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

TRANSFORMATIONS OF THE PLANE

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carried into the circle
(8) 12 + 12 = p2
The roles of the two planes are merely reversed. Adopting
this procedure obviates the necessity of solving (1) for the
values of x, y to be substituted in (2). For now (2) is replaced
by (8) and x' and y' are given by (1). Thus one step in the
process is eliminated. The others remain unchanged.
Another simplification arises in factoring a transformation
of the form
(9) (-XI- x+3y+4,
(9~) ly'=-3x- y-2.
Instead of proceeding as in ~ 6, we set
I = X + 4, ( X= x+3y,
-+4' and -
y =y-2, a y=-3x — y.
Thus (9) is the product of the transformation (1) and the
translation which carries the origin into the point (4, - 2).
EXERCISES
Factor the following transformations, using the simplified
method.
1. x'=x+3y, y'= -3x-y.
2. x= 3x -2y, y'= - 2x - 3y.
3. x' =Sx -+1y, y =10x+2y.
4. x'=6x+18y-2, y'= 17x+y+3.
5. x'=lllx+4y+1, y' =52x+ 78y-7.
8. Simple Shears. In a rectangle with its center at the
origin and with its sides parallel to the coordinate axes, draw
the lines parallel to the axis of x. Twist this rectangle as
shown in the figure, leaving the line along the axis of x fixed
and sliding each parallel line along itself into a new position.
Thinking of the lines as representing the edges of a pack of
cards or of a block of paper is an aid in visualizing the motion.