesian co-ordinates.
We represent a curve of the order m by an equation (*≬ x, y, z)m = 0,
the function on the left hand being a homogeneous rational and
integral function of the order m of the three co-ordinates (x, y, z);
clearly the number of constants is the same as for the equation
(*≬ x, y, 1)m = 0 in Cartesian co-ordinates.
The theorem of duality is considered and developed, but chiefly in regard to its metrical applications, by Michel Chasles in the Mémoire de géométrie sur deux principes généraux de la science, la dualité et l’homographie, which forms a sequel to the Aperçu historique sur l’origine et le développement des méthodes en géométrie (Mém. de Brux. t. xi., 1837).
We now come to Julius Plücker; his “six equations” were given in a short memoir in Crelle (1842) preceding his great work, the Theorie der algebraischen Curven (1844). Plücker first gave a scientific dual definition of a curve, viz.; “A curve is a locus generated by a point, and enveloped by a line—the point moving continuously along the line, while the line rotates continuously about the point”; the point is a point (ineunt.) of the curve, the line is a tangent of the curve. And, assuming the above theory of geometrical imaginaries, a curve such that m of its points are situate in an arbitrary line is said to be of the order m; a curve such that n of its tangents pass through an arbitrary point is said to be of the class n; as already appearing, this notion of the order and class of a curve is, however, due to Gergonne. Thus the line is a curve of the order 1 and class 0; and corresponding dually thereto, we have the point as a curve of the order 0 and class 1.
Plücker, moreover, imagined a system of line-co-ordinates (tangential co-ordinates). (See Geometry: Analytical.) The Cartesian co-ordinates (x, y) and trilinear co-ordinates (x, y, z) are point-co-ordinates for determining the position of a point; the new co-ordinates, say (ξ, η, ζ) are line-co-ordinates for determining the position of a line. It is possible, and (not so much for any application thereof as in order to more fully establish the analogy between the two kinds of co-ordinates) important, to give independent quantitative definitions of the two kinds of co-ordinates; but we may also derive the notion of line-co-ordinates from that of point-co-ordinates; viz. taking ξx + ηy + ζz = 0 to be the equation of a line, we say that (ξ, η, ζ) are the line-co-ordinates of this line. A linear relation aξ + bη + cζ = 0 between these co-ordinate determines a point, viz. the point whose point-co-ordinates are (a, b, c); in fact, the equation in question aξ + bη + cζ = 0 expresses that the equation ξx + ηy + ζz = 0, where (x, y, z) are current point-co-ordinates, is satisfied on writing therein x, y, z = a, b, c; or that the line in question passes through the point (a, b, c). Thus (ξ, η, ζ) are the line-co-ordinates of any line whatever; but when these, instead of being absolutely arbitrary, are subject to the restriction aξ + bη + cζ = 0, this obliges the line to pass through a point (a, b, c); and the last-mentioned equation aξ + bη + cζ = 0 is considered as the line-equation of this point.
A line has only a point-equation, and a point has only a line-equation; but any other curve has a point-equation and also a line-equation; the point-equation (*≬ x, y, z)m = 0 is the relation which is satisfied by the point-co-ordinates (x, y, z) of each point of the curve; and similarly the line-equation (*≬ ξ, η, ζ)n = 0 is the relation which is satisfied by the line-co-ordinates (ξ, η, ζ) of each line (tangent) of the curve.
There is in analytical geometry little occasion for any explicit use of line-co-ordinates; but the theory is very important; it serves to show that in demonstrating by point-co-ordinates any purely descriptive theorem whatever, we demonstrate the correlative theorem; that is, we do not demonstrate the one theorem, and then (as by the method of reciprocal polars) deduce from it the other, but we do at one and the same time demonstrate the two theorems; our (x, y, z.) instead of meaning point-co-ordinates may mean line-co-ordinates, and the demonstration is then in every step of it a demonstration of the correlative theorem.
