Proceedings

of the Tomsk Technological Institute

of Emperor Nicholas II

Vol. 24. 1911. No. 4.

S. P. Gomellia.

On the Question of Constructing Projections of Mutual Intersections of Surfaces of Revolution of the Second Order.

With a table of drawings.

1–9.

On the Construction of Projections of Mutual Intersections of Second-Order Surfaces of Revolution

The methods of descriptive geometry, as one of the branches of knowledge belonging to the domain of the mathematical sciences, must, in solving any given question, follow the principle firmly established in these sciences of maintaining the greatest possible mathematical precision. For the various kinds of graphical constructions in the field of descriptive geometry, this reduces to so-called geometrical precision, i.e. to the determination, by means of compass and ruler, of the intersections of straight lines and circles. Thus, methods of such a kind, in which the construction of curves by points is allowed, and only then are the points of intersection with these curves of other lines determined, no longer being geometrically exact, should be excluded from the graphical solutions of problems in descriptive geometry, since exact solutions of them exist.

However, in all likelihood, owing to the practical orientation of the teaching of descriptive geometry as a science that has wide application in applied knowledge, due attention is not always paid, in the exposition of some of its sections, to the observance of geometrical precision. Such allowances could be justified if, thanks to them, certain simplifications were obtained in the solutions of problems; unfortunately, this is not always the case.

One may point to an entire section of descriptive geometry, concerning the question of constructing the mutual intersections of second-order surfaces of revolution, where the indicated departures from geometrical precision have acquired the right of citizenship—whereas there is every possibility of applying simpler and geometrically exact methods of solution, based on the characteristic properties of these surfaces. The description of these methods for constructing the projections of mutual intersections of second-order surfaces of revolution constitutes the subject of the present article.

§ 1. All problems on finding the mutual intersection of second-order surfaces of revolution may be subdivided into two cases: the first case—when the axes of rotation of the given surfaces lie in one plane, i.e. when they intersect one another at a finite ...

as before, parallel to one of the projection planes—say, the vertical one—while one axis of rotation is placed perpendicular to the other projection plane (the horizontal one), and then both surfaces are intersected by auxiliary planes \(P'_v\) (Fig. 3) parallel to this latter plane. From the intersection of the given surfaces by any one of these auxiliary planes—say \(P'_v\)—we obtain, on the horizontal projection plane, circles mutually intersecting with other curves of the second order, whose common points \(x_0\), \(x\) also belong to the horizontal projection of the required line of intersection of the given surfaces of revolution. Transferring these points in the direction of projection onto the vertical traces of the plane \(P'_v\), we obtain the corresponding vertical projections \(x'_0\), \(x'\). In the same way we determine the remaining points of the required line of intersection of the given surfaces.

Since the method just described for constructing projections of the lines of intersection of second-order surfaces of revolution reduces to determining the intersection of a curve of the second order with a circle whose center does not lie on the axis of the former—an intersection that cannot be constructed geometrically exactly with compass and straightedge—the method indicated, generally speaking, will not be geometrically exact, not to mention that all the constructions in the described manner of solving the question are to a considerable degree complicated.

§ 3. The essence of the method proposed below for constructing mutual intersections of second-order surfaces of revolution consists in the fact that both given surfaces are intersected by a series of parallel planes perpendicular to the plane of the axes of rotation (regardless of whether the axes of the given surfaces lie in one plane or in two parallel planes), and the position of these auxiliary planes must be such that both lines of their intersection with the given second-order surfaces of revolution would be similar ellipses with mutually parallel axes. In setting up the problem, one of the projection planes (suppose, the vertical) is placed parallel to the plane of the axes of rotation and, consequently, perpendicular to the auxiliary secant planes; the other projection plane (in the present case the horizontal) is placed so that the intersections of the given surfaces with the auxiliary planes are projected onto it as circles. With such an arrangement of the projection planes and of the auxiliary secant planes, the problem reduces to a case analogous to the case when the axes of rotation of the given surfaces are parallel to one another. Indeed, in both cases the horizontal projections of the points belonging to the required intersection are obtained from the mutual intersection of circles, and the vertical projections—by transferring the horizontal projections of the required points in the direction of—

onto the vertical traces of the auxiliary secant planes.

§ 4. Before proceeding to determine the inclination of the secant auxiliary planes to the axes of rotation of the given surfaces, let us note that, from the point of view of the similarity of plane sections of some surface of revolution of the second order, it is immaterial whether we cut the given surfaces of revolution or surfaces similar to them, but with axes of rotation parallel to the given ones; thus, in determining the inclination of the secant auxiliary surfaces, what matters to us is only the relative dimensions of both axes of each of the surfaces of revolution and the mutual inclination of the axes of rotation. Therefore one may imagine an arbitrary spherical surface to which two surfaces of revolution, similar to the given ones and having axes of rotation parallel to the given ones, are tangent; if some plane cuts these new surfaces of revolution along lines similar to one another, then every plane parallel to it will also cut the given surfaces along lines likewise similar to these sections.

