Graphics Reference
In-Depth Information
Figure 6.15.
Computing the distance from a
point to a plane.
is the
unique
point of
X
which is closest to
P
.
Proof.
Exercise 6.6.2.
If one has a normal vector to a plane, then the formula for the distance of a point
to it is much simpler.
6.6.3 Formula.
The distance d from a point
P
to a plane
X
that contains the point
Q
and has normal vector
N
is given by
N
N
(
)
=
d
=
dist
PX
,
QP
•
.
(6.22)
The point
N
N
N
N
=-
Ê
Ë
ˆ
¯
A
P
QP •
(6.23)
is the
unique
point of
X
that is closest to
P
.
Proof.
See Figure 6.15. The vector
wQP•
N
N
N
N
=
Ê
Ë
ˆ
¯
is the orthogonal projection of
QP
onto
N
. Therefore, d =|
w
|. Define
N
N
N
N
=+
Ê
Ë
ˆ
¯
B
Q
QP •
.
Then
A
=
Q
+
BP
=
Q
+
QP
-
QB
=
P
-
w
is the unique point of
X
that is closest to
P
because
AP
is orthogonal to the plane (Theorem 4.5.12 in [AgoM05]).
Formula 6.6.3 can be restated in terms of coordinates as follows:
