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
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.
Suppose that any particular system of point-coordinates, in which lx + my + nz = 0 may represent any straight line, is before us: notice that not only are trilinear and areal coordinates such systems, but Cartesian coordinates also, since we may write x/z, y/z for the Cartesian x, y, and multiply through by z. The line is exactly assigned if l, m, n, or their mutual ratios, are known. Call (l, m, n) the coordinates of the line. Now keep x, y, z constant, and let the coordinates of the line vary, but always so as to satisfy the equation. This equation, which we now write xl + ym + zn = 0, is satisfied by the coordinates of every line through a certain fixed point, and by those of no other line; it is the equation of that point in the line-coordinates l, m, n.
Line-coordinates are also called tangential coordinates. A curve is the envelope of lines which touch it, as well as the locus of points which lie on it. A homogeneous equation of degree above the first in l, m, n is a relation connecting the coordinates of every line which touches some curve, and represents that curve, regarded as an envelope. For instance, the condition that the line of coordinates (l, m, n), i.e. the line of which the allied point-coordinate equation is lx + my + nz = 0, may touch a conic (a, b, c, f, g, h) (x, y, z)2 = 0, is readily found to be of the form (A, B, C, F, G, H) (l, m, n)2 = 0, i.e. to be of the second degree in the line-coordinates. It is not hard to show that the general equation of the second degree in l, m, n thus represents a conic; but the degenerate conics of line-coordinates are not line-pairs, as in point-coordinates, but point-pairs.
The degree of the point-coordinate equation of a curve is the order of the curve, the number of points in which it cuts a straight line. That of the line-coordinate equation is its class, the number of tangents to it from a point. The order and class of a curve are generally different when either exceeds two.
26. The system of line-coordinates allied to the areal system of point-coordinates has special interest.
The l, m, n of this system are the perpendiculars ξ, η, ζ of § 24; and x′ξ + y ′η + z′ζ = 0 is the equation of the point of areal coordinates (x′, y ′, z′), i.e. is a relation which the perpendiculars from the vertices of the triangle of reference on every line through the point, but no other line, satisfy. Notice that a non-homogeneous equation of the first degree in ξ, η, ζ does not, as a homogeneous one does, represent a point, but a circle. In fact x′ξ + y ′η + z′ζ = R expresses the constancy of the perpendicular distance of the fixed point x′ξ + y ′η + z′ζ = 0 from the variable line (ξ, η, ζ), i.e. the fact that (ξ, η, ζ) touches a circle with the fixed point for centre. The relation in any ξ, η, ζ which enables us to make an equation homogeneous is not linear, as in point-coordinates, but quadratic, viz. it is the relation {aξ, bη, cζ}2 = 4Δ2 of § 24. Accordingly the homogeneous equation of the above circle is
Every circle has an equation of this form in the present system of line-coordinates. Notice that the equation of any circle is satisfied by those coordinates of lines which satisfy both x′ξ + y ′η + z′ζ = 0, the equation of its centre, and {aξ, bη, cζ}2 = 0. This last equation, of which the left-hand side satisfies the condition for breaking up into two factors, represents the two imaginary circular points at infinity, through which all circles and their asymptotes pass.
There is strict duality in descriptive geometry between point-line-locus and line-point-envelope theorems. But in metrical geometry duality is encumbered by the fact that there is in a plane one special line only, associated with distance, while of special points, associated with direction, there are two: moreover the line is real, and the points both imaginary.
II. Solid Analytical Geometry.
27. Any point in space may be specified by three coordinates. We consider three fixed planes of reference, and generally, as in all that follows, three which are at right angles two and two. They intersect, two and two, in lines x′Ox, y ′Oy, z′Oz, called the axes of x, y, z respectively, and divide all space into eight parts called octants. If from any point P in space we draw PN parallel to zOz′ to meet the plane xOy in N, and then from N draw NM parallel to yOy ′ to meet x′Ox in M, the coordinates (x, y, z) of P are the numerical measures of OM, MN, NP; in the case of rectangular coordinates these are the perpendicular distances of P from the three planes of reference. The sign of each coordinate is positive or negative as P lies on one side or the other of the corresponding plane. In the octant delineated the signs are taken all positive.
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| Fig. 57. | Fig. 58. |
In fig. 57 the delineation is on a plane of the paper taken parallel to the plane zOx, the points of a solid figure being projected on that plane by parallels to some chosen line through O in the positive octant. Sometimes it is clearer to delineate, as in fig. 58, by projection parallel to that line in the octant which is equally inclined to Ox, Oy, Oz upon a plane of the paper perpendicular to it. It is possible by parallel projection to delineate equal scales along Ox, Oy, Oz by scales having any ratios we like along lines in a plane having any mutual inclinations we like.
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| Fig. 59. |
For the delineation of a surface of simple form it frequently suffices to delineate the sections by the coordinate planes; and, in particular, when the surface has symmetry about each coordinate plane, to delineate the quarter-sections belonging to a single octant. Thus fig. 59 conveniently represents an octant of the wave surface, which cuts each coordinate plane in a circle and an ellipse. Or we may delineate a series of contour lines, i.e. sections by planes parallel to xOy, or some other chosen plane; of course other sections may be indicated too for greater clearness. For the delineation of a curve a good method is to represent, as above, a series of points P thereof, each accompanied by its ordinate PN, which serves to refer it to the plane of xy. The employment of stereographic projection is also interesting.
28. In plane geometry, reckoning the line as a curve of the first order, we have only the point and the curve. In solid geometry, reckoning a line as a curve of the first order, and the plane as a surface of the first order, we have the point, the curve and the surface; but the increase of complexity is far greater than would hence at first sight appear. In plane geometry a curve is considered in connexion with lines (its tangents); but in solid geometry the curve is considered in connexion with lines and planes (its tangents and osculating planes), and the surface also in connexion with lines and planes (its tangent lines and tangent planes); there are surfaces arising out of the line—cones, skew surfaces, developables, doubly and triply infinite systems of lines, and whole classes of theories which have nothing analogous to them in plane geometry: it is thus a very small part indeed of the subject which can be even referred to in the present article.
In the case of a surface we have between the coordinates (x, y, z) a single, or say a onefold relation, which can be represented by a single relation ƒ(x, y, z) = 0; or we may consider the coordinates expressed each of them as a given function of two variable parameters p, q; the form z = ƒ(x, y) is a particular case of each of these modes of representation; in other words, we have in the first mode ƒ(x, y, z) = z − ƒ(x, y), and in the second mode x = p, y = q for the expression of two of the coordinates in terms of the parameters.
