On Congruences of Circles {1.2} with Multiple Focal Points

M. R. Weintraub

(Presented by the Department of Higher Mathematics)

General congruences of circles in three-dimensional Euclidean space belong to a two-parameter family of planes. Such congruences of circles possessing multiple focal points were studied in [1] and [2].

It is of interest to consider a congruence in which the planes of the circles form a one-parameter family (a congruence {1.2}). In [3] the focal properties of such a congruence were considered.

In the present paper congruences {1.2} with multiple focal points are studied.

1. Basic geometric objects associated with a congruence

In three-dimensional Euclidean space \(E_3\), consider a congruence of circles whose planes form a one-parameter family (a congruence {1.2}). Choosing the Pfaff forms \(\omega_3^1\) and \(\Theta^0\) [4] as independent, we write the system of differential equations of this congruence with respect to a canonized frame [5] in the form:

\[ \omega_1=\Gamma_{11}\omega_3^1+\Gamma_{10}\Theta^0,\quad \omega_2=\Gamma_{21}\omega_3^1,\quad \omega_3^2=\Lambda_{31}\omega_3^1, \]
\[ \omega^3=\Lambda_{01}\omega_3^1,\quad \omega_1^2=\Gamma_1\omega_3^1+\Gamma_0\Theta^0. \tag{1} \]

Here \(\Gamma_{10}>0\), by virtue of the orientation of the vector \(\bar e_1\) of the frame.

The closure of system (1) has the form:

\[ [d\Gamma_{11}\omega_3^1]+[d\Gamma_{10}\Theta^0] = (\Gamma_0\Gamma_{11}-\Gamma_{21}\Gamma_0)[\omega_3^1\Theta^0], \]
\[ [d\Gamma_{21}\omega_3^1] = (\Gamma_0\Gamma_{21}+\Gamma_0\Gamma_{11}-\Gamma_{10}\Gamma_1)[\omega_3^1\Theta^0], \]
\[ [d\Lambda_{31}\omega_3^1] = (\Gamma_0+\Gamma_0\Lambda_{31}^2)[\omega_3^1\Theta^0], \tag{2} \]
\[ [d\Lambda_{01}\omega_3^1] = (\Gamma_{10}+\Gamma_0\Lambda_{01}\Lambda_{31})[\omega_3^1\Theta^0], \]
\[ [d\Gamma_1\omega_3^1]+[d\Gamma_0\Theta^0] = \Gamma_0\Gamma_1\Lambda_{31}[\omega_3^1\Theta^0]. \]

Analysis of system (2) by means of the Bakhvalov criterion [6] shows that congruences {1.2} exist and are determined with arbitrariness of 2 functions of 2 arguments.

The coordinate submanifold \(\omega_3^1=0\) is a one-parameter family of circles incident with one plane. The charac-

characteristic points of the circle in the canonical frame are determined in the form:
\[ \begin{aligned} \bar A_1&=\bar A+\frac{1}{\Gamma_{10}}\,\bar e_1+ \frac{\sqrt{R^2\Gamma_{10}^2-1}}{\Gamma_{10}}\,\bar e_2,\\ \bar A_2&=\bar A+\frac{1}{\Gamma_{10}}\,\bar e_1- \frac{\sqrt{R^2\Gamma_{10}^2-1}}{\Gamma_{10}}\,\bar e_2. \end{aligned} \tag{3} \]

From (3) follows the geometric meaning of the invariant \(\Gamma_{10}\) as the value inverse to the abscissa of the characteristic points of the circle.

Denoting by \(ds=(\Gamma_{10}\Theta^\circ)|_{\omega_3^1=0}\) the element of arc of the line of centers of the circles of the submanifold \(\omega_3^1=0\), we obtain:
\[ \varkappa=\frac{\Gamma_0}{\Gamma_{10}}, \tag{4} \]
where \(k\) is the curvature of this curve.

The submanifold \(\Theta^0=0\) is a one-parameter family of circles of constant radius, since \(dR=-\frac{1}{R}\Theta^0\).

Let us consider some geometric images invariantly associated with the congruence \(\{1.2\}\).