Let us suppose that the axes of rotation of these auxiliary surfaces of revolution tangent to the sphere lie in a plane parallel to the vertical plane of projection; then the lines of contact of the spherical surface with the surfaces of revolution will project onto the vertical plane of projection as two straight lines \(a'b'\) and \(c'd'\) (Fig. 4). The points of intersection of these lines of tangency (circles), which project onto the vertical surface as coincident points \(t_0'\), \(t'\), at which the straight lines \(a'b'\) and \(c'd'\) intersect, will be common points of contact for the surfaces of revolution under consideration, so that planes drawn through these points and tangent to the sphere will also be tangent planes to each of the surfaces of revolution. If now we take yet some third point of intersection of the surfaces of revolution, for example the point \(n'\), lying at the intersection of their vertical outlines, then the plane \(t_0't'n'\) \((t_0tn)\) drawn through these three points will contain two intersections with the second-order surfaces of revolution under consideration and two tangents to these intersections (from the intersection of the plane \(t_0't'n'\), \((t_0tn)\), with the tangent planes). But since these intersections of the second-order surfaces of revolution are curves of the second order having three common points and two common tangents, these curves merge into one and, consequently, the planes \(t_0't'n'\), \((t_0tn)\), and \(t_0't'm'\), \((t_0tm)\), drawn through the points of tangency \(t_0't'\) of the surfaces of revolution and the sphere, and also through the intersection of their outlines, coincide with the line of mutual intersection of the surfaces of revolution, which represents a curve of the second order.

Thus, the direction found for the secant planes is the very direction sought, for which the intersections of both

of the given surfaces of revolution are similar. With the arrangement of the vertical plane of projection adopted by us, these auxiliary cutting planes will be vertical-projecting, and, consequently, the direction of their traces is determined by the points of mutual intersection of the vertical outlines of the surfaces of revolution tangent to the sphere. It is not difficult to see that, for each case of intersection of two second-order surfaces of revolution, there exist two directions of cutting auxiliary planes, each giving two series of similar curves; moreover, always, although for one of these directions, closed curves are obtained in the sections, i.e. ellipses, which, with a proper choice of the direction of the horizontal plane of projection, are projected onto it as circles.

The latter presents no particular difficulty, since the major axes of the ellipses \(n'k'\), \(l'm'\), obtained from the intersection of the given surfaces of revolution by the auxiliary planes indicated above, by virtue of the arrangement of the projections adopted by us, will be parallel to the vertical plane of projection and will be projected onto it in their true size. And since it is possible, for any section, to determine also the size of the minor axis of the ellipse, the required projection of the major axis of the ellipse onto the horizontal plane is thereby determined as equal to it, under which this ellipse is projected as a circle. Knowing the vertical projection of the major axis of the ellipse, parallel to the vertical plane, and its horizontal projection, we obtain the direction of projection onto the horizontal plane of projection as the direction of the leg of a right triangle in which the position of the hypotenuse and the length of the other leg are given.

§ 5. Thus, the problem of finding the direction of the cutting auxiliary planes can be reduced to determining the intersection of two outlines of second-order surfaces of revolution. Such outlines may be: 1) two parallel straight lines, 2) two intersecting straight lines, 3) a circle, 4) a parabola, 5) a hyperbola, and 6) an ellipse, corresponding respectively to the following surfaces of revolution: cylinder, cone, sphere, paraboloid, hyperboloid, and ellipsoid*).

Let us consider all possible cases of mutual intersection of second-order surfaces of revolution and indicate the simplest methods for determining the direction of the cutting planes.

a) If among the intersecting surfaces of revolution there is a cylindrical or conical surface, then the intersection of their outlines is reduced to the intersection of a second-order curve with a straight line—a problem solved exactly with the aid of compass and ruler; since the methods of constructing the intersections of these lines are generally known, there is no need to give them here.

