PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE NAMED AFTER S. M. KIROV,
ORDER OF THE OCTOBER REVOLUTION AND ORDER OF THE RED BANNER OF LABOR

Vol. 261　　　　　　　　　　　　　　　　　　　　　　　　　1975

THE USE OF CIRCULAR SECTIONS OF QUADRIC SURFACES IN SOLVING CERTAIN PROBLEMS OF DESCRIPTIVE GEOMETRY

E. E. MANASHEROV

(Presented at the scientific seminar of the departments of descriptive geometry of the Tomsk higher educational institutions)

In solving a number of problems in descriptive geometry, a well-known property of quadric surfaces is used: if a plane intersecting a given surface passes parallel to the plane of one of the families of circular sections, then the figure of the section of the surface is a circle.

With the use of circular sections the following problems can be solved: a) constructing the missing projection of a point belonging to a surface; b) constructing a line and a plane tangent to a surface; c) constructing sections of a surface by a plane and the points of intersection of a line with a surface; d) mutual intersection of surfaces.

Tangent Plane to a Surface

Example 1. Suppose that at a point \(A\) (the frontal projection is given) of the surface of an elliptic paraboloid (Fig. 1) it is required to draw a tangent plane.

Through the point \(A\) pass two circles (\(a\) and \(b\)); the tangents to them determine the required tangent plane, i.e., the problem reduces to constructing two tangent lines \(n\) and \(m\).

The tangent \(n\) to the circle \(a\) is constructed in the following way. The auxiliary projection \(\overline{A}_2\) is found at the intersection of the arc \(a_2\) with the perpendicular erected from \(A_2\) to \(a_2\). The perpendicular \(A_2N_2\) to \(O_2A_2\) is the auxiliary projection \(n_2\) of the tangent \(nn_2 \equiv a_2\). The horizontal projection \(A_1\) is determined by laying off from \(\Lambda\) along the vertical line of connection the segment \(K_1A_1 = A_2\overline{A}_2\); the projection \(N_1\) lies on \(\Lambda\). The straight line \(N_1A_1 \equiv n_1\) determines the horizontal projection of the tangent \(n\) to the circle \(a\). Similarly, the tangent \(m\) to the circle \(b\) is constructed.

Naturally, this method of constructing the tangent plane is suitable for all quadric surfaces that have families of circular sections [1, 2, 3]. Moreover, the tangent plane may be drawn both through a point chosen on the surface itself and outside it.

The problem of constructing a tangent plane to the above-mentioned quadric surfaces may be solved by transforming the given—

Fig. 1.

Fig. 2.

of the given surface into a surface of revolution and then solving the problem by the known method for a surface of revolution. Let us give one example, using an affine transformation of space.

Example 2. Let the surface of an elliptic paraboloid be given (Fig. 2). It is required, from the given frontal projection \(M_2\), to determine the horizontal projection \(M_1\) of a point \(M\) belonging to the surface, and to construct the tangent plane to the surface of the elliptic paraboloid at this point. With the aid of a collineation of space [4], we transform the surface of the elliptic paraboloid into a surface of revolution. The frontal plane \(\Lambda\) is taken as the plane of collineation; the direction of collineation is perpendicular to the plane \(\Lambda\). In order to transform the elliptic base into a circle, we carry out the following construction. Through the points \(A_1\) and \(B_1\) we draw the straight line \(l_1\) to its intersection with \(\Lambda_1\) and find the double point of the transformation \(K_1 \equiv \overline{K}_1\). From this point we draw the collinear straight line \(\overline{l}_1\) at an angle of \(45^\circ\) to \(\Lambda_1\) and find the points \(\overline{A}_1\) and \(\overline{B}_1\). The points \(\overline{C}_1\) and \(\overline{D}_1\) are determined without special constructions. The point \(M_1\) is found as the missing projection of the point \(M\) on the transformed surface—the paraboloid of revolution. The point \(M_1\) is found as the point collinear to \(\overline{M}_1\). From \(\overline{M}_1\) we draw a straight line parallel to \(\overline{l}_1\) to its intersection with \(\Lambda_1\), and then parallel to \(l_1\) to its intersection with the direction of collineation from the point \(\overline{M}_1\). To construct the tangent plane, we draw tangent lines: \(n\) to the horizontal section—the parallel—and \(m\) to the meridional section—the parabola of the surface of revolution. Then, by the inverse transformation, we find the position of the tangents. Here the frontal projections \(n_2\) and \(m_2\) remain unchanged; \(m_1\) passes through \(M_1\) to the center of the ellipse. The straight line \(n_1\) is constructed as collinear to \(\overline{n}_1\). The tangent \(\overline{n}_1\) is drawn at the point \(\overline{M}_1\) of the circle—the section of the surface of the paraboloid of revolution by the plane \(\Gamma\). The straight lines \(n\) and \(m\) determine the required tangent plane \(T\) at the point \(M\) of the surface.

