Graphics Reference
In-Depth Information
Y
2
T
a
b
n
t
c
C
R
1
1
1
X
Z
Figure 6.23.
The plane's equation is
p
=
t
+
a
+
b
where
a
=
i
b
=
k
t
=
i
+
2
j
c
=
i
+
j
+
k
R
=
1
Calculate
n
:
i j k
100
001
=
×
=
=−
n
a
b
j
and
n
ˆ
=−
j
Therefore,
=
−
j
·
i
+
2
j
−
i
−
j
−
k
=
1
As
1, the sphere touches the plane.
The parametric plane equation lends itself to calculating the circle of intersection when the
sphere intersects the plane, so let's calculate the radius r of the intersecting circle. Figure 6.24(a)
shows a cross section of the relevant geometry from which, using the theorem of Pythagoras,
=
R
=
CP
2
R
2
=
r
2
+
but as
CP
=
√
R
2
r
=
−
2
(6.37)