On Congruences of Circles Possessing a Doubled Family of Canal Surfaces

M. R. Vaintrub

(Presented by the Department of Higher Mathematics of the Tomsk Polytechnic Institute)

1. In the paper [1] a metric characterization is given of congruences of circles possessing one or two families of canal surfaces.

In the present note we consider congruences \(\{2.2\}\) (congruences of circles whose planes depend on two parameters) possessing a doubled family of canal surfaces.

1. The closed system of differential equations of the congruence \(\{2.2\}\) with respect to the canonical frame (see [2]) is written in the form:

\[ \begin{gathered} \omega_i=\Lambda_{ij}\omega_3^j,\qquad \omega_1^2=b_j\omega_3^j,\\ da_0=\Lambda_{0j}\omega_3^j,\qquad \omega^3=\Lambda_1\omega_3^1,\\ [d\Lambda_{01}\omega_3^1]+[d\Lambda_{02}\Lambda_3^2] =-\{b_1\Lambda_{01}+b_2\Lambda_{02}\}[\omega_3^1\omega_3^2],\\ [db_1\omega_3^1]+[db_2\omega_3^2] =-\{b_1^2+b_2^2+1\}[\omega_3^1\omega_3^2],\\ [d\Lambda_1\omega_3^1] =\{\Lambda_{21}-\Lambda_{12}-b_1\Lambda_1\}[\omega_3^1\omega_3^2],\\ [d\Lambda_{11}\omega_3^1]+[d\Lambda_{12}\omega_3^2] =\{b_1(\Lambda_{22}-\Lambda_{11})-b_2(\Lambda_{12}+\Lambda_{21})\}[\omega_3^1\omega_3^2],\\ [d\Lambda_{21}\omega_3^1]+[d\Lambda_{22}\omega_3^2] =\{-b_1(\Lambda_{21}+\Lambda_{21})+b_2(\Lambda_{11}-\Lambda_{22})-\Lambda_1\}[\omega_3^1\omega_3^2], \end{gathered} \tag{1} \]

where the Pfaff forms \(\omega_i\equiv\omega^i,\ \omega_j^i,\ i,j=1,2\), satisfy the skew-symmetry conditions and the Cartan structural equations ([3], pp. 137–138) of three-dimensional Euclidean space \(E_3\).

Each of the coordinate submanifolds \(\omega_3^i=0\) is a one-parameter family of circles and is characterized by the fact that the tangent vector to the spherical image of the ruled surface described by the axes of the circles is the vector \(e_j\) of the frame (here and below: \(j\ne i\)).

Let us denote by \(\lambda_{ir}, p_i, r_i\), respectively, the abscissa of the striction point, the distribution parameter, and the cosine of the distribution of the ruled surface described by the axes of the circles along \(\omega_3^i=0\). Applying the corresponding computational formulas [4], we obtain:

\[ \lambda_{ir}=\Lambda_{ij},\qquad p_i=\Lambda_{ij},\qquad r_i=-b_j. \tag{2} \]

For the invariants \(\lambda_{0i}\) we have the following relative characteristic: the vanishing of each of the invariants \(\lambda_{0i}\) distinguishes, respectively, congruences of circles of constant radius along the directions \(\omega_3^i=0\).

1. The foci and focal directions of the congruence of axes are determined by the system

\[ (\rho-\Lambda_{11})\omega_3^1-\Lambda_{12}\omega_3^2=0, \]
\[ -\Lambda_{21}\omega_3^1+(\rho-\Lambda_{22})\omega_3^2=0, \tag{3} \]

where \(\rho\) is the abscissa of the focus of a ray of the congruence.

Eliminating from (3) the ratio of the Pfaff forms \(\omega_3^1:\omega_3^2\), we obtain the equation for determining the coordinates of the foci of this congruence:

\[ \rho^2-(\Lambda_{11}+\Lambda_{22})\rho+\Lambda_{11}\Lambda_{22}-\Lambda_{12}\Lambda_{21}=0. \tag{4} \]

1. Directly from the theorem of R. M. Geidelman, proved in [1], there follows a necessary condition for the existence of congruences \(\{2.2\}\) possessing a doubled family of canal surfaces: if a congruence of circles belonging to a two-parameter family of planes possesses a doubled family of canal surfaces, then the congruence of its axes is parabolic.

If the congruence of axes \((\bar A \bar e_3)\) is parabolic, then the condition holds

\[ (\Lambda_{11}-\Lambda_{22})^2+4\Lambda_{12}\Lambda_{21}=0, \tag{5} \]

and its doubled focus coincides with the point

\[ \bar F=\bar A+\frac{\Lambda_{11}+\Lambda_{22}}{2}\bar e_3. \tag{6} \]

To the focus (6) there corresponds the focal direction

\[ \Omega=2\Lambda_{21}\omega_3^1-(\Lambda_{11}-\Lambda_{22})\omega_3^2=0. \tag{7} \]

The doubled family of canal surfaces of the congruence \(\{2.2\}\), if it exists, envelops the focal sphere with center at the point (6), and its equation is equation (7). Analytically this is expressed in the form

\[ (\bar M-\bar F,d\bar M)\equiv 0 \pmod{\Omega}, \tag{8} \]

where \(\bar M\) is the radius vector of an arbitrary point of a circle of the congruence, determined by the formula

\[ \bar M=\bar A+R(\bar e_1\cos\varphi+\bar e_2\sin\varphi). \tag{9} \]

Here \(\varphi\) is the angle formed by the radius vector \(\overline{AM}\) and the axis \(\bar e_1\).

