Graphics Reference
In-Depth Information
Since —f(x,y,z) = (-2x,2y,2z), —f( p ) = (
,0,4) is a normal vector to S at p . It follows
-23
that the tangent plane is defined by
(
) (
(
) - (
)
) =
-
2304
,,
xyz
,,
302
,,
0
or
-+-=
3270
xz
.
From the point of view of finding normals to surfaces it is therefore advantageous
to have an implicit representation of it. Unfortunately, except for quadric surfaces,
surfaces are usually presented via parameterizations. Finding an implicit representa-
tion given a parameterization is a nontrivial problem that is addressed in Chapter 10
in [AgoM04].
It should be pointed out that many surfaces of revolution are implicit surfaces, so
that the gradient method for finding normals and tangent planes applies to them. For
example, consider the surface of revolution defined by equation (12.2). It is easy to
show that this surface