Graphics Reference
In-Depth Information
Figure 14.1.
Distance from a point to a curve.
first find the critical points of the function h(u,v) in equation (14.1). These are the
points where both of its partial derivatives vanish, that is, we must solve
∂
∂
h
u
(
)
=-
2
pq p
•
¢=
0
,
∂
∂
h
v
(
)
(14.3)
=-
2
pq q
•
¢=
0
.
Notice again that, geometrically, we are looking for points
p
and
q
on the curves with
the property that either
p
=
q
or the line through
p
and
q
is orthogonal to the tan-
gents of the curves at those points. See Figure 14.2(a). Finally, we also have to check
the distances from each endpoint of a curve to the other curve.
14.2.1
Example.
To find the distance between the curves defined by
()
=
(
)
2
()
=
(
)
p u
u u
,
and
q v
v
,
24
v
+
.
Solution.
See Figure 14.2(b). Equation (14.1) translates into
(
)
(
)
(
)
=-
2
2
huv
,
u vu
,
- -
24
v
•
u vu
-
,
-
24
v
-
.
It is easy to check that the solutions to the equations
∂
∂
h
u
(
)
(
[
(
)
()
]
=
2
)
=-+--
2
=- --
2
uvu
,
2
v
4
•
1 2
,
u
2
uv u
2
v
4 2
u
0
and
∂
∂
h
v
(
)
[
(
)
]
=
2
(
)
=-
2
=- --
2
uvu
,
2
v
4
•
--
1
,
2
2
uv
-
+
2
u
-
2
v
-
4
0