Since

\[ d\bar M=(\omega^1+dR\cos\varphi-R\sin\varphi\,d\varphi-R\omega_1^2\sin\varphi)\bar e_1+ \]
\[ +(\omega^2+dR\sin\varphi+R\cos\varphi\,d\varphi+R\omega_1^2\cos\varphi)\bar e_2+ \tag{10} \]
\[ +(\omega^3+R\omega_1^3\cos\varphi+R\omega_2^3\sin\varphi)\bar e_3, \]

then, by virtue of (8), we shall have

\[ R\cos\varphi(\omega^1+\rho\omega_3^1)+R\sin\varphi(\omega^2+\rho\omega_3^2)+ \]
\[ +R\,dR-\rho\omega^3\equiv 0 \pmod{\Omega}. \tag{11} \]

Taking into account (1), (5), (6), and (7), we finally obtain

\[ (\Lambda_{11}-\Lambda_{22})[2\Lambda_{01}+(\Lambda_{11}+\Lambda_{22})\Lambda_1]+4\Lambda_{02}\Lambda_{21}=0. \tag{12} \]

Combining (5) and (12), we obtain two independent conditions characterizing the congruence \(\{2.2\}\) with a doubled family of canal surfaces.

Analysis of system (1) by means of S. V. Bakhvalov’s criterion (for example, [5], p. 44) shows that the arbitrariness of existence of such congruences is 2 functions of 2 arguments.

1. Let us consider some special classes of congruences {2.2} possessing a doubled family of canal surfaces.

1) Let us require that the doubled family of canal surfaces be the envelope of a focal sphere whose center coincides with the center of the circle of the congruence. From (6) we have:

\[ \Lambda_{11}+\Lambda_{22}=0. \tag{13} \]

Conditions (5) and (12) take, respectively, the form

\[ \Lambda_{11}^{2}+4\Lambda_{12}\Lambda_{21}=0 \tag{14} \]

and

\[ V_{01}\Lambda_{11}+\Lambda_{02}\Lambda_{21}=0. \tag{15} \]

From (2), in view of (13), it follows that the throat points of the ruled surfaces described by the axes of the circles of the congruence along \(\omega_{3}^{1}=0\) and \(\omega_{3}^{2}=0\) are symmetric with respect to the center of the focal sphere.

2) Let us require that the doubled family of canal surfaces of the congruence {2.2} be a family of tubular surfaces. This means that the congruence {2.2} must be a congruence of circles of constant radius. For such congruences

\[ \Lambda_{01}=\Lambda_{02}=0. \tag{16} \]

Analysis of equations (5) and (12) shows that here two possibilities arise:

a) case:

\[ \Lambda_{01}=\Lambda_{02}=0,\qquad \Lambda_{11}+\Lambda_{22}=0. \tag{17} \]

Equation (5) reduces to the form: \(\Lambda_{11}^{2}+\Lambda_{12}\Lambda_{21}=0\). In this case the center of the focal sphere enveloped by the doubled family of tubular surfaces of the congruence {2.2} coincides with the center of the circle.

b) case:

\[ \Lambda_{01}=\Lambda_{02}=0,\qquad \Lambda_{11}=\Lambda_{22}. \tag{18} \]

From (5) it follows that \(\Lambda_{12}\Lambda_{21}=0\). Consequently, either the conditions

\[ \Lambda_{01}=\Lambda_{02}=0,\qquad \Lambda_{11}=\Lambda_{22},\qquad \Lambda_{12}=0, \tag{19} \]

or the conditions

\[ \Lambda_{01}=\Lambda_{02}=0,\qquad \Lambda_{11}=\Lambda_{22},\qquad \Lambda_{21}=0. \tag{20} \]

hold.

From (2) it follows that, when each of the conditions (19) and (20) is satisfied, the ruled surfaces described by the axes of the circles along \(\omega_{3}^{1}=0\) and \(\omega_{3}^{2}=0\), respectively, are tori with common edge of regression

\[ \bar{s}=\bar{A}_{11}+\Lambda_{11}e_{3}. \]

Each of the special classes considered exists and is determined with arbitrariness of 1 function of 2 arguments.

REFERENCES

1. R. M. Geidelman. Metric characterization of congruences of circles possessing families of canal surfaces, Uchenye zapiski MGU, vol. XII, no. 4 (76), 1957.

2. M. R. Vaintrub. On congruences of circles in Euclidean space, Proceedings of Tomsk University, vol. 191, series Mechanics and Mathematics, 143–149, 1967.

3. S. P. Finikov. Method of Exterior Forms of Cartan. Moscow–Leningrad, 1948.

4. R. N. Shcherbakov. Construction of the metric theory of a complex of lines by means of the reperage of ruled submanifolds, Proceedings of Tomsk University, vol. 155, mathematical series, 3–24, 1961.

5. R. N. Shcherbakov. Course of Affine and Projective Differential Geometry. Tomsk, 1960.