In the case of a curve we have between the coordinates (x, y, z) a twofold relation: two equations ƒ(x, y, z) = 0, φ(x, y, z) = 0 give such a relation; i.e. the curve is here considered as the intersection of two surfaces (but the curve is not always the complete intersection of two surfaces, and there are hence difficulties); or, again, the coordinates may be given each of them as a function of a single variable parameter. The form y = φ(x), z = ψ(x), where two of the coordinates are given in terms of the third, is a particular case of each of these modes of representation.
29. The remarks under plane geometry as to descriptive and metrical propositions, and as to the non-metrical character of the method of coordinates when used for the proof of a descriptive proposition, apply also to solid geometry; and they might be illustrated in like manner by the instance of the theorem of the radical centre of four spheres. The proof is obtained from the consideration that S and S′ being each of them a function of the form x2 + y2 + z2 + ax + by + cz + d, the difference S−S′ is a mere linear function of the coordinates, and consequently that S−S′ = 0 is the equation of the plane containing the circle of intersection of the two spheres S = 0 and S′ = 0.
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| Fig. 60. |
30. Metrical Theory.—The foundation in solid geometry of the metrical theory is in fact the before-mentioned theorem that if a finite right line PQ be projected upon any other line OO′ by lines perpendicular to OO′, then the length of the projection P′Q′ is equal to the length of PQ into the cosine of its inclination to P′Q′—or (in the form in which it is now convenient to state the theorem) the perpendicular distance P′Q′ of two parallel planes is equal to the inclined distance PQ into the cosine of the inclination. The principle of § 16, that the algebraical sum of the projections of the sides of any closed polygon on any line is zero, or that the two sets of sides of the polygon which connect a vertex A and a vertex B have the same sum of projections on the line, in sign and magnitude, as we pass from A to B, is applicable when the sides do not all lie in one plane.
31. Consider the skew quadrilateral QMNP, the sides QM, MN, NP being respectively parallel to the three rectangular axes Ox, Oy, Oz; let the lengths of these sides be ξ, η, ζ, and that of the side QP be = ρ; and let the cosines of the inclinations (or say the cosine-inclinations) of ρ to the three axes be α, β, γ; then projecting successively on the three sides and on QP we have
and
whence ρ2 = ξ2 + η2 + ζ2, which is the relation between a distance ρ and its projections ξ, η, ζ upon three rectangular axes. And from the same equations we obtain α2 + β2 + γ2 = 1, which is a relation connecting the cosine-inclinations of a line to three rectangular axes.
Suppose we have through Q any other line QT, and let the cosine-inclinations of this to the axes be α′, β′, γ′, and δ be its cosine-inclination to QP; also let ρ be the length of the projection of QP upon QT; then projecting on QT we have
And in the last equation substituting for ξ, η, ζ their values ρα, ρβ, ργ we find
which is an expression for the mutual cosine-inclination of two lines, the cosine-inclinations of which to the axes are α, β, γ and α′, β′, γ′ respectively. We have of course α2 + β2 + γ2 = 1 and α′2 + β′2 + γ′2 = 1; and hence also
= (βγ′ − β′γ)2 + (γα′ − γ′α)2 + (αβ′ − α′β)2;
so that the sine of the inclination can only be expressed as a square root. These formulae are the foundation of spherical trigonometry.
32. Straight Lines, Planes and Spheres.—The foregoing formulae give at once the equations of these loci.
For first, taking Q to be a fixed point, coordinates (a, b, c), and the cosine-inclinations (α, β, γ) to be constant, then P will be a point in the line through Q in the direction thus determined; or, taking (x, y, z) for its coordinates, these will be the current coordinates of a point in the line. The values of ξ, η, ζ then are x − a, y − b, z − c, and we thus have
| x − a | = | y − b | = | z − c | (= ρ), |
| α | β | γ |
which (omitting the last equation, = ρ) are the equations of the line through the point (a, b, c), the cosine-inclinations to the axes being α, β, γ, and these quantities being connected by the relation α2 + β2 + γ2 = 1. This equation may be omitted, and then α, β, γ, instead of being equal, will only be proportional, to the cosine-inclinations.
Using the last equation, and writing
these are expressions for the current coordinates in terms of a parameter ρ, which is in fact the distance from the fixed point (a, b, c).
It is easy to see that, if the coordinates (x, y, z) are connected by any two linear equations, these equations can always be brought into the foregoing form, and hence that the two linear equations represent a line.
