Graphics Reference
In-Depth Information
If we rotate half of a parabola about its axis, then we get a
paraboloid of revolution
(also called an elliptic paraboloid). See Figure 12.5(c). If we do the same thing to a
hyperbola, we get a
hyperboloid of revolution
(also called a
hyperboloid of one sheet
).
12.2.4 Example.
Let
S
be the paraboloid of revolution obtained by rotating the
part of the parabola x = y
2
, y ≥ 0, about the x-axis. We want to find the tangent plane
and normal to
S
at
p
= (4,0,2).
Solution.
The standard parameterization for
S
is
)
=
(
)
(
px
,
q
x
,
x
cos ,
q
x
sin
q
.
Since p(4,p/2) = (4,0,2) and
∂
∂
p
x
1
1
)
=
Ê
Ë
ˆ
¯
(
x
,
q
1
,
cos ,
q
sin
q
2
x
2
x
∂
∂q
p
(
)
(
x
,
q
)
=-
0
,
x
sin ,
q
x
cos
q
,
evaluating these vectors at (4,p/2) and taking the cross product gives us that (1/2,0,
-2) is a normal vector, so that
(
)
∑
(
(
)
-
(
)
)
=
120 2
,,
-
xyz
,,
402
,,
0
,
or
x-+=
44 ,
is the equation for the tangent plane.
12.3
Quadric Surfaces and Other Implicit Surfaces
Like the conic curves, quadric surfaces are an important shape for CAGD. A quadric
surface is a subset of points (x,y,z) in
R
3
which satisfies a general quadratic equation
of the form
2
2
2
ax
+++++++++=.
by
cz
dxy
exz
fyz
gx
hy
iz
j
0
(12.9)
A complete classification of the solutions to equation (12.9) can be found in Section
3.7 in [AgoM04] and we shall not repeat it here. Omitting the degenerate cases, one
gets the basic ellipsoids, cylinders, cones, paraboloids, and hyperboloids. Figure
12.5(a) and (c) showed a sphere (a special case of an ellipsoid) and paraboloid. Figure
12.6 shows examples of the others. Clearly, many mechanical parts have such shapes.
In an interesting paper, Goldman ([Gold83]) analyzes quadrics that are surfaces of
