Straight lines and conics are represented in trilinear and in areal, because in Cartesian, coordinates by equations of the first and second degrees respectively, and these degrees are preserved when the equations are made homogeneous. What must be said about points infinitely far off in order to make universal the statement, to which there is no exception as long as finite distances alone are considered, that every homogeneous equation of the first degree represents a straight line? Let the point of areal coordinates (x′, y′, z′) move infinitely far off, and mean by x, y, z finite quantities in the ratios which x′, y′, z′ tend to assume as they become infinite. The relation x′ + y′ + z′ = 1 gives that the limiting state of things tended to is expressed by x + y + z = 0. This particular equation of the first degree is satisfied by no point at a finite distance; but we see the propriety of saying that it has to be taken as satisfied by all the points conceived of as actually at infinity. Accordingly the special property of these points is expressed by saying that they lie on a special straight line, of which the areal equation is x + y + z = 0. In trilinear coordinates this line at infinity has for equation aα + bβ + cγ = 0.
On the one special line at infinity parallel lines are treated as meeting. There are on it two special (imaginary) points, the circular points at infinity of § 19, through which all circles pass in the same sense. In fact if S = O be one circle, in areal coordinates, S + (x + y + z)(lx + my + nz) = 0 may, by proper choice of l, m, n, be made any other; since the added terms are once lx + my + nz, and have the generality of any expression like a′x + b′y + c′ in Cartesian coordinates. Now these two circles intersect in the two points where either meets x + y + z = 0 as well as in two points on the radical axis lx + my + nz = 0.
24. Let us consider the perpendicular distance of a point (α′, β′, γ′) from a line lα + mβ + nγ. We can take rectangular axes of Cartesian coordinates (for clearness as to equalities of angle it is best to choose an origin inside ABC), and refer to them, by putting expressions p − x cos θ − y sin θ, &c., for α &c.; we can then apply § 16 to get the perpendicular distance; and finally revert to the trilinear notation. The result is to find that the required distance is
(lα′ + mβ′ + nγ′) / {l, m, n},
where {l, m, n}² = l² + m² + n² − 2mn cos A − 2nl cos B − 2lm cos C.
In areal coordinates the perpendicular distance from (x′, y′, z′) to lx + my + nz = 0 is 2Δ(lx′ + my′ + nz′)/{al, bm, cn}. In both cases the coordinates are of course actual values.
Now let ξ, η, ζ be the perpendiculars on the line from the vertices A, B, C, i.e. the points (1, 0, 0), (0, 1, 0), (0, 0, 1), with signs in accord with a convention that oppositeness of sign implies distinction between one side of the line and the other. Three applications of the result above give
ξ/l = 2Δ / {al, bm, cn} = η/m = ζ/n;
and we thus have the important fact that ξx′ + ηy′ + ζz′ is the perpendicular distance between a point of areal coordinates (x′y′z′) and a line on which the perpendiculars from A, B, C are ξ, η, ζ respectively. We have also that ξx + ηy + ζz = 0 is the areal equation of the line on which the perpendiculars are ξ, η, ζ; and, by equating the two expressions for the perpendiculars from (x′, y′, z′) on the line, that in all cases {aξ, bη, cζ}² = 4Δ².
25. Line-coordinates. Duality.—A quite different order of ideas may be followed in applying analysis to geometry. The notion of a straight line specified may precede that of a point, and points may be dealt with as the intersections of lines. The specification of a line may be by means of coordinates, and that of a point by an equation, satisfied by the coordinates of lines which pass through it. Systems of line-coordinates will here be only briefly considered. Every such system is allied to some system of point-coordinates; and space will be saved by giving prominence to this fact, and not recommencing ab initio.