7. Singularities of a Curve. Plücker’s Equations.—The above dual generation explains the nature of the singularities of a plane curve. The ordinary singularities, arranged according to a cross division, are
| Proper. | Improper. | |
| Point-singularities— | 1. The stationary point, cusp or spinode; | 2. The double point or node; |
| Line-singularities— | 3. The stationary tangent or inflection; | 4. The double tangent; |
arising as follows:—
1. The cusp: the point as it travels along the line may come to rest, and then reverse the direction of its motion.
3. The stationary tangent: the line may in the course of its rotation come to rest, and then reverse the direction of its rotation.
2. The node: the point may in the course of its motion come to coincide with a former position of the point, the two positions of the line not in general coinciding.
4. The double tangent: the line may in the course of its motion come to coincide with a former position of the line, the two positions of the point not in general coinciding.
It may be remarked that we cannot with a real point and line obtain the node with two imaginary tangents (conjugate or isolated point or acnode), nor again the real double tangent with two imaginary points of contact; but this is of little consequence, since in the general theory the distinction between real and imaginary is not attended to.
The singularities (1) and (3) have been termed proper singularities, and (2) and (4) improper; in each of the first-mentioned cases there is a real singularity, or peculiarity in the motion; in the other two cases there is not; in (2) there is not when the point is first at the node, or when it is secondly at the node, any peculiarity in the motion; the singularity consists in the point coming twice into the same position; and so in (4) the singularity is in the line coming twice into the same position. Moreover (1) and (2) are, the former a proper singularity, and the latter an improper singularity, as regards the motion of the point; and similarly (3) and (4) are, the former a proper singularity, and the latter an improper singularity, as regards the motion of the line.
But as regards the representation of a curve by an equation, the case is very different.
First, if the equation be in point-co-ordinates, (3) and (4) are in a sense not singularities at all. The curve (*≬ x, y, z)m = 0, or general curve of the order m, has double tangents and inflections; (2) presents itself as a singularity, for the equations dx(*≬ x, y, z)m = 0, dy(*≬ x, y, z)m = 0, dz(*≬ x, y, z)m = 0, implying (*≬ x, y, z)m = 0, are not in general satisfied by any values (a, b, c) whatever of (x, y, z), but if such values exist, then the point (a, b, c) is a node or double point; and (1) presents itself as a further singularity or sub-case of (2), a cusp being a double point for which the two tangents becomes coincident.
In line-co-ordinates all is reversed:—(1) and (2) are not singularities; (3) presents itself as a sub-case of (4).
The theory of compound singularities will be referred to farther on.
In regard to the ordinary singularities, we have
| m, | the | order, |
| n | ” | class, |
| δ | ” | number of double points, |
| κ | ” | ” cusps, |
| τ | ” | ” double tangents, |
| ι | ” | ” inflections; |
and this being so, Plücker’s “six equations” are
| (1) | n = m (m − 1) − 2 δ − 3κ, |
| (2) | ι = 3m (m − 2) − 6δ − 8κ, |
| (3) | τ = ½m (m − 2) (m2 − 9) − (m2 − m − 6) (2δ + 3κ) + 2δ (δ − 1) + 6δκ + 92κ (κ − 1), |
| (4) | m = n (n − 1) − 2τ − 3ι, |
| (5) | κ = 3n (n − 2) − 6τ − 8ι, |
| (6) | δ = ½n (n − 2) (n2 − 9) − (n2 − n − 6) (2τ + 3ι) + 2τ (τ − 1) + 6τι + 92ι (ι − i). |
It is easy to derive the further forms—
| (7) | ι − κ | = 3 (n − m), |
| (8) | 2 (τ − δ) | = (n − m) (n + m − 9), |
| (9) | ½m (m + 3) − δ − 2κ | = ½n (n + 3) − τ − 2ι, |
| (10) | ½ (m − 1) (m − 2) − δ − κ | = ½ (n − 1) (n − 2) − τ − ι, |
| (11, 12) | m2 − 2δ − 3κ | = n2 − 2τ − 3ι, = m + n,— |
