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

CHAPTER XV

TRANSFORMATIONS OF THE PLANE. STRAIN
1. Translations. Definition. By a translation of a plane
region S is meant a displacement of S whereby each point of
S is carried in a given (fixed) direction by one and the same given distance. Thus, when a window is
raised, a pane of glass in the window
FIG. 1 - experiences a translation.
It is not important what particular
region S is considered. Indeed, it is usually desirable to
consider the whole unbounded plane as S. The essential
thing is the above law which connects the initial position of
an arbitrary point of S with its final position.
Analytic Representation. Let P: (x, y) be an arbitrary
point of the plane, and let P': (x', y') be the point into which
P is carried by the translation. Let a
and b be respectively the projections of v P.(,y)
the directed line-segment PP' on the a
axes of x and y. Then P:(x,y)
( x' x- X a,
(1) l y, y+. FIG. 2
I y =y + b.
These formulas are the same as those which represent a
transformation of coordinates, the new axes being parallel to
the old and having the same respective directions. But the
interpretation of the formulas is wholly different. There, the
point P remained unchanged. It had new coordinates assigned to it by referring it to a new set of axes. Here, the
330