Naturally, with other quadric surfaces of general form, the problem of constructing a tangent plane and of finding the missing projection of a point can be solved in an analogous way.

The problem of constructing a tangent plane to the surface of a triaxial ellipsoid is also solved by using the related transformation (stretching or compression) of the surface of the triaxial ellipsoid into a spherical surface and back [5].

Example 3. A surface of a triaxial ellipsoid is given (Fig. 3). It is required to construct the tangent plane to the surface at the point \(S\) (the frontal projection \(S_2\) on it is given).

Fig. 3.

First, from the frontal projection \(S_2\), the horizontal projection \(S_1\) of the point \(S\) is constructed. Here, in essence, a double related transformation is applied. The triaxial ellipsoid is first transformed into a biaxial one (of revolution). The horizontal plane is taken as the plane of affinity. The horizontal projection of the surface remains unchanged, while the frontal projection is transformed into a circle with radius equal to the major semiaxis \(OA\). In this process the frontal projection of the point \(S_2\) passes into \(\overline{S}_2\). Then the ellipsoid of revolution is transformed into a sphere of radius \(OA\). In the second transformation the frontal plane is taken as the plane of affinity. The transformed horizontal projection \(S_1\) will be determined as a point belonging to the sphere. By the inverse transformation (compression) of the sphere into an ellipsoid of revolution, we find the required horizontal projection \(S_1\) of the point \(S\).

The construction of the tangent plane at the point \(S\) is reduced to the construction of two tangent straight lines. One line—\(k\)—is drawn tangent to the horizontal section (without constructing the ellipse section itself), and the other—\(l\)—is tangent to the section that passes through the middle axis of the triaxial ellipsoid. Each of the elliptical sections is transformed into a circle; tangent lines to the circles are constructed, and by the inverse transformation the tangents \(t\) and \(l\) are found. \(D_2S_2 \equiv l_2;\ l_1 = D_1S_1;\ t_2\) is drawn through \(S_2\) parallel to \(O_2A_2;\ t_1 \equiv S_1N_1\). Thus, the tangent plane \(T \equiv t \times l\) has been constructed.

The advantage of the method presented is that it does not require construction of the outline of the surface.

However, the method proposed by the author for using circular sections, as is clear from what was set out earlier, is simpler and possesses sufficient generality.

Intersection of a Surface by a Plane and a Line

Example 4. Let the surface of an elliptic paraboloid be given (Fig. 4), and let it be required to construct the section of the surface by a frontal-projecting plane. The frontal-projecting plane \(\Sigma\) is parallel to the plane of the family of circular sections \(\Lambda\), and therefore intersects

the given surface in a circle. The circle will be obtained in the section of the surface by a plane \(\Sigma'\), parallel to the plane of another family of circular sections \(\Lambda'\). This applies to a second-order surface having two systems of circular sections.

