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

352

ANALYTIC GEOMETRY

If the line one unit above the x-axis slides to the right
through the distance k, then the line y units above the x-axis,y will evidently slide to the,o-y~ - P right through the distance Icy,
while a line which is, say, 2........ o units below the x-axis will
_o_) -. 2/ slide a distance 2 c to the
left. In other words, the
/ ___ algebraic distance through
r~- e7 /Iy /k which each line slides is
o / x equal to k times the algebraic
distance of the line from the
fiZ7_ a2x-axis.
FIG. 14 This is true, also, of the
motion of each point of the
rectangle, since the lines slide as units. If, then, the whole
plane is twisted according to this law, an arbitrary point (x, y)
will be carried along a parallel to the axis of x through the
algebraic distance ky. Hence
(1) { 'x = x+ Icy,
are the equations of the transformation.
Thus far we have assumed that k is positive. It may
equally well be negative. Then points above the axis of x are
shifted to the left, and points below it to the right.
If the sliding were along parallels to the axis of y, the transformation would be
X, -I X,
(2) y'=i+y,
where I is any constant, not zero.
These transformations are known as simple shears, and the
motions which they generate are called shearing motions.
Example 1. Subject the curve

(3)

y = x3+2x