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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.
| 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.
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.
| Fig. 60. |