Graphics Reference
In-Depth Information
r
2
0.
BD • BD
D•BD
=
=
In other words,
2
2
(
)
2
db d r
-
+=
1
2
2
(
)
+=.
dd
-
b d
0
11
2
If follows that
22
br
b
-
r
b
22
d
=
and
d
=±
br
-
.
1
2
Since b corresponds to |
AB
| in the original problem, our solution translates into the
stated one in world coordinates.
6.8.4 Formula.
Consider two circles in the plane centered at points
A
and
B
with
radii r
1
and r
2
, respectively. Assume that |
AB
|>r
1
+ r
2
and r
1
> r
2
. Let
D
i,±
and
E
i,±
be
the points where the four lines
L
i
that are tangent to both of these circles intersect
the circles. See Figure 6.23(a). Then
(
)
rr
-
r
r
2
2
i
12
i
(
)
DA
=+
u
±
AB
-
rr
-
v
,
i
,
±
i
12
AB
AB
(
)
rr
+
r
r
i
12
i
2
2
(
)
EA
=+
e
u
±
AB
-
rr
+
v
,
i
,
±
i
i
12
AB
AB
where
A
1
=
A
,
A
2
=
B
, e
1
=+1, e
2
=-1, and
u
and
v
are the orthonormal vectors
Figure 6.23.
Lines tangent to two circles.