*) The axis of rotation of which is the major axis of the generating ellipse.

b) If among the intersecting surfaces of revolution there is a spherical one, or if the axes of rotation of both surfaces are parallel to one another, then, by bringing the center or the axis of this surface into coincidence with the axis of rotation of the other given surface, we shall see that the points of their intersection will always lie in a plane perpendicular to the axes of rotation; consequently, the direction of the auxiliary secant planes in intersections of second-order surfaces of revolution having parallel axes, and also in intersections with a sphere, must always be perpendicular to the axes of rotation of the given surfaces.

c) If among the intersecting second-order surfaces of revolution there is a paraboloid of revolution, then, in order to determine the line of intersection, there is no need to find the intersection of the outlines, since one can solve much more simply the question of the direction of the secant planes, if one takes into account the property of the paraboloid of revolution which consists in the fact that the projection of any plane section of a paraboloid of revolution onto a plane normal to its axis is a circle.*) Thus, in the present case it is necessary to draw the auxiliary secant planes in such a way that, solely from their intersection with the other given surface, one obtains, as projections on the horizontal plane (perpendicular to the axis of rotation of the paraboloid), circles; moreover, when the axes of rotation of the given surfaces are parallel to the vertical plane of projection, then, in order to determine the direction of the auxiliary secant planes, we construct on the horizontal plane of projection (Fig. 5) a circle tangent to the horizontal outline of the other of the given surfaces and having its center \(c,c'\) somewhere on the horizontal projection of its axis of rotation. Taking this circle to be the horizontal projection of the intersection of the second of the given surfaces of revolution with the desired auxiliary secant plane, we draw to the circle two tangent horizontally-projecting lines \(mm'\) and \(kk'\), in whose directions we find the points \(l', m', k'\) and \(n'\) of the vertical outline of the surface of revolution through which pass the traces of the secant planes sought, \(P'_v\). Thus in this case also the problem is reduced to the intersection of straight lines with two straight lines, or of straight lines with a curve of the second order.

d) If the second of the given surfaces of revolution is also a paraboloid of revolution, then, as is not difficult to see, it follows from the property of these surfaces that the traces of the auxiliary secant planes on the vertical plane of projection must be parallel or perpendicular to the bisector which divides in half

*) Since the cylinder of revolution possesses the same property, everything said here about the paraboloid of revolution also holds for the cylinder of revolution.

the angle between the vertical projections of the axes of the given surfaces of revolution.

d) If among the mutually intersecting surfaces there is a hyperboloid of revolution, then, in finding the direction of the auxiliary secant surfaces, one may make use of the well-known property of its asymptotic cone, namely that, when a hyperboloid of revolution (of one sheet or of two sheets) and its asymptotic cone are intersected by any plane, curves similar to one another are obtained in the sections. Thus, in finding the direction of the traces of the auxiliary secant planes, one may always use, instead of the hyperboloid of revolution, its asymptotic cone of revolution, which considerably simplifies the problem.

e) It remains now to consider the case of the intersection of two ellipsoids of revolution; in this case the problem reduces to the construction of the intersection of two ellipses having a common center and equal minor axes.

This problem is solved quite simply if these ellipses are regarded as curves arising from the intersection of two equal cylinders of revolution with the vertical plane of projection—whereby the diameters of these cylinders are equal to the minor axis of the ellipses, and their axes of revolution intersect at a point lying in the vertical plane of projection. It is not difficult to observe that, for such an arrangement, the points of intersection of the ellipses lie in the bisector planes with respect to the axes of the cylinders, and, consequently, the direction of the traces of the auxiliary secant planes is determined by the lines of intersection of these bisector planes with the vertical plane of projection.

Thus, whatever second-order surfaces of revolution are given, it is always possible to find, geometrically exactly, without even drawing their outlines, the necessary direction of the auxiliary secant planes; it is always possible to determine, with respect to the axes of revolution of the given surfaces, the position of the horizontal plane of projection and to obtain geometrically exactly first the horizontal, and then the vertical, projections of the points of intersection of the given surfaces of revolution.

Moreover, one and the same method is applicable both to surfaces of revolution whose axes lie in one plane, and to surfaces whose axes lie in two parallel planes *).

§ 6. Let us now consider several examples.

a) To find the line of intersection of a cone of revolution and an ellipsoid of revolution, the axes of which are situated in two planes parallel

*) An exception is presented by an ellipsoid of revolution for which the minor axis of the generating ellipse serves as the axis of revolution.

vertical plane of projection (Fig. 6), the axes of the generating ellipse, the angle of the generatrix of the cone with its axis, and the position of the axes of rotation being given.

