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

336

ANALYTIC GEOMETRY

2. Show that the effect of performing transformation (1)
and then transformation (2) is to leave the plane unchanged.
4. Reflections in the Axes. Let the plane be reflected in
the axis of x. In other words, let it be rotated through 180~
about the axis of x. Let P: (x, y) be an
arbitrary point, and let P':(x', y') be the
point into which P is carried. Then,
obviously,,0 ^^y /^ \ X - x
o, x (1) y=__y.
Similarly, a reflection in the axis of y is
represented by the formulas:
FIG. 7 (2) - x
The condition that a curve be symmetric in one of the axes
(cf. Ch. V, ~ 2) is obtained at once from these transformations.
Thus the curve C will be symmetric in the axis of x if the
curve C', into which C is carried by (1), is the same curve as C;
and the test for this is, that the equation of C be essentially
unchanged when the transformation (1) is performed on it.
For example, if C is the curve
y4 + X2 2 y2 + x3,
its equation is unchanged by (1), and hence C is symmetric in
the axis of x. But it is changed by (2), and C is, therefore,
not symmetric in the axis of y.
Isogonal Transformations. A transformation is said to be
equiangular or isogonal if the angle which any two intersecting
curves, C1 and C0, make with each other is the same as the
angle which the transformed curves, C'1 and C'2, make with
each other.
All of the transformations considered thus far are evidently
isogonal. We turn now to a transformation which is not.