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

CHAPTER XIX

THE PLANE
1. Surfaces and Equations. Example 1. The equation
x=5
is satisfied by the coordinates of those points and only those
points which lie in the plane parallel to the (y, z)-plane and
5 units in front of it. We say that the equation represents
this plane.
Example 2. Consider the equation
x = y.
The points in the (x, y)-plane whose coordinates satisfy it are
the points of the line L bisecting the angle between the positive x- and y-axes. Since z is unrestricted by the equation,
the points in space whose coordinates satisfy it are the points
which lie directly above or below L, or are on L, i.e. the
points of the vertical plane through L. The equation, then,
represents this plane.
Example 3. The equation
x2 + 2 = 25
represents, in the (z, x)-plane, a circle, C, with its center at the
origin and of radius 5. But the equation does not restrict in
any way the value of y. Consequently, it represents in space
the circular cylinder formed by drawing through each point
of the circle C a line parallel to the axis of y and extending
indefinitely in both directions.
Surfaces. The planes and the cylinder represented by the
three equations considered are known as surfaces; the cylinder
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