Graphics Reference
In-Depth Information
Since p(2,p/2) =
p
, it follows that
∂
∂
p
y
∂
∂q
p
(
)
=
(
)
(
)
=-
(
)
22 001
,
p
, ,
and
22 020
,
p
,
,
are a basis for our tangent plane. Finally,
∂
∂
p
y
∂
∂q
p
(
)
¥
(
)
=
(
)
22
,
p
22 200
,
p
, ,
is a normal vector to the plane, so that its equation is
(
)
∑
(
(
)
-
(
)
)
=
200
,,
xyz
,,
302
,,
0
,
or
x -30.
Next, if we rotate a semicircle about an axis whose endpoints lie on the axis, then
we get a
sphere
. In the case of the semicircle of radius r about the origin, we can
parameterize its points with the map
Æ
(
)
Œ
[
]
f
r
cos , sin
f
r
f
for
f
0
,
p
.
This leads to the following parameterization of the sphere of radius r about the origin:
(
)
=
(
)
p
fq
,
r
cos , sin
f
r
f
cos , sin
q
r
f
sin
q
.
(12.5)
See Figure 12.5(a). The partial derivatives
∂
∂f
p
(
)
=-
(
)
fq
,
r
sin , cos
f
r
f
cos , cos
q
r
f
sin
q
(12.6)
∂
∂q
p
(
)
=-
(
)
fq
,
0
,
r
sin
f
sin , sin
q
r
f
cos
q
define the tangent planes except at the two poles (±r,0,0) where they vanish. The
tangent planes at those two points have to be handled as a special case unfortunately.
If we rotate a circle about an axis and if this circle does not meet the axis, then we
get a
torus
. As a special case, let
T
be the torus obtained by rotating the circle of radius
r with center (0,R,0), r < R, about the x-axis. See Figure 12.5(b). Here is another natural
way to visualize the standard parameterization of
T
. Let
P
q
be the plane through the x-
axis that makes an angle q with the x-y plane. This plane intersects
T
in a circle
C
q
with
center R
u
q
, where
u
q
= (0,cos q, sin q) is the unit vector in the y-z plane that makes an
angle q with the y-axis. Parameterizing the points of
C
q
by the angle f that the ray from
the center of
C
q
makes with the x-axis corresponds to the map
