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

TRANSFORMATIONS OF THE PLANE

337

EXERCISE
Show that the equations of the inverse of the reflection (1)
[or (2)] are precisely of the same form as the equations of the
reflection. A transformation for which this is true is said to
be involutory.
5. Simple Elongations and Compressions. Let the plane be
stretched directly away from the axis of x, so that each point
is carried, along a parallel to the
axis of y, to twice its original dis-
tance from the axis of x (Fig. 8).
Evidently, the analytic condition 0
is that. 1 — x
x' = x, y' = 2y. \ /
More generally, if a point
P: (x, y) is to be carried to I timesG. 8
its original distance from the axis
of x, where 1 may have any positive constant value, not unity,
the transformation will be given by the formulas:
(1) 1-.
y ly.
When I is greater than unity, these formulas represent an
elongation; when 1 is less than unity, they represent a compression.
If the elongation is away from the axis of y or the compression is toward it, then
(2 x'= kx,
(2) I,
where k is greater than unity in the first case and less than
unity, but positive, in the second.
These transformations were discussed geometrically in Ch.
XIV, ~ 7. There we called them one-dimensional, or simple,
elongations and compressions; or, jointly, one-dimensional
strains.