The plane \(\Gamma\) is also a projecting plane, but it is parallel neither to the plane \(\Lambda\) nor to the plane \(\Lambda^1\). Therefore, to construct the section one must use auxiliary planes, drawing them parallel to the plane \(\Lambda\) or \(\Lambda^1\). The auxiliary planes intersect the surface in circles, and the plane \(\Gamma\) in front-projecting straight lines. The intersection of a circle \(n\) and a straight line gives us two points \(C\) and \(D\) of the section. The highest point \(A\) and the lowest point \(B\) of the section are essentially determined, since their frontal projections are already present in the drawing. The remaining points of the section are constructed analogously to the points \(C\) and \(D\). In the section by the plane \(\Gamma\) an ellipse is obtained, which could have been constructed by its axes. The major axis \(A_2B_2\) is given; the minor axis is determined as the chord of the circle whose plane passes through the ellipse being determined. Thus one can construct every elliptical section that is perpendicular to the plane of the ombilic diameters. An ombilic diameter is a straight line on which the centers of the circular sections of one family are located. On the surface, in the general case, there are two ombilic diameters. The ombilic diameters form a plane, which we shall call the plane of the ombilic diameters. This plane is also a plane of symmetry of the surface.

Fig. 4.

Example 5. A surface of a triaxial ellipsoid is given (Fig. 5). It is required to construct the section of the surface by the plane \(\Sigma\) \((a \times b)\), which is a plane in general position. In this case as well we use auxiliary planes, which we draw parallel to the plane of the circular section \(\Lambda\). In the given case the plane \(\Lambda\) is parallel to the horizontal plane of projections \(\Pi_1\). The auxiliary plane \(\Gamma\) intersects the surface of the triaxial ellipsoid in a circle \(m\), and the plane \(\Sigma\) in the horizontal line \(h\); \(\Gamma_2 \equiv m_2 \equiv h_2\). Since the straight line is a horizontal, \(h_2 \parallel a_2\), and \(h_1 \parallel a_1\). The intersection of the straight line \(h\) with the circle \(m\) gives two points \(1\) and \(2\) of the section. The remaining points of the section, \(3, 4\), and others, are constructed in an analogous manner.

The problem of constructing plane sections of second-order surfaces of general type, having two families of circular sections, is solved—

Fig. 5.

is also performed with the use of auxiliary projection onto a plane parallel to the plane of one of the families of circular sections [6, p. 88].

Example 6. A general one-sheeted hyperboloid is given (Fig. 6). It is required to construct the section of the surface by a plane \(\Lambda\), specified by the triangle \(KLM\). As the auxiliary plane, the profile plane of projection \(\Pi_3\) is taken; as the direction of projection, the side \(LM\) of the triangle. Since the planes of one family of circular sections are parallel to \(\Pi_3\), the circles are projected onto this plane without any distortion, and their centers lie on the straight line \(1_3 2_3\)—the auxiliary projection of the umbilic diameter. The plane \(\Lambda\) \((KLM)\) is projected as the straight line \(\Lambda_3\). The intersections of the auxiliary projections of the circles and the plane \((\Lambda_3)\) give the projections of the required points of the section. By inverse projection we transfer them to the given projections. The drawing shows the construction of the point \(J\). The circle tangent to \(\Lambda_3\) gives the projection \(N_3\) of the extreme right-hand point \(N\) of the section.

Fig. 6.

Example 7. Let the surface of an elliptic paraboloid and a straight line \(m\) be given (Fig. 4). It is required to determine the points of intersection of the straight line with the surface. The problem is solved by the general method. Through the straight line \(m\) a plane \(\Gamma\) is drawn, and the section is found; its intersection with the straight line \(m\) gives the two required points 1 and 2.

If the straight line \((l)\) is parallel to the plane of circular sections, then the construction is simplified, since the auxiliary plane \((\Sigma)\) gives a circle in section, which, in intersection with the given straight line \((l)\), gives the two required points (3 and 4). The constructions for this problem are shown in Fig. 4.

Suppose it is required to construct the points of intersection of the straight line \(l \equiv KM\) (Fig. 6) with the surface of a general one-sheeted hyperboloid. This problem can be solved by means of auxiliary projection onto the plane \(\Pi_3\). \(\Pi_3\) is chosen parallel to the plane of one of the families of circular sections. Through the straight line \(l\) we draw a plane \(\Lambda\) \((KLM)\), which is projecting with respect to the new image plane. The line of section of the surface by the plane \(\Lambda\), in intersection with the straight line \(l\), gives the required points of intersection 4 and 5.

