On Canal Surfaces of Complexes of Circles in Three-Dimensional Euclidean Space
Abstract Generated abstract
This paper studies three-parameter complexes of circles in three-dimensional Euclidean space whose circle planes form an h-parameter family, with h equal to 1, 2, or 3. Using Cartan’s moving frame method and Pfaffian forms, it describes the differential equations of such complexes in a canonical frame and analyzes when a one-parameter submanifold through a circle is a canal surface. The argument derives the condition that neighboring positions of a circle lie on a common sphere and reduces the existence question to a characteristic equation of degree h. It concludes that a complex of type h,3 possesses h generally distinct families of canal surfaces.
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ON CANAL SURFACES OF COMPLEXES OF CIRCLES IN THREE-DIMENSIONAL EUCLIDEAN SPACE
M. R. VAINTRUB
(Presented by the Department of Higher Mathematics)
- In [1] it is proved that complexes of circles in three-dimensional conformal space possess three, in the general case distinct, families of canal surfaces.
Below we consider the question of the existence of families of canal surfaces passing through the circles of complexes in three-dimensional Euclidean space.
- Consider, in three-dimensional Euclidean space \(E_3\), a three-parameter manifold of circles whose planes form an \(h\)-parameter, \(h=1,2,3\), family (a complex \((h,3)\)). To each circle of this complex we attach a local orthonormal frame \((\bar A,\bar e_1,\bar e_2,\bar e_3)\), placing the origin \(\bar A\) and the vectors \(\bar e_1,\bar e_2\) of it in the plane of the circle. The infinitesimal displacement of the frame is determined by the equations
\[ d\bar A=\omega^a \bar e_a,\quad d\bar e_a=\omega_a^b \bar e_b,\quad a,b,c=1,2,3. \tag{1} \]
where
\[ \omega_a^b=-\omega_b^a, \tag{2} \]
and the Pfaffian forms \(\omega^a,\ \omega_a^b\) satisfy the Cartan structure equations [2]:
\[ D\omega^a=[\omega^c\omega_c^a],\quad D\omega_a^b=[\omega_a^c\omega_c^b]. \tag{3} \]
Place the origin \(\bar A\) of the frame at the center of the current circle of the complex. Then the equations of this circle with respect to the frame so chosen will be written in the form:
\[ (x^1)^2+(x^2)^2+2a_0=0,\qquad x^3=0. \tag{4} \]
The system of equations of the circle adjacent to (4) with respect to the frame \((\bar A,\bar e_1,\bar e_2,\bar e_3)\) is written in the form:
\[ (x^1+dx^1)^2+(x^2+dx^2)^2+2(a_0+da_0)=0, \tag{5} \]
\[ x^3+dx^3=0. \]
Requiring that equations (4) and (5) determine one and the same circle, and determining here the differentials \(dx^a\) from the stationarity conditions [3]
\[ dx^a=-x^b\omega_b^a-\omega^a \tag{6} \]
of a point of Euclidean space, we find that the six Pfaffian forms \(\omega^i,\ \omega_j^i,\ \omega^3,\ \Theta^0\equiv da_0,\ i,j=1,2,\) are the primary forms of the circ—
ness in \(E_3\). Three Pfaff forms \(\omega^i,\ \omega^3\) determine the position of the plane of the circle in space.
- In what follows we shall assume that \(\omega_1^i \equiv \Theta^i,\ -R\Theta^\circ \equiv \Theta^3,\ \omega_3^i \equiv \omega^i\); the indices \(\alpha,\ \beta\) take \(h\) definite values from the numbers \(1,2,3\); the indices \(\alpha',\ \beta'\) take all the remaining values from this set of numbers.
Since a complex of circles \(\{h,3\}\) is considered, among the six primary Pfaff forms \(\omega^\alpha,\ \Theta^\alpha\) there are \(h\) linearly independent forms determining the position of the plane of a circle of the complex. Let these be the forms \(\omega^\alpha\). As the remaining \((3-h)\) linearly independent forms of the complex we take the forms \(\Theta^{\alpha'}\). Then the system of differential equations of the complex \(\{h,3\}\) is written in the form:
\[ \omega^{\alpha'}=A_{\beta}^{\alpha'}\omega^\beta,\qquad \Theta^\alpha=B_{\beta}^{\alpha}\omega^\beta+C_{\beta'}^{\alpha}\Theta^{\beta'}. \tag{7} \]
- The following theorem holds.
Theorem. A complex of circles \(\{h,3\}\) in three-dimensional Euclidean space \(E^3\) possesses \(h\), in the general case distinct, families of canal surfaces.
Proof. Let the complex \(\{h,3\}\) be referred to its canonical frame [4]. By imposing two relations between the independent forms \(\omega^\alpha,\ \Theta^{\alpha'}\), we single out a certain one-parameter submanifold passing through a circle of the complex. If this submanifold is a family of canal surfaces, then two infinitely close points \(M\) and \(M'\) of the circle lie (under displacement of the circle in the canal surface) on one and the same sphere with center at some point
\[ \overline{F}=\overline{A}+\rho\overline{e}_3, \tag{8} \]
that is,
\[ (\overline{M}-\overline{F},\, d\overline{M})=0. \tag{9} \]
We shall determine the position of an arbitrary point \(\overline{M}\) of the circle of the complex by means of the angle \(\varphi\) which the radius vector \(\overline{AM}\) makes with the axis \(\overline{e}_1\):
\[ \overline{M}=\overline{A}+R(\overline{e}_1\cos\varphi+\overline{e}_2\sin\varphi). \tag{10} \]
Taking into account that
\[ \begin{aligned} d\overline{M}={}&\overline{e}_1\{\Theta^1+dR\cos\varphi-R\sin\varphi\,d\varphi-R\omega_2^1\sin\varphi\}\\ &+\overline{e}_2\{\Theta^2+dR\sin\varphi+R\cos\varphi\,d\varphi+R\omega_2^1\cos\varphi\}\\ &+\overline{e}_3\{\omega^3-R\omega^1\cos\varphi-R\omega^2\sin\varphi\}, \end{aligned} \tag{11} \]
condition (9) can be written in the form:
\[ R(\Theta^2-\rho\omega^2)\sin\varphi +R(\Theta^1-\rho\omega^1)\cos\varphi +\rho\omega^3-\Theta^3=0. \tag{12} \]
Equations (12) must be satisfied identically with respect to \(\varphi\). Consequently, the coefficients of \(\sin\varphi\), \(\cos\varphi\), and the free term must be equal to zero:
\[ \Theta^a-\rho\omega^a=0. \tag{13} \]
Substituting the values of \(\omega^{\alpha'}\) and \(\Theta^\alpha\) from (7), we shall have
\[ \Theta^{\alpha'}-\rho A_{\beta}^{\alpha'}\omega^\beta=0, \]
\[ C_{\beta'}^{\alpha}\Theta^{\beta'}+B_{\beta}^{\alpha}\omega^\beta-\rho\omega^\alpha=0. \tag{14} \]
Eliminating from equations (14) the relation of the linearly independent Pfaff forms, we obtain the equation
\[ \left| \begin{array}{cccc} B_1^1-\rho & \cdot & \cdot & B_h^1 \\ \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot \\ B_1^h & \cdot & \cdot & B_h^h-\rho \end{array} \right|=0 \tag{15} \]
of degree \(h\) with respect to \(\rho\).
The theorem is proved.
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