Graphics Reference
In-Depth Information
Figure 14.3.
Distance from a point to a
surface.
Equivalently, we can look for a
q
so that the vector
p
-
q
is parallel to the normal
n
q
(u,v), that is,
(
(
)
)
¥
(
)
=
p
-
quv
,
n uv
,
0
(14.5)
q
If there are more than one solution to equations (14.4) or (14.5), then we need to
check the distances between all those points and
p
and pick the closest. After that
we will also have to find and check the distance from
p
to the four edge curves of
q|∂([a,b] ¥ [c,d]). The need for that is shown in Figure 14.3. Solving equations (14.4)
and (14.5) would determine the point
q
1
in the figure, but the closest point is actually
q
2
.
Curve-surface Distance.
To find the distance between a parameterized curve
[
]
Æ
R
3
pab
:
,
11
and a parameterized surface
[
]
¥
[
]
Æ
R
3
qab
:
,
cd
,
22
22
we need to solve the equations
(
)
¥¥
(
)
=-
(
)
¥=
pq
-
q
q
pq n
0
u
v
q
(
)
pq p
-
•
¢=
0
.
(14.6)
See Figure 14.4. In addition, we also need to check the distances of p(a
1
) and p(b
1
) to
q(u,v) and the distance of q|∂([a
2
,b
2
] ¥ [c
2
,d
2
]) to p(u).
Surface-surface Distance.
To find the distance between parameterized surfaces
[
]
¥
[
]
Æ
R
3
[
]
¥
[
]
Æ
R
3
pab
:
,
cd
,
and
qab
:
,
cd
,
11
11
22
22
we need to solve the equations
