Graphics Reference
In-Depth Information
3 Straight Lines
3.1 Introduction
Now that we have explored the basics of vector analysis, let us apply some of these ideas to the
description of straight lines. There are various ways of describing straight lines; these include
the normal form of the line equation y
=
mx
+
c,
the general form of the line equation ax
+
by
+
c
=
0,
the Cartesian form of the line equation ax
+
by
=
c,
and the parametric form of the line equation
p
=
t
+
v
.
ˆ
The first three only apply to 2D straight lines, whereas the fourth definition applies to 2D and
3D straight lines. So let us begin with this latter equation.
3.2 The parametric form of the line equation
The first objective is to find a way to move along a straight line using a scalar parameter.
Fortunately, vectors provide such a mechanism in the form of the operation
v
, which scales
ˆ
the unit vector
v
with the scalar . But we also need to relate our position relative to some
reference point, which must be included in our definition. We begin as follows.
Figure 3.1 shows a line that passes through the points T and P and is parallel with the vector
ˆ
v
. From this scenario we can state
ˆ
−
OP
=
−
OT
+
−
TP
Let
v
ˆ
=
the unit vector defining the line's direction,
=
ˆ
a scalar that acts as a parameter to position us anywhere along the direction of
v
,
T
=
a reference point on the line,
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