Mutual Intersection of Planes

Let us consider several examples of constructing the line of intersection of surfaces using circular sections. Suppose that a general second-order surface intersects a faceted surface. In constructing the line of intersection of a second-order surface that has two systems of circular sections with a faceted surface, the problem essentially reduces to constructing the line of intersection of each face (plane) with the surface.

Example 8. A surface of a single-cavity elliptic hyperboloid and a triangular pyramid are given (Fig. 7). It is required to construct the line of intersection of the given surfaces. The problem is solved with the aid of auxiliary planes parallel to the plane of the circular sections. The auxiliary plane \(\Sigma\) intersects the hyperboloid in the circle \(m\), and the pyramid in the triangle \(ABC\). The circle and the triangle intersect in this case at two sought points \(D\) and \(E\). These points are easily found on the new projection plane \(\Pi_5\), which is parallel to \(\Sigma\), since \(m_5\) is a circle,
\(D_5 = A_5B_5 \times m_5\),
\(E_5 = A_5C_5 \times m_5\). By inverse projection we find the frontal and horizontal projections of the points \(D\) and \(E\). The remaining points are found in an analogous manner.

Fig. 7.

Example 9. The surfaces of a triaxial ellipsoid and an elliptic hyperboloid are given, the planes of whose circular sections are directed in the same way (Fig. 8, a). It is required to construct the line of intersection of these surfaces. In this case the planes of the umbilic diameters of both surfaces coincide and are parallel to \(\Pi_2\). We choose the auxiliary plane \(\Sigma\) parallel to the circular sections; therefore the sections of the surfaces are two circles, whose intersection gives two common points 1 and 2.

The solution of the problem is given in one projection by rotation about the straight line joining the centers of the circles of the sections. We place the planes of both circles parallel to the plane \(\Pi_2\), which immediately gives us the auxiliary projections of points 1 and 2. By inverse rotation we find the frontal projections \(1_2 \equiv 2_2\) of the sought points. On the auxiliary projection only arcs of circles are used. The remaining points of the line of intersection are constructed in the same way.

Fig. 8.

The method presented makes it possible to construct the line of intersection of surfaces that have a common plane of symmetry parallel to the plane of the drawings. If the planes of symmetry of the surfaces were parallel to one another but did not coincide, then an additional projection would have to be constructed onto a plane parallel to the planes of the circular sections. In doing so, it is necessary to take into account the distance between the planes of symmetry. This method is general for surfaces with identically directed circular sections.

In an analogous way, particular cases of the problem of intersecting the surfaces of an elliptic paraboloid and an elliptic cone (Fig. 8, b), an elliptic cone and a cone of revolution (Fig. 8, c), and a paraboloid and a cone of revolution (Fig. 8, d) have been solved. In all these cases both intersecting surfaces have identically directed circular sections. In the first two examples both surfaces are second-order surfaces having two systems of circular sections; in the third, one of the surfaces is a surface of revolution; in the fourth, both surfaces are surfaces of revolution. In the case of a second-order surface of revolution we have two systems of coincident circular sections.

The problem is simplified if one of the second-order surfaces is a spherical surface. Parallel to the planes of the circular sections of the surface that intersects the sphere, we draw an auxiliary plane. In this case both surfaces will be intersected along circles, since every plane section of a spherical surface is a circle. Thus, this problem is reduced to the case of intersecting surfaces with identically directed circular sections.

By the method described earlier, problems have been solved concerning the intersection of the surfaces of a triaxial ellipsoid, an elliptic cone, and a torus with a spherical surface (Fig. 9, a, b, c). We assume that, for the spherical surface, every plane passing through its center is a plane of umbilical diameters. In the first and third examples the surfaces have a common plane of umbilical diameters; in the sec-

