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

CHAPTER XI

TRANSFORMATION OF COORDINATES
1. Parallel Axes. It sometimes happens that it is desirable
to shift from a given system of Cartesian axes to a new
y, / system of axes having the same direcp tions as the old, but with a different
io,_ I X origin.
Let P be any point of the plane;
~~i ' I let the coordinates of P, referred to
0..== the old axes, be (x, y), and let the
FIG. 1 coordinates be (x', y') with respect to
the new axes. Let the new origin,
0', have the coordinates (xo, yo) in the old system. Then it is
easy to show that
( ) Xy-Y+Yo;
Y = Y' + Y0;
or
I Xl = X - x,,
(2)
( y' = y - Yo.
For, consider the line-segment OP and the broken line, O0'P,
which has the same extremities. Then
Proj. OP= Proj. 00 + Proj. O'P,
no matter what direction is chosen, along which the projection
is to take place (Introduction, ~ 3).
If the direction is taken, first, as the positive axis of x and
then, again, as the positive axis of y, we obtain, by applying
the definition of coordinates (Ch. I, ~ 1), the equations (1).
Example 1. Find the equation of the curve
(3) y2+2y-4x+9 =0,
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