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and
ˆ
n
=
1
as shown in Fig. 6.20.
Now the shortest line between the sphere's centre C and the plane is when the line is
perpendicular to the plane. Naturally, this line is parallel with the plane's normal
n
and
ˆ
is represented vectorially by
−
CP. The length of
−
CP:
−
CP
, informs us whether the sphere
intersects, touches, or misses the plane:
−
CP
<Rintersect condition
−
CP
=
R touch condition
−
CP
>Rmiss condition
As
1, d represents the perpendicular distance from the origin to the plane.
Therefore, we can state that
ˆ
n
=
−
CP
=
n
ˆ
(6.34)
where is a scalar.
Y
P
CP
C
n
c
p
O
d
y
C
Z
x
C
z
C
X
Figure 6.20.
The position vector
p
is defined as
+
−
CP
p
=
c
=
c
+
n
ˆ
(6.35)
but as
n
ˆ
·
p
=
d
n
ˆ
·
p
=ˆ
n
·
c
+
n
ˆ
·ˆ
n
=
d
