Graphics Reference
In-Depth Information
As r is mid-way between p and q , we have
=
1
2 p
+
1
2 q
r
and
q
=
2 r
p
(3.56)
Substituting Eq. (3.52) in Eq. (3.56) gives
q
=
2 t
+
v
p
If v is a unit vector
v , then
ˆ
v
·
p
t
(3.57)
q
=
2 t
+
ˆ
v
p
(3.58)
Let's test Eq. (3.4) with a simple example.
Figure 3.26 illustrates a line whose direction is given by
ˆ
=
1
2
+
v
i
j . T 10 is some point
on the line with position vector t
=
i , and P 11 is some arbitrary point with position vector
p
=
i
+
j .
Y
P (1,1)
p
1
n
v
T
t
Q
1
X
Figure 3.26.
Using Eq. (3.57) gives
1
2
1
2
v
·
p
t
=
i
+
j
·
i
+
j
i
=
1
2
=
Using Eq. (3.58) gives
2 i
j
1
2
1
2
q
=
2 t
+
v
ˆ
p
=
+
i
+
i
j
=
2 i
i
+
j
i
j
=
0
Therefore, the reflected point is Q00.
3.10 A line perpendicular to a line through a point
Here is another geometric problem. Given a line m and a point P, what is the equation of a
straight line that passes through P and is perpendicular to m? The algebraic solution is relatively
easy to solve, so let's consider a vector solution.
 
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