Graphics Reference
In-Depth Information
We begin by defining
p
as follows for the two lines:
p
=
r
+
a
p
=
s
+
b
where P x
P
y
P
is the point of intersection and
p
its position vector.
Therefore,
r
+
a
=
s
+
b
(6.1)
To eliminate , we multiply Eq. (6.1) throughout by
b
⊥
, as this creates the term
b
⊥
·
b
, which
is zero:
b
⊥
·
b
⊥
·
b
⊥
·
b
⊥
·
b
⊥
·
r
+
a
=
s
+
b
=
s
Therefore,
b
⊥
·
−
s
r
=
b
⊥
·
a
where
b
⊥
=−
y
b
i
+
x
b
j
Therefore,
−
y
b
i
+
x
b
j
·
x
S
−
x
R
i
+
y
S
−
y
R
j
=
−
y
b
i
+
x
b
j
·
x
a
i
+
y
a
j
x
b
y
S
−
y
R
−
y
b
x
S
−
x
R
=
(6.2)
x
b
y
a
−
x
a
y
b
To eliminate from Eq. (6.1), we multiply throughout by
a
⊥
, as this creates the term
a
⊥
·
a
,
which is also zero:
a
⊥
·
a
⊥
·
a
⊥
·
a
⊥
·
a
⊥
·
r
+
a
=
s
+
b
=
r
a
⊥
·
r
−
s
=
a
⊥
·
b
where
a
⊥
=−
y
a
i
+
x
a
j
Therefore,
−
y
a
i
+
x
a
j
·
x
R
−
x
S
i
+
y
R
−
y
S
j
=
−
y
a
i
+
x
a
j
·
x
b
i
+
y
b
j
x
a
y
S
−
y
R
−
y
a
x
S
−
x
R
=
(6.3)
x
b
y
a
−
x
a
y
b
