* ...:
THE COMPOSITION OF FINITE ROTATIONs
* ABOUT PARALLEL AXES . . . .
º
, t -
By ProFEssoR A1. ExANDER ZIWET

;
;
t
t;
|
l
n
i
ii
!
i
* * *
!; *t
* * *
:::::::::)
** A tº its
:::::: t
* $ &
& t #####:::::
#: !º
{{#if::::::: g
tº a ######:::::
::::: }}}#3:::::::
{; i * :::::::::::
* º
* * !: ! *** ***,\ ,
t $4 ::::, ;
* * * *
#
§
*:
* *
tº º
* * *
tº
* *
$g
$
#:
::::#;
*] § $4 tº
{#
#
$? § {
#:
#::::
g
l
}* tº... . .
* , a . . . .
{
:
it
|
;
y
;
}
i
§
§
i
}
#
§ }:
i
;
:
§
*
§
tº it

NOTE ON THE COMPOSITION OF FINITE
ROTATIONS ABOUT PARALLEL AXES.
Q BY PROFESSOR ALEXANDER ZIWET.
1. It is well known that the succession of two finite rotations
of a rigid plane figure in its plane (or, what amounts to the
same, of a rigid body about parallel axes), say a rotation of
angle 6' about a point O' followed by a rotation 6" about O", is
equivalent to a single rotation of angle 0 = 0 + 6" about a
point O. The center O is found as the intersection of the lines
obtained by turning O'O" about O' through an angle — $6' and
O'O' about O' through -- $6".
As a clockwise rotation of angle b is equivalent to a counter-
clockwise rotation of angle 27 — b, the angles of rotation can
all be made of the same sense, say counterclockwise; and as a
rotation of angle b-i-2kT is equivalent to the rotation of angle b,
the angles can be confined to values between 0 and 27t.
The construction of the point O is therefore always possible
unless 0 = 0 + 6" is E 0 (mod 27). In this limiting case the
center lies at infinity, and the resultant displacement is a trans-
lation whose vector is readily determined.
2. When there are more than two successive rotations, the
resultant of the first and second rotations can be compounded
by the same method with the third rotation, and so on. It is
apparent that the angle of the final resultant rotation is the sum
of the angles of the given rotations, but the construction of the
center becomes rather complicated. It is the object of the
present note to indicate a convenient method for finding this
center, by applying vector methods systematically.
QA
(a 23
, 2 × 2 v.
209 - COMPOSITION OF ROTATIONS. [Feb.,
3. A single rotation, 6 about O' (Fig. 1), carries any point
A of the rigid figure from the initial position A, to the final
FIG. 1. FIG. 2.
position A1. Taking the fixed point O' as origin and putting
A – O' = ro, A – O' = r., we have the vector equation
rl = e”'r,
which means that the operator e” applied to the vector r, turns.
it in the plane through the angle 6'.") This is of course merely
another form of stating the ordinary method of complex numbers.
Two successive rotations, 6 about O' and 6" about O’ (Fig.
2), carry the point A from A, to A, and from A, to A. With
O' — O' = cl, A, - O' = r, we have A, - O' = r + c,
A, - O' = r., + c, whence
r, + c = €”(r) + c,),
or, replacing r, by its value from the preceding equation,
*º-sº e 9//+6/) 6//
r, = e^ ro + e”"c, – c. ^.
In the case of three successive rotations, 6 about O', 6" about
O", 6" about O'", putting O" – O'" = c, A, - O' = r, we have
i0/// -
rs -- C, + C, = €”(r, + c, -- c.),
whence eliminating r, we find
I3 = *or, + **oc, + e”c, –– C3,
where c, - – c – c. = O'" — O'.
Similarly we find for four successive rotations, 6 about O',
..., 6" about O'"
* G. Peano, Gli elementi di calcolo geometrico, Torino, 1891, p. 19.


