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

490 ANALYTIC GEOMETRY
8. Parametric Representation of a Curve. The Straight Line.
Given a directed straight line passing through
the point PO: (xo, o, Zo) and having the direction
\tP cosines cos a, cos s, cos y. Let P: (x, y, z) be an
arbitrary point of the line other than P0 and let
r\ 0 r be the algebraic distance from PO to P, positive
if the direction from P0 to P is that of the line
/ o and negative if this direction is opposite to that
of the line.
FIG. 7 The projections of PoP, each divided by r, are
equal respectively to the direction cosines of the
line, by Ch. XVIII, ~ 1, Th. 2. Thus
x - x cos a, Y -Y cos/O, -- Z=cos y.
r r r
These equations can be put into the form
(1) x=x+rcos a, = yo + r cos/, z = o - +rcosy.
Equations (1) give the coordinates (x, y, z) of the point P on
the given line at the arbitrary distance r from PO. If r is
allowed to vary through all values, positive, zero, and negative, P takes on all positions on the line, and always its
coordinates are given by equations (1). These equations, then,
represent the line. Since they express the coordinates of the
point P: (x, y, z) tracing the line in terms of the auxiliary
variable, or parameter; r, we call them a parametric representation of the line.
If the line is determined by the point Po: (X, yo, zo) and its
direction components 1, m, n, we have, according to (2), ~ 3,
the following parametric representation
(2) x = Xo + pl, y =Yo + pm, z = zo + pn.
The parameter p is not, in general, equal to the distance from
P0 to P: (x, y, z), but is merely proportional to this distance.+
* Cf. footnote, p. 479.