Graphics Reference
In-Depth Information
sions. It is based on the intuitive geometric idea that a line is defined by a point and
a direction.
(The
point-direction-vector
definition of a line) Any subset
L
of
R
n
Definition.
of
the form
{
}
pv
+
tt
Œ
R
,
(1.5a)
where
p
is a fixed point and
v
is a fixed nonzero vector in
R
n
, is called a
line
(
through
p
). The vector
v
is called a
direction vector
for the line
L
. By considering the compo-
nents of a typical point
x
=
p
+ t
v
in
L
separately, one gets equations
xptv
xptv
=+
=+
1
1
1
2
2
2
.
.
.
x
=+
p
tv
,
t
Œ
R
,
(1.5b)
n
n
n
that are called the
parametric equations for the line
.
In the case of the plane, it is easy to show that the two definitions of a line agree
(Exercise 1.2.2). The definition based on the equation in (1.1) is an
implicit
defini-
tion, meaning that the object was defined by an equation, whereas the definition using
(1.5a) is an
explicit
definition, meaning that the object was defined in terms of a
para-
meterization
. We can think of t as a time parameter and that we are walking along
the line, being at the point
p
+ t
v
at time t.
Note that the direction vector for a line is not unique. Any nonzero multiple of
v
above would define the same line. Direction vectors are the analog of the slope of a
line in higher dimensions.
1.2.1. Example.
To describe the line
L
containing the points
p
= (0,2,3) and
q
= (-2,1,-1).
Solution.
The vector
pq
= (-2,-1,-4) is a direction vector for
L
and so parametric
equations for
L
are
x
=-
=-
=-
2
t
y
2
34
t
z
t
1.2.2. Example.
Suppose that the parametric equations for two lines
L
1
and
L
2
are:
L
:
x
=-
1
t
L
:
x
= +
2
t
1
2
y
=+
=- +
2
t
y
=-
=- +
12
2
t
z
1
t
z
t
(1.6)
Do the lines intersect?
Solution.
We must solve the equations