Secondly, taking for greater simplicity the point Q to be coincident with the origin, and α′, β′, γ′, p to be constant, then p is the perpendicular distance of a plane from the origin, and α′, β′, γ′ are the cosine-inclinations of this distance to the axes (α′2 + β′2 + γ′2 = 1). P is any point in this plane, and taking its coordinates to be (x, y, z) then (ξ, η, ζ) are = (x, y, z), and the foregoing equation p = α′ξ + β′η + γ′ζ becomes
which is the equation of the plane in question.
If, more generally, Q is not coincident with the origin, then, taking its coordinates to be (a, b, c), and writing p1 instead of p, the equation is
and we thence have p1 = p − (aα′ + bβ′ + cγ′), which is an expression for the perpendicular distance of the point (a, b, c) from the plane in question.
It is obvious that any linear equation Ax + By + Cz + D = O between the coordinates can always be brought into the foregoing form, and hence that such an equation represents a plane.
Thirdly, supposing Q to be a fixed point, coordinates (a, b, c), and the distance QP = ρ, to be constant, say this is = d, then, as before, the values of ξ, η, ζ are x − a, y − b, z − c, and the equation ξ2 + η2 + ζ2 = ρ2 becomes
which is the equation of the sphere, coordinates of the centre = (a, b, c), and radius = d.
A quadric equation wherein the terms of the second order are x2 + y2 + z2, viz. an equation
can always, it is clear, be brought into the foregoing form; and it thus appears that this is the equation of a sphere, coordinates of the centre −12A, −12B, −12C, and squared radius = 14(A2 + B2 + C2) − D.
33. Cylinders, Cones, ruled Surfaces.—If the two equations of a straight line involve a parameter to which any value may be given, we have a singly infinite system of lines. They cover a surface, and the equation of the surface is obtained by eliminating the parameter between the two equations.
If the lines all pass through a given point, then the surface is a cone; and, in particular, if the lines are all parallel to a given line, then the surface is a cylinder.
Beginning with this last case, suppose the lines are parallel to the line x = mz, y = nz, the equations of a line of the system are x = mz + a, y = nz + b,—where a, b are supposed to be functions of the variable parameter, or, what is the same thing, there is between them a relation ƒ(a, b) = 0: we have a = x − mz, b = y − nz, and the result of the elimination of the parameter therefore is ƒ(x − mz, y − nz) = 0, which is thus the general equation of the cylinder the generating lines whereof are parallel to the line x = mz, y = nz. The equation of the section by the plane z = 0 is ƒ(x, y) = 0, and conversely if the cylinder be determined by means of its curve of intersection with the plane z = 0, then, taking the equation of this curve to be ƒ(x, y) = 0, the equation of the cylinder is ƒ(x − mz, y − nz) = 0. Thus, if the curve of intersection be the circle (x − α)2 + (y − β)2 = γ2, we have (x − mz − α)2 + (y − nz − β)2 = γ2 as the equation of an oblique cylinder on this base, and thus also (x − α)2 + (y − β)2 = γ2 as the equation of the right cylinder.
If the lines all pass through a given point (a, b, c), then the equations of a line are x − a = α(z − c), y − b = β(z − c), where α, β are functions of the variable parameter, or, what is the same thing, there exists between them an equation ƒ(α, β) = 0; the elimination of the parameter gives, therefore, ƒ(x − az − c, y − bz − c) = 0; and this equation, or, what is the same thing, any homogeneous equation ƒ(x − a, y − b, z − c) = 0, or, taking ƒ to be a rational and integral function of the order n, say (*) (x − a, y − b, z − c)n = 0, is the general equation of the cone having the point (a, b, c) for its vertex. Taking the vertex to be at the origin, the equation is (*) (x, y, z)n = 0; and, in particular, (*) (x, y, z)2 = 0 is the equation of a cone of the second order, or quadricone, having the origin for its vertex.
34. In the general case of a singly infinite system of lines, the locus is a ruled surface (or regulus). Now, when a line is changing its position in space, it may be looked upon as in a state of turning about some point in itself, while that point is, as a rule, in a state of moving out of the plane in which the turning takes place. If instantaneously it is only in a state of turning, it is usual, though not strictly accurate, to say that it intersects its consecutive position. A regulus such that consecutive lines on it do not intersect, in this sense, is called a skew surface, or scroll; one on which they do is called a developable surface or torse.
Suppose, for instance, that the equations of a line (depending on
