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[Reprinted from The American Mataematical Monthly, Vol. XXXV,' Apri!,M9l? |'\ J *? '
NOTE ON SOME APPLICATIONS OF A GEOMETRICAL TRANS-
FORMATION TO CERTAIN SYSTEMS OF SPHERES.^
By dr. henry W. STAGER, Fresno, California.
Ill a paper in Volume VI of the Proceedings of the Edinburgh Mathematical
Society, Professor Allardice considers the transformation in piano:
Let I be a fixed straight line, C any plane curve, t a tangent to C meeting /
in X and making an angle 6 with /; then C, the transformed curve of C, is the
envelope of a straight line t' through X making an angle with I, where 6 and
are connected by the relation
tan |0 = k tan ^6.
The length of the tangent from any point in I to C remains unaltered by the trans-
formation. A similar method is applicable in space and the following paper
gives some of the results in its applications to certain systems of spheres. The
method of transformation is as follows:
Let a be a fixed plane, S any surface, and ^ a plane tangent to S intersecting
a in the line i and making with a an angle 6. If (3' be a plane through i making
with a an angle 0, determined by the relation
(1) tan ^(f) = k tan hd,
then the envelope of j8' will be defined as the transformed surface of S, or, more
simply, the "transform of S." ^..
From the given relation, tan ^0 — k tan |0, and the identity,
_ 2 tan (/)/2
we find that
2k tan d
(2) tan (j)
± (1 - F) Vl + tan2 d + 1 + F '
It is evident from the nature of this relation that for every value of k there are
two values of tan 0, according as we consider the angle afi as d or (6 -{- ir).
For simplicity, we will use Cartesian coordinates to find the equation of the
I transform of a given sphere from its equation, and we will take the .r^-plane as
the plane of transformation, a. Let 6 be the angle formed by the .r?/-plane and
the plane ux + vy + ivz — 1 = 0, and be the angle formed by the a-z/-plane
and the plane ux + vy + 2^12 — 1 = 0. (Since the intercepts of both planes on
the .T-axis and on the y-ax\s are equal, respectively l/u and l/v, they intersect
the .T2/-plane in the same line, as required by the transformation.)
' Presented to the San Francisco Section of the American Mathematical Society, April
12, 1913.
' 1*55 •' APPUCATIONS OF A GEOMETRICAL TRANSFORMATION.
From the formula for the angle between two planes,
_ (J?(7^ - B'Cf + { CA' - CAY + {AB' - A'Bf
tan- - ^^j, _^ ^^, _^ ^^,^2
we have
*?/ ■ I ■ ill" 97 I I -i***
tan- = ^ — and tan^ (b = , — .
Substituting these values in equation (2) and solving for w, we obtain
(1 + /r)wi ± (1 - F) a/m2 _^ ^^+^2
(3) ^. = 2^ ,
or,
,^^ , 2(1 + A;4)Wi2 + (1 - F)2(W2 + ^2) _i. 2^,(1 _ Jt4) Vw2 + ^2 _|. ^^2
(4) ^2 = _^^ ,
Now, let the center of the sphere lie in the 2-axis and its equation will be
(5) x^+y''+z'' + 2nz+ d= 0.
Expressing the condition that the plane ux -\- vy -\- wz — 1 = he tangent
to this sphere, we have
(6) (d - 71^1^ + (d - n^y + dw'^ + 2nw +1 = 0,
which is the tangential equation of the sphere.
Substituting the values of w and w- found in (3) and (4) and then simplifying
and factoring the result, we have the equation of the transform of the given sphere
[{(F + lyd - 2n^{¥ + 1) + 2n{h^ - 1) -in^ - d]{iv' + v") + Ulc^w^-
+ 4F + 4/j{n(P + 1) + (F- - 1) 4n^ - d]wA X [{(/r + l)-(i
(7) ,
- 2n\k' + 1) - 2/t(/.-' - 1) V/i2 - d^iv" + r) + Ukhci'^ + 4^^
+ 4/j{n(F + 1) _ (/,2 _ 1) V^i^ _ d]ic?[ = 0.
This equation breaks up into two equations which are the tangential equations
of two spheres, and may be written in the form
(8) yli(w2 ^ ^2) ^ Q^^,2 _^ 2NiWi + D = 0;
(9) ^2(^2 + v"") + Cwi" + 2A^2Wi + D= 0.
The Cartesian equations of these forms are
(10) {CD - Ni')x^ + (CD - iVi2)7/2 + AiDz"" + .4iC + 2yli.Yi2 = 0;
(11) {CD - N2V + {CD - N.})y'' + AnDz- + J-.C + 2/I2.V02 = 0,
where, from the actual values of the coefficients involved.
TO CERTAIN SYSTEMS OF SPHERES. 156
(CD - Ni") = AiD, and {CD - Ni") = A^D.
Simplifying and substituting, we have finally
(12) 4F(.t2 + 2/2 + S-) + ik{n(F" + 1) + (/v- - 1) Vw^ - d}z + 4<fF = 0;
(13) 4F(a:2 + r + S") + ^k{n{k'' + 1) - (F - 1) ^^-^}z + 4dA;2 = Q:
the equations of two spheres whose centers lie in the z-axis. Hence, in general,
a sphere is transformed into two spheres whose centers lie in a line perpendicular
to the plane of transformation and passing through the center of the given sphere.
Now, let p represent the radius of the given sphere and 5 the distance of its
center from the plane of transformation; also, let r' and r" and d' and d" repre-
sent respectively the radii and distances from the plane of transformation of the
centers of the transformed spheres. The equations of the latter may then be
written
(14) x' + 2/2 _^ .2 _^ [_ 5(^. + 1//,) ^ (/, _ i|^.)^], + d=0;
(15) x^ + r + 2' + [- 8{k + l/k) - {k - \lk)p\z + (i = 0:
whence the resulting relations:
(^' 4- / = (5 - p)k; d" + t" = (5 + p)k;
(16)
d' - r' = (5 + p)lk; d" - r" = (5 - p)lk.
We will next consider for what values of k the transformed spheres will
degenerate into points. It is evident that we must consider each case separately.
In the first case we have
r' =
-^{(5+p) -kH8- p)}\=0,
whence
(17) 'F- = (5 + p)/(5 - p).
Here we have two real values for k, provided the given sphere does not cut the
plane of transformation; and a zero or infinite value for k if the given sphere is
tangent to that plane. The corresponding values of the distance are
d' = T V52 - p
2
or in terms of the original coefficients,
(18) d' = q= V^.
In the second case, by similar methods, we find that
(19) P = (5 - p)/(5 + p),
and
(20) d" = =F V52 - p\ or, d" = T V^.
157 APPLICATIONS OF A GEOMETRICAL TRANSFORMATION,
It follows that, in general, a sphere may be transformed into a point for any
one of four values of k, the points being the same in pairs, and all four being equi-
distant from the plane of transformation,, ope pair on each side. It is to be
especially noted that while each of the two transformed spheres may degenerate
into points, both spheres cannot become points at the same time, unless either
p = 0, or 5 = 0; i. e., when the given sphere is itself a point and this point may
be transformed into its image with respect to the plane of transformation, or
when the center of the given sphere lies in the plane of transformation and the
points are imaginary. We may also note that a point can be transformed into
only one sphere.
We have already seen, equations (12) and (13), that for the same value of h
a given sphere may be transformed into two distinct spheres. These two spheres
evidently result from the two values of 0, according as we consider the angle
a/3 as d or {B-\- -k). In case the angle is B we will say that the transformed
sphere is traced out by a direct movement, and where the angle is {Q + tt) we
will say that it is traced out by an indirect, or inverse, movement. It is evident
that if we transform two spheres, hoth directly, or both inversely, the direct
common tangent planes to the two spheres become common tangent planes to
the transformed spheres, while if one sphere is transformed directly and the
other inversely, the transverse common tangent planes become common tangent
planes to the transformed spheres.
Forming the equation of the radical plane of the given sphere and either of
its transforms, we have
z = 0,
which is also the plane of transformation. It follows that the plane of transforma-
tion is the radical plane of the given sphere and its transforms, and that the
length of a tangent line from any point in the plane of transformation to the
sphere is unaltered by the transformation. Hence the distance between the
points of contact of common tangent planes to two spheres remains unaltered.
This is the important property of the transformation. By transforming several
spheres into points at the same time, relations and properties of spheres may be
obtained directly from known relations between points.
For two or more spheres to be transformed simultaneously into points, it is
evidently sufficient that the values of Ic- be equal. Considering the case of two
spheres with radii p' and p" and with the distances of their centers from the
plane of transformation, 5' and 5", respectiveh', we have
p = (5' - p')/(5' + p') and P = (d" - p")l{h" + p");
whence
(5' - p')/(5' + p') = (5" - p")l{b" + p"), or 575" = p'lp".
This is the condition that the plane of transformation pass through the direct
center of similitude of the two spheres. Using the reciprocal values for k"^ we
obtain the same result. Finally, let
TO CERTAIN SYSTEMS OP SPHERES. 158
F = (5' - p')/(5' + p') and P = (5" + p")/(5" - p")\
whence
575" = - p'Ip":
the condition that the plane of transformation pass througli the inverse center
of simihtude of the two spheres. We conclude that two spheres may be trans-
formed simultaneously into points if the plane of transformation contain a
center of similitude of the two spheres. Likewise, if the plane of transformation
contain any one of the four axes of similitude of three spheres, these spheres will
all be transformed into points by the same transformation. Further, if the plane
of transformation be any one of the eight planes of similitude of four spheres/
the four spheres may be transformed into points simultaneously; and, finally,
any number of spheres which have a common plane of similitude may be so
transformed. Two spheres on the same side of the plane of transformation will
be transformed, either both directly, or both inversely; while of two on opposite
sides, one will be transformed directly, the other inversely.
By giving k every possible value a given sphere may be transformed into an
infinite series of spheres which, by virtue of the properties of the plane of trans-
formation as the radical plane of a sphere and its transforms, is cut orthogonally
by a system of spheres. Then, since the given sphere is one of the series, if we
transform the whole system for the same value of h, it will transform into itself.
It may be noted at this point that the results here obtained by analytic
methods may readily be obtained by synthetic methods, and in the sequel they
will be interpreted in accordance with the particular problem under consideration.
In all problems which follow it w^ill be necessary to associate only those common
tangent planes and distances between points of contact of common tangent
planes to two spheres which determine the centers of similitude in the planes
of similitude used as planes of transformation. Thus we will associate with the
plane of similitude passing through the six direct centers of similitude of four
spheres the external common tangent planes of each pair of spheres, and only
the external common tangent planes of these spheres.
Some Applications of the jNIethod.
1. The condition that four planes passing through a point should be tangent
to the same sphere is given by the equation
tan \ (j)' tan ^ </>" = tan ^ d' tan ^ 6",
where 0', cf)", 6', 6" are the angles formed by the given planes and the plane
determined by the lines of intersection of the two pairs of opposite planes.
1 The twelve centers of similitude of four spheres lie in sets of six in a plane: the six direct
centers lie in a plane; the three direct centers of any three spheres lie in a plane with the three
inverse centers not paired with them; and any two direct centers, using each sphere only once,
lie in a plane with the four inverse centers not paired with them. Thus there are eight planes
of similitude, one containing all the direct centers; four containing three direct and three inverse
centers; and three containing two direct and four inverse centers.
159
APPLICATIONS OF A GEOMETRICAL TRANSFORMATION.
Transform the sphere into a point with the external diagonal plane as the
plane of transformation, and let the angles of the transformed planes with this
plane of transformation be and 6, corresponding respectively to the angles
4)' , (f)" and 6', 6". We then have the following relations:
tan I = A- tan \ <^'; tan \ d = k tan h 6';
tan I (0 + tt) = /.• tan | (j>"; tan ^ (d + t) = k tan | d".
Eliminating cj), 6, h frdin these equations, we have the given condition.
2. If five spheres have a common plane of similitude they may be transformed
into five points by the same transformation. Therefore the distances between
the points of contact of the common tangent planes of each pair of spheres satisfy
the relation connecting the ten straight lines joining five points in space.
Let di, do, dz, • • ■ f/io be the distances. Then the relation is given by the
determinant^
rfr
d-z'
^3^
d:~
di'
d/
d,'
d^'
d,'
d,'
de
d<f
dz'
d,'
ds'
d,o'
d^
d,'
d,'
d,^
1
1
1
1
1
0.
3. If, in addition to the conditions of No. 2, the five spheres also touch a
sixth sphere, the ten distances will satisfy the additional relation connecting five
points lying on the surface of a sphere.
Transforming the five spheres into points as above, they must lie on the
surface of the transform of the sixth sphere and will therefore satisfy the addi-
tional relation given by the determinant^
d^
d-f
d^
d^
d,'
d^
d,'
d,'
di"
d,'
di
d,-^
d,'
d,'
^8^
did
d,'
d,'
d,'
c/io^
= 0.
4. The radius 11 of a given sphere S tangent to four given spheres may be
found in terms of R, the distance 5 of the center of iS from the plane containing
the six direct centers of similitude of the four spheres, the distances between
the j)oints of contact of common tangent planes to pairs of spheres, and the
' Cayluy, "CuUcclcd Mathematical Papers," Volume I, p. 1 et seq.
::•
:\'
TO CERTAIN SYSTEMS OF SPHERES. 160
radius and distance from its center to the plane of similitude of any one of the
given spheres.
If we transform the four spheres into points by the same transformation,
these four points will lie on the transform of S. Hence
^ _ 1 / (2aa')[n(- aa' + bh ' + cc')]
^' ~ 2 \l.Wa'\W + c2 - a2 + h''' + c'^ - a'^)] - {cC-W - i:a'b"c") '
where a, a', etc., are the distances between the points of contact of the common
tangent planes of two pairs of spheres with no sphere in common, and where 2
and TT represent cyclo-symmetrical functions only. But
R' =
where
^ {5(P-1)T(F+1)/^}
21c
F = (^^ _ ri)l(di + n), or P = (di + ri)l(di - n),
from equations (17), and (19).
Hence
R=\- 5(ri/(^i) ± 2{(^dx'-r,')fd}W\.
5. The locus of the center of a sphere S such that the distances between the
points of contact of the common tangent planes to S and four fixed spheres are
equal is a straight line perpendicular to a plane of similitude of the four spheres.
Transforming the fixed spheres into points, the transformed spheres S' will
form a system of spheres Avith a common center; namely, the center of the
sphere determined by the four points. INIoreover, all the spheres S' with real
tangents to the four points lie within this sphere; consequently the distance
between the points of contact cannot exceed the radius of this sphere, and hence
it is a maximum when S' becomes a point; i. e., when the four fixed spheres and S
have a common plane of similitude. The theorem follows directly from the fact
that the system S' is a system of concentric spheres. There are eight cases for con-
sideration corresponding to the eight planes of similitude of the four fixed spheres.
6. The centers of the spheres tangent to four given spheres lie in pairs on
eight straight lines passing through the radical center and perpendicular to one
of the planes of similitude of the four spheres.
This is only a special case of the preceding. It is evident that the radical
plane of any pair of the tangent spheres is the plane of similitude, which is
perpendicular to the line joining the centers of the two spheres. The results
of this theorem can be employed in the construction of the spheres tangent to
four given spheres.
7. If >S is a sphere tangent to four given spheres, which are tangent to one
another, the five spheres have common planes of similitude.
Transforming the four spheres into points, the points will coincide, since each
sphere touches one of the others. But the four points must lie on S', the trans-
161 APPLICATIONS OF A GEOMETKICAL TRANSFORMATION.
form of S, and therefore S' must also reduce to a point simultaneously with the
other four; hence the result.
8. The locus of the centers of all spheres which touch a given sphere and
have a common plane of similitude with it is an ellipsoid of revolution.
Let r be the radius and d be the distance from the common plane of similitude
of the center of the given sphere, and let p and 8 be the radius and distance
respectively of the variable sphere. Transform the spheres into points for the
same value of k and we have
7 (J
d' = ^ (k + lA) -lik- l/k) • 6' = 2 (/. + 1/^) - 2 ^^' ~ ^'^'^'
But, since all the spheres are tangent to the given sphere, the points into which
they transform are coincident, and therefore, d' = 5'. Hence
{k - HMr +p) = (^d+b- Y^^^ {k + m,
or, (r + p)/(5 + c) = e, where e and c are constants.
Therefore, the locus of the centers of the spheres is the locus of a point,
the ratio of whose distances from the center of the given sphere to its distance
from a plane parallel to and at the distance c from the common plane of similitude
is constant; i. e., an ellipsoid of revolution.
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