Sophus Lie has pointed out a different analogy with sphere geometry. Suppose, in fact, that the equation of a sphere of radius r is
x² + y² + z² + 2ax + 2by + 2cz + d = 0,
so that r² = a² + b² + c² − d; then introducing the quantity e to make this equation homogeneous, we may regard the sphere as given by the six coordinates a, b, c, d, e, r connected by the equation a² + b² + c² − r² − de = 0, and it is easy to see that two spheres touch, if the polar form 2aa1 + 2bb1 + 2cc1 − 2rr1 − de1 − d1e vanishes. Comparing this with the equation x1² + x2² + x3² + x4² − 2x5x6 = 0 given above, it appears that this sphere geometry and line geometry are identical, for we may write a = x1, b = x2, c = x3, r = x4δ − 1, d = x5, e = ½x6; but it is to be noticed that a sphere is really replaced by two lines whose coordinates only differ in the sign of x4, so that they are polar lines with respect to the complex x4 = 0. Two spheres which touch correspond to two lines which intersect, or more accurately to two pairs of lines (p, p′) and (q, q′), of which the pairs (p, q) and (p′, q′) both intersect. By this means the problem of describing a sphere to touch four given spheres is reduced to that of drawing a pair of lines (t, t′) (of which t intersects one line of the four pairs (pp′), (qq′), (rr′), (ss′), and t′ intersects the remaining four). We may, however, ignore the accented letters in translating theorems, for a configuration of lines and its polar with respect to a linear complex have the same projective properties. In Lie’s transformation a linear complex corresponds to the totality of spheres cutting a given sphere at a given angle. A most remarkable result is that lines of curvature in the sphere geometry become asymptotic lines in the line geometry.
Some of the principles of line geometry may be brought into clearer light by admitting the ideas of space of four and five dimensions.
Thus, regarding the coordinates of a line as homogeneous coordinates in five dimensions, we may say that line geometry is equivalent to geometry on a quadric surface in five dimensions. A linear complex is represented by a hyperplane section; and if two such complexes are in involution, the corresponding hyperplanes are conjugate with respect to the fundamental quadric. By projecting this quadric stereographically into space of four dimensions we obtain Klein’s analogy. In the same way geometry in a linear complex is equivalent to geometry on a quadric in four dimensions; when two lines intersect the representative points are on the same generator of this quadric. Stereographic projection, therefore, converts a curve in a linear complex, i.e. one whose tangents all belong to the complex, into one whose tangents intersect a fixed conic: when this conic is the imaginary circle at infinity the curve is what Lie calls a minimal curve. Curves in a linear complex have been extensively studied. The osculating plane at any point of such a curve is the nul-plane of the point with respect to the complex, and points of superosculation always coincide in pairs at the points of contact of stationary tangents. When a point of such a curve is given, the osculating plane is determined, hence all the curves through a given point with the same tangent have the same torsion.
The lines through a given point that belong to a complex of the nth degree lie on a cone of the nth degree: if this cone has a double line the point is said to be a singular point. Similarly, Non-linear complexes. a plane is said to be singular when the envelope of the lines in it has a double tangent. It is very remarkable that the same surface is the locus of the singular points and the envelope of the singular planes: this surface is called the singular surface, and both its degree and class are in general 2n(n − 1)², which is equal to four for the quadratic complex.
The singular lines of a complex F = 0 are the lines common to F and the complex
| δF | δF | + | δF | δF | + | δF | δF | = 0. | |||
| δl | δλ | δm | δμ | δn | δν |
As already mentioned, at each line l of a complex there is an infinite number of tangent linear complexes, and they all contain the lines adjacent to l. If now l be a singular line, these complexes all reduce to straight lines which form a plane pencil containing the line l. Suppose the vertex of the pencil is A, its plane a, and one of its lines ξ, then l′ being a complex line near l, meets ξ, or more accurately the mutual moment of l′, and is of the second order of small quantities. If P be a point on l, a line through P quite near l in the plane a will meet ξ and is therefore a line of the complex; hence the complex-cones of all points on l touch a and the complex-curves of all planes through l touch l at A. It follows that l is a double line of the complex-cone of A, and a double tangent of the complex-curve of a. Conversely, a double line of a cone or curve is a singular line, and a singular line clearly touches the curves of all planes through it in the same point. Suppose now that the consecutive line l′ is also a singular line, A′ being the allied singular point, a′ the singular plane and ξ′ any line of the pencil (A′, a′) so that ξ′ is a tangent line at l′ to the complex: the mutual moments of the pairs l′, ξ and l, ξ are each of the second order; hence the plane a′ meets the lines l and ξ′ in two points very near A. This being true for all singular planes, near a the point of contact of a with its envelope is in A, i.e. the locus of singular points is the same as the envelope of singular planes. Further, when a line touches a complex it touches the singular surface, for it belongs to a plane pencil like (Aa), and thus in Klein’s analogy the analogue of a focus of a hyper-surface being a bitangent line of the complex is also a bitangent line of the singular surface. The theory of cosingular complexes is thus brought into line with that of confocal surfaces in four dimensions, and guided by these principles the existence of cosingular quadratic complexes can easily be established, the analysis required being almost the same as that invented for confocal cyclides by Darboux and others. Of cosingular complexes of higher degree nothing is known.
Following J. Plücker, we give an account of the lines of a quadratic complex that meet a given line.