…in the example under consideration, the planes of the umbilic diameters do not coincide; they are parallel to \(\Pi_2\). Moreover, for the sphere they are closer to the frontal plane of projection by the amount \(a\) (Fig. 9, б). The points \(B\) and \(C\) of the line of intersection of the surfaces of the sphere and the elliptic cone are constructed as follows. The auxiliary plane \(\Sigma\) intersects the surfaces in the circles \(m\) and \(n\). On the circle \(n\), the diameter \(i\), which is a frontal, is chosen as the axis of rotation. By rotating about this axis, the plane \(\Sigma\) is brought into coincidence with the frontal level plane. In this case the transformed frontal projections of the circles \(\overline{m}_2\) and \(\overline{n}_2\) will be projected without distortion, and their centers \(\overline{1}_2\) and \(\overline{2}_2\) are displaced by the amount \(a\). The intersection of these circles gives the auxiliary projections \(\overline{B}_2\) and \(\overline{C}_2\) of the points \(B\) and \(C\). By inverse rotation we find the frontal projections \(B_2\) and \(C_2\) of the required points of the line of intersection. The remaining points of the line of intersection are constructed analogously. The points \(D\) and \(E\) on the contour generator \(l\) of the cone are constructed with the aid of an auxiliary plane taken to be the plane of the umbilic diameters of the elliptic cone. This plane, passing through \(l\), intersects the spherical surface in the circle \(k\), which is displaced from the plane of the circle of the principal meridional section by the amount \(a\). The projection \(k_2\) of the circle \(k\) on the plane \(\Pi_2\) is projected without distortion. Its center is \(O_2\), and its radius is equal to \(R\). The intersection of \(k_2\) and \(l_2\) gives the frontal projections of the required points; \(D_2 = k_2 \times l_2\) and \(E_2 = k_2 \times l_2\).

Fig. 9.

Thus, the enumerated cases of intersection of two surfaces may be reduced to a single type of problem—with equally directed circular sections—and solved by a general method, using projecting auxiliary planes.

Example 10. The surfaces of an elliptic paraboloid and a torus are given (Fig. 10, а). It is required to construct their line of intersection. The plane of the umbilic diameters of the elliptic paraboloid is the plane of symmetry of both surfaces. We shall use the well-known method of eccentric spherical auxiliary surfaces. From the center of the circular section \(m\), \(m_2 \equiv \Sigma\), of the surface of the elliptic paraboloid, we restore a perpendicular until it intersects the axis \(i\) of the torus at the point \(N\). From this point, as from a center, we draw a spherical auxiliary surface whose radius is \(R = A_2N_2\). The spherical surface intersects the surface of the elliptic paraboloid in the circle \(m\), and the surface of the torus in the circle \(n\). The circles \(m\) and \(n\), as belonging to one spherical surface, intersect in two required points \(B\) and \(C\) \((B_2 \equiv C_2)\). The remaining points of the line of intersection are constructed in a similar manner.

In the same way, problems of the intersection of surfaces have been solved: of an elliptic paraboloid and a ring (Fig. 10, б), of an elliptic cone and

Fig. 10.

of a cylinder of revolution (Fig. 10c), a “skew cone” [7, p. 329] and a torus (Fig. 10d), and a ring and a torus (Fig. 10e). In these examples the intersecting surfaces have a common frontal plane of symmetry.

Let us dwell on the construction of the line of intersection of the surfaces of a “skew cone” and a torus. The plane curve \(k\) is the locus of the centers of the circular sections (Fig. 10d). Take a point \(C\) on this curve and join it with the vertex \(S\). The plane \(\Sigma\), passing through \(C\) and perpendicular to \(SC\), intersects the surface of the “skew cone” in the circle \(m\) \((m_2 \equiv \Sigma_2)\). The line \(SC\), in its intersection with the axis of the torus \(i\), gives us the center \(N\) of the intermediary spherical surface. This spherical surface intersects the surface of the torus in the circle \(n\) \((n_2)\), and the surface of the “skew cone” in the circle \(m\) \((m_2)\). The intersection of \(n_2\) and \(m_2\) gives the frontal projections \(A_2 \equiv B_2\) of the required points \(A\) and \(B\) of the line of intersection. The remaining points of the line of intersection are constructed in the same way. If from the point \(S_2\) a ray is drawn parallel to \(i_2\), then we obtain the circular section \(a\) with center \(D\), whose plane will be parallel to the plane of the circular sections of the torus. If the axis \(i\) passes through the vertex \(S\), then the point \(S\) may be taken as the center of concentric spherical surfaces, since every spherical surface drawn from the center \(S\) intersects the surface of the “skew cone” in a circle [7].

