Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

350

ANALYTIC GEOMETRY

Consequently, a rotation of the (x, y)-plane about 0 through
- 45~:
x +y -- x + y
(4) x- =, yl1 = Y
(/2 V2
brings OL to lie along the positive axis of x. And a rotation
of the (x', y')-plane about 0 through + 45~:
X' - y' x' + y'
(5) x1- = -8, Y1 ' —,
does the same for OL'.
The next step is to find the transformation carrying (xl,-y1)
directly into (x1/, y1). This we do by eliminating x, y and
/x, y' from (1), (4), (5). Thus,
= -(x + 3y + 3x + y)= 4 x + 4xl,
1' = (x +3y-3x-y)_2 -x+=2y,
or
(6) X11 = 4 X1, Y1l = 2 yl.
Finally, we solve (5) for x' and y':
<7' / _ Ad ye=- +.Y1 -1
(7) XI - ___I _l - XI
The transformation (1) is now seen to be the product of the
transformations (4), (6), and (7); (4)
xI,) carries (x, y) into (x1, Yi) by a rotation
about 0 through - 45~; (6) carries
(x, yi) into (xi', y1') by two one-di-(x^ (x-1y) mensional strains; (7) carries (x/, yl')
(x'y') into (x', y') by another rotation about
-o' X 0 through - 45~.
FIG. 13 Simplifications in Technique. Instead of seeking the ellipse in the
(x', y')-plane into which the circle (2) is carried, we might
equally well ask for the ellipse in the (x, y)-plane which is