Graphics Reference
In-Depth Information
It is obvious from Eqs. (3.3) and (3.4) that when
=
v
, x
=
x
2
and y
=
y
2
, and represents a
point (x,y)
xy
Let us follow this parametric description of a straight line with the derivation of the Cartesian
form of the line equation.
v
units along the line from P
1
·
3.3 The Cartesian form of the line equation
We already know that the Cartesian form of the line equation is ax
c, but what is not
obvious is the geometric significance of a, b, and c. Fortunately, vector analysis provides an
insight into these terms, which is the objective of this section.
We begin by setting out to discover the coordinates of any point P xy on a straight line.
The strategy used may not immediately appear intuitive, but try following it, and hopefully the
approach will become obvious.
+
by
=
Step 1
Construct a line away from the origin O containing the point P xy and its associated position
vector
p
, as shown in Fig. 3.4.
P
p
O
Figure 3.4.
We now need to construct a second path from O to P.
Step 2
Construct a line from O to a point C on the line such that
−
OC is perpendicular to the line. It
is convenient to make
−
OC a scalar multiple of a unit vector
b
j
. i.e.,
−
OC
n
ˆ
=
a
i
+
=
c
n
, where c
ˆ
is some scalar as shown in Fig. 3.5.