The type of problem shown in Fig. 10 is found in textbooks [7, 8, 9] and in other literature on descriptive geometry.

Fig. 11.

Example 11. The surfaces of a one-sheeted elliptic hyperboloid and an elliptic cone are given; their line of intersection is to be constructed. The planes of the umbilic diameters of both surfaces coincide and are parallel to \(\Pi_2\). We carry out an affine transformation of space in such a way that one of the surfaces is transformed into a surface of revolution [4]. Then we arrive at a special case (Fig. 10) and, using circular sections, solve the problem with the aid of eccentric intermediary spherical surfaces.

In our example (Fig. 11), after the transformation, the surfaces obtained are a one-sheeted hyperboloid of revolution and an elliptic cone. The construction of the points \(A\) and \(B\) \((A_2 \equiv B_2)\) is clear from the drawing. The remaining points are determined analogously.

Since the frontal plane was taken as the plane of affinity, the frontal projection after the transformation did not change. Therefore,

the frontal projections of the line of intersection before and after the transformation coincide. From the frontal projections it is easy to determine the horizontal projections of the points of the line of intersection with the aid of the generators of the conical surfaces.

In this case there is no need to construct the line of intersection on the transformed horizontal projection. This projection is needed essentially only to determine the direction of the families of circular sections of the elliptic cone after the transformation.

The example given shows that, by using circular sections, the problem of the mutual intersection of quadric surfaces of general form whose planes of ombilic diameters coincide can also be solved.

The investigation has shown that the use of circular sections simplifies the solution of the problems listed and makes it possible to approach their solution from a general point of view. The latter is also positive from a methodological point of view. Although not all possible variants have been considered in the work, they nevertheless convince us of the expediency of using circular sections in solving certain positional problems of descriptive geometry.

LITERATURE

1. E. E. Manasherov. On drawing tangent planes to certain surfaces. Abstracts of reports of the scientific-technical conference of TashIIT, Tashkent, Publishing House of the Tashkent Institute of Railway Transport Engineers, 1963.

2. E. E. Manasherov. On drawing tangent planes to certain surfaces. Problems of Descriptive Geometry and Engineering Graphics. Proceedings of TashIIT, issue 39, Tashkent. Publishing House of the Academy of Sciences of the Uzbek SSR, 1966.

3. I. V. Tsvitsinsky. Tangent planes to quadric surfaces having families of circular sections. In: Applied Geometry and Engineering Graphics, issue 2, Kiev, Budivelnyk Publishing House, 1966.

4. I. S. Dzhaparidze. Geometric transformations of space and their applications in descriptive geometry. In: Methods of Descriptive Geometry and Its Applications. Collection of articles edited by N. F. Chetverukhin. Moscow, Gostekhizdat, 1955.

5. V. V. Ryzhkov. Descriptive geometry of curved lines and surfaces. Problems of Modern Descriptive Geometry. Collection of articles edited by N. F. Chetverukhin. Moscow–Leningrad, Gostekhizdat, 1947.

6. I. V. Tsvitsinsky. Homological transformation of quadric surfaces having families of circular sections, and its application. Problems of the Theory and Practice of Descriptive Geometry. Collection of articles of the Kiev Civil Engineering Institute, Kiev, 1960.

7. N. A. Popov. A Course in Descriptive Geometry. Moscow–Leningrad, Gostekhizdat, 1947.

8. N. F. Chetverukhin, V. S. Levitsky, Z. I. Pryanishnikova, A. M. Tevlin, G. I. Fedotov. A Course in Descriptive Geometry. Moscow–Leningrad, Gostekhizdat, 1956.

9. N. N. Pshenichny, M. I. Repina, L. I. Marchenko. Descriptive Geometry. Moscow, Sovetskaya Nauka Publishing House, 1956.