Construction. On the vertical projection of the minor axis of the ellipsoid we describe, as on a diameter, a circle to which we draw two tangents parallel to the generatrices of the vertical outline of the cone; having determined, by a geometric construction, the four points of intersection of the outline of the ellipsoid with these straight lines, we join these points in pairs by straight lines (intersecting one another), which will give the direction of the traces of vertically projecting auxiliary secant planes. If now, from the ends of the diameters of the ellipse parallel to the secant plane, we draw tangents to the circle constructed by us, then these determine the direction of projection. The subsequent construction consists in the following: drawing a series of auxiliary secant planes parallel to the direction found, we transfer to the horizontal plane the centers of the chords of the ellipse obtained from the intersection of the outline of the ellipsoid by the traces of these planes; then we construct on it circles whose diameters are respectively equal to the horizontal projections of the chords. In exactly the same way we construct the horizontal projections of the sections of the cone by the auxiliary planes, and obtain a second system of circles; the intersections of these circles with the corresponding circles obtained earlier give a series of points belonging to the horizontal projection of the required line of intersection of the given surfaces. Having transferred these points in the direction of projection onto the traces of the auxiliary secant planes, we obtain the corresponding vertical projections of the required line.

b) To find the intersection of two ellipsoids of revolution whose axes neither intersect one another nor are parallel (Fig. 7).

Construction. We arrange the vertical plane of projection parallel to the planes of the axes of rotation of the given ellipsoids. On the vertical plane of projection, from the point \(u'\), by which the center of one of the ellipsoids is projected, as from a center, we construct the axes of an ellipse similar to the vertical outline of the other given ellipsoid, so that the minor axis of this ellipse is equal to the minor axis of the outline of the first ellipsoid, and the major axis \(u't'\) is parallel to the axis of rotation of the other ellipsoid. To find the points of intersection of these concentric ellipses, we construct the axes of two mutually intersecting cylinders of revolution of equal diameter, whose traces on the vertical plane of projection are the above-mentioned pair of concentric ellipses. The vertical projections of the axes of these cylinders coincide with the major axes of the ellipses \(e'u'\) and \(u't'\), while the horizontal projections can easily be constructed if we bring each axis of a cylinder by rotation about the centers of the ellipses \(uu'\) into a plane parallel to the ver-

tical plane of projection. In this position the angle of inclination of the horizontal projection of the axis of any cylinder \(d\) is determined from the condition: \(\sin a=\frac{a}{b}\), where \(a\) and \(b\) are the axes of the ellipse; thus, having constructed the projections for the displaced position of the axes, we bring them back to the previous position corresponding to the axes of the concentric ellipses we have constructed. Having constructed planes that bisect the angles between the axes of the auxiliary cylinders \(e'u'eu\), \(t'u'tu\), we find their vertical traces \(P'_v\), which will also be parallel to the traces of the auxiliary vertically projecting planes intersecting the given ellipsoids of revolution in curves projected onto the horizontal plane of projection as circles. In all other respects the construction proceeds analogously to the construction indicated in the first example.

c) To find the intersection of a paraboloid of revolution and a one-sheeted hyperboloid of revolution, whose axes are situated in two planes parallel to the horizontal plane of projection (Fig. 8).

Construction. On the horizontal plane of projection we construct the outline of the asymptotic cone of the hyperboloid of revolution, and then also its vertical projection on a plane perpendicular to the axis of the given paraboloid. On the vertical projection of the cone we describe an arbitrary circle tangent to the generators of its outline, and we transfer the points of its intersection \(l'\) and \(k'\) with the vertical projections of the horizontal outline of the cone onto the generators of the horizontal outline of the latter in the direction of projection. The diagonals connecting the opposite vertices of the quadrilateral thus obtained will give the direction of the auxiliary horizontally projecting planes \(P_h\), whose intersections with the given surfaces of revolution are projected onto the vertical plane of projection as circles.

In what follows, the constructions for finding the projections of the line of intersection of the given surfaces are analogous to the constructions given in the first example.—

Thus the method described above for constructing the projections of the mutual intersection of second-order surfaces of revolution not only makes it possible to obtain a geometrically exact solution of the problem in those cases where the ordinary method of construction yields an inexact solution, but at the same time is also a simplification in the sense of carrying out the problem, since the essence of the proposed construction reduces to determining the intersection of two circles in those cases where, under the usual solution, one has to find intersections of circles with other second-order curves constructed by points.

S. P. Gomella.

\(ab, a'b', cd, c'd'\) — axes of the given ellipsoid.

\(fe, f'e'\) — axis of the given cone.

\(x, x'\) and \(x_0, x'_0\) — the sought points of intersection.

… projections of intersections of surfaces of revolution of the second order.

Fig. 3.

Fig. 4.

\(P N^\circ \mathfrak{B} K\) — direction of projection for secant planes \(\parallel n'k'\).

\(\mathfrak{B} m \mathfrak{B} d\) — direction of projection for secant planes \(\parallel m'l'\).

The horizontal projections are not shown in the drawing.

Fig. 7.

\(a'b', ab\) — axis of rotation of the paraboloid.

\(e'd', ed\) — axis of rotation of the hyperboloid.

\(B.\)

Fig. 8.