1. The congruence of axes \((\bar A\bar e_1)\) of the frame. One focus of this congruence coincides with the center of the circle, and to it corresponds the focal direction
   \[ \omega_3^1=0. \tag{5} \]
   The second focus coincides with the point
   \[ \bar F=\bar A+\Lambda_{01}\bar e_1. \tag{6} \]
   To this focus corresponds the focal direction
   \[ (\Gamma_{21}+\Gamma_1\Lambda_{01})\omega_3^1+\Gamma_0\Lambda_{01}\Theta^\circ=0. \tag{7} \]

Denoting by \(\rho_\psi\) the abscissa of the center of the ray of the congruence \((\bar A\bar e_1)\), by virtue of formula (6) we obtain the geometric characteristic of the invariant \(\Lambda_{01}\):
\[ \Lambda_{01}=2\rho_\psi . \tag{8} \]

1. The congruence of axes \((\bar A\bar e_2)\) of the frame. One focus of this congruence coincides with the point
   \[ \Pi=\bar A+\frac{\Gamma_{10}}{\Gamma_0}\bar e_2 \tag{9} \]
   and to it corresponds the focal direction (5). The second focus is determined by the formula
   \[ \bar F=\bar A+\frac{\Lambda_{01}}{\Lambda_{31}}\bar e_2. \tag{10} \]
   To the focus (10) corresponds the focal direction
   \[ (\Gamma_{11}\Lambda_{31}-\Lambda_{01}\Gamma_1)\omega_3^1+ (\Gamma_{10}\Lambda_{31}-\Lambda_{01}\Gamma_0)\Theta^\circ=0. \tag{11} \]

2. The congruence of axes \((\bar A\bar e_3)\). This congruence is cylindrical, with its own focus at the point
   \[ \bar F=\bar A-\frac{\Gamma_{21}}{\Lambda_{31}}\bar e_3, \tag{12} \]

to which there corresponds the focal direction

\[ \left(\Gamma_{11}-\frac{\Gamma_{21}}{\Lambda_{31}}\right)\omega_3^1+\Gamma_{10}\Theta^\circ=0. \tag{13} \]

Denoting by \(\varphi\) the angle between the axis \(\bar e_1\) and the tangent vector to the spherical image of the congruence \((\bar A\bar e_3)\), we obtain:

\[ \Lambda_{31}=\operatorname{tg}\varphi. \tag{14} \]

1. The characteristic of the one-parameter family of planes \(x^3=0\), containing the circles of the congruence, is determined by the equations:

\[ x^1+\Lambda_{31}x^2-\Lambda_{01}=0,\qquad x^3=0. \tag{15} \]

1. The envelope \((M_{13})\) of the planes \(x^2=0\) coincides with the focal surface described by the proper focus (12) of the congruence \((\bar A\bar e_3)\).

2. The radius vector of the current point of the envelope \((M_{23})\) of the planes \(x^1=0\) is determined in the form:

\[ \bar M_{23}=\bar A+\frac{\Gamma_{10}}{\Gamma_0}\bar e_2+ \left(\Gamma_{11}-\frac{\Gamma_1\Gamma_{10}}{\Gamma_0}\right)\bar e_3. \tag{16} \]

Congruences \(\{1.2\}\) with multiple focal points.

Recall [3] that two foci \(F_1\) and \(F_2\) of a congruence \(\{1.2\}\) are the characteristic points (3) of the circles incident to one plane. To these two foci there corresponds one focal direction (5). The two remaining foci

\[ \bar F_3=\bar A+\frac{1}{1+\Lambda_{31}^2} \left[(\Lambda_{01}-\Lambda_{31}\Lambda)\bar e_1+ (\Lambda_{01}\Lambda_{31}+\Lambda)\bar e_2\right] \tag{17} \]

and

\[ \bar F_4=\bar A+\frac{1}{1+\Lambda_{31}^2} \left[(\Lambda_{01}+\Lambda_{31}\Lambda)\bar e_1+ (\Lambda_{01}\Lambda_{31}-\Lambda)\bar e_2\right], \tag{18} \]

where

\[ \Lambda=\sqrt{R^2(\Lambda_{31}^2+1)-\Lambda_{01}^2} \tag{19} \]

are the points of intersection of the characteristic (15) with the circle of the congruence. To the focus (17) there corresponds the focal direction

\[ [\Gamma_{11}(\Lambda_{01}+\Lambda)+ \Gamma_{21}(\Lambda_{01}\Lambda_{31}+\Lambda)]\omega_3^1+ \Gamma_{10}(\Lambda_{01}+\Lambda)\Theta^\circ=0, \tag{20} \]

and to the focus (18), the direction

\[ [\Gamma_{11}(\Lambda_{01}-\Lambda)+ \Gamma_{21}(\Lambda_{01}\Lambda_{31}-\Lambda)]\omega_3^1+ \Gamma_{10}(\Lambda_{01}-\Lambda)\Theta^\circ=0. \tag{21} \]

Let us consider some classes of congruences of circles \(\{1.2\}\) with multiple focal points.

1. The class \(\Gamma_{10}=\dfrac{1}{R}\) exists, is determined with arbitrariness of 1 function of 2 arguments, and is characterized by the coincidence of the foci \(F_1\) and \(F_2\).

2. The class \(\Gamma_{10}=\dfrac{1}{R},\ \Gamma_0=0\) exists, is determined with arbitrariness of 5 functions of 1 argument, and is characterized by the fact that the foci \(F_1\) and \(F_2\) coincide, while the surface described by this double focus degenerates into a curve. Since \(\Gamma^0=0\), it follows that: a) the congruence of axes \((\bar A\bar e_2)\) is cylindrical (follows from (9)); b) the envelope of the family of planes \(x^1=0\) is a developable surface (follows from

(16); c) the planes \(x^2=0\) form a one-parameter family whose characteristic is the line

\[ x^1\Gamma_1+\Gamma_{21}=0,\qquad x^2=0. \tag{22} \]

The last assertion follows from the fact that the forms \(\omega,\omega_3^2,\omega^2\), which determine the position of the plane \(x_2=0\) in space, become functions of one parameter.

1. The class \(R^2(1+\Lambda_{31}^2)-\Lambda_{01}^2=0\) exists, is determined with an arbitrary function of 2 arguments, and is characterized by the coincidence of the foci \(F_3\) and \(F_4\). In view of (8) and (14) it follows that the radius of a circle of such a congruence is equal to twice the length of the projection of the segment \(AH\) of the axis \(e_1\) onto the tangent to the spherical image of the congruence \((\bar A e_3)\).

2. The class \(\Lambda_{01}=R,\ \Lambda_{31}=0\) is characterized by the fact that the characteristic (15) touches the circle of the congruence at the point

\[ \overline{M}_1=A+Re_1. \tag{23} \]

From system (2) two finite relations follow: \(\Gamma_0=0\) and \(\Gamma_{10}=\dfrac{1}{R}\). Consequently, the point (23) is the fourth focus of the congruence \(\{1.2\}\). The congruence of axes \((\bar A e_3)\) is bicylindrical. This class exists and is determined with an arbitrary 3 functions of 1 argument.

1. The class \(\Gamma_{10}=\dfrac{1}{R},\Lambda_{01}=R\) is characterized by the fact that the foci \(F_1,F_2,F_4\) coincide. From the fourth equation of system (2) it follows that: \(\Gamma_0\Lambda_{31}=0\). If \(\Gamma_0=0\), then the point (23) is the fourth focus. If \(\Lambda_{31}=0\), then the class under consideration exists and is determined with an arbitrary 4 functions of 1 argument. The tangent to the focal curve described by the triple focus of this congruence lies in the plane of the circle of the congruence.

Let us note that if the foci \(F_1,F_2\), and \(F_3\) coincide, then the point (23) is always the fourth focus of the congruence \(\{1.2\}\).

References

1. Gambier B. Congruence de cercles: points focaux et curfaces focales. C. R. Acad. Sci., Paris, 195, 928—390, 1932.

2. Vincensini, P. Points focaux des cercles d’une congruence. C. R. Acad. Sci., Paris, 195, 1359—1361, 1932.

3. M. R. Vaintrub. On congruences of circles in Euclidean space. Proceedings of Tomsk University, Ser. Mechanics and Mathematics, vol. 191, 1432—149, 1967.

4. M. R. Vaintrub. On canal surfaces of complexes of circles in three-dimensional Euclidean space (the present collection).

5. M. R. Vaintrub. Differential geometry of \(n\)-parametric manifolds of circles in Euclidean space. Materials of the Third Interuniversity Conference on Problems of Geometry, Kazan, 27, 1967.

6. S. V. Bakhvalov. Remarks on the method of the moving trihedron. Mat. sb., 7 (49):2, 321—326, 1940.