1908.] COMPOSITION OF ROTATIONS. 210
— ai(9'V+0"--974-97 t(6 iW--97'--9” i(6iv-E67) $6iv
r, = e” Jr. -- eº"+ }c, +e” C,-H C” Cº-FC,
where c + c, -i- c, -- c, = 0. The vectors c, . . , c, are equal to
the sides of the quadrilateral formed by the centers O', . . , O";
these centers are points of the fixed plane, not of the moving
figure. The initial and final radii vectores ro, r, of A are drawn
from O'. If any other point Q of the plane were taken as
origin, it would only be necessary to replace ro, r, by r. — q,
r — q, where rº, r are the radii vectores of A drawn from Q,
and q = O' — Q. -
4. In the expression found for ro all terms after the first are
independent of the particular point A of the figure. Their
sum represents a vector s, which can be written
S = cº"(e."(ºc, + C) + C) + c,.
This form indicates the most convenient way of constructing
the vector s, (Fig. 3): turn O' — O' = c, about O' through 6"
and add c, ; then turn the sum so obtained about O'" through
6" and add c, ; finally turn the vector so obtained about O"
through 6" and add c, ; this gives the vector s, = 0, - O'
which evidently represents the displacement of that point of the
rigid figure which originally coincided with O'.
Denoting the angle of the resultant rotation by 6, so that
6 = 0 + 6" + 6" + 6"(mod 27t) and introducing the vector s,
we have the simple result
— 236
r, = €”ro + S,.
5. It is obvious that in the case of m, successive rotations we
have similarly
I, = e'r, —H. Sº,
where 6 = 0 + 6" + . . . -- 0° and
S =
ºl.
gº) { & E & [eº"(eiº"c, + C.) + cal + . . . -- Cº-1} + C.
Thus, the final radius vector of any point of the rigid figure is
found by turning its initial radius vector about O' through an
angle 0 = 0 + 6" + . . . -- 0° and adding to it the vector s.
In other words, the resultant displacement is resolved into the
rotation of angle 6 about O' and the translation s, of O'.
The center O of the equivalent single rotation being a fixed
point its radius vector r is found by putting r, = r =r which
211 COMPOSITION OF ROTATIONS. [Feb., 1908.
gives
— e”)r = s : - = __ oi(tr–6)/2
(1 – e’”)r=s, or r 2 sin #9° S.
This means that the point O is the vertex of the isosceles tri-
angle whose base is O'O, and whose angle at O is 0 if 0 < T
and is 2T – 6 if 6 × T. In the former case the sense O'O,O
is positive, in the latter it is negative. If 6 = T, the resultant
displacement is a reversal (Umwendung) about the midpoint of
O'O. If 6 = 0 (mod 27t) the resultant displacement reduces
to the translation s. *
If, in particular, the angles of rotation 0", 6", . . , 6") are
equal, respectively, to the exterior angles of the polygon of
centers at O'", O", ..., O", the vector s, has, as appears at once
from its construction (Fig. 3), the direction and sense of c, and
a length equal to the perimeter of the polygon O O" . . . O".
If, in addition, 6' is equal to the exterior angle at O' and the
polygon is convex, the resultant rotation is zero, and the resul-
tant displacement reduces to the translation S.
ANN ARBOR, December, 1907.
;
:
".
:
©
:
:
:
:
.*
:

º ||||||||||||||
5 O51 16 1902
Sample of
GAYLORD BROS.”
PAMPHLET BINDER
Made Of PHOTO= MOUNT
The size of this binder is
4. X ? / inches.
2- inch.
Space at back
Color
('Olor ('loth . "
:
|
i
f
;
|
-
/
DIRECTIONS.
Moisten the gum ned surfaces, insert magazine,
close the binder, and press firmly for a no ment to
make adhesion secure.
remove the cover, moisten the gum mod surfaces
of the binding strip and insert the contents.
Then past tº the magazine covers previously de-
tached, to the ungu in med surfaces of the binding
strip. Some librarians prefer to paste the maga-
zine covers to the outside covers of the bin cler.
This will allow the title of the magazine to be
seen at a glance.
-
-
-º
•G
--
º
-t
º
:
GAYLORD BROS.,
Syracuse, N. Y.
|
i
i
§
;*g
#
;
§:
#:
§: f
§
ººº
::
ſ
i;
!';
; :
H
11,
ºf
* } !!!"
º tº

.
:
§
º-
º
º
t º
º º
§
sº
1 s :
&
ſ
º
tº
Fº
º
*... f.
§º
º
º
*º
º
sº
* .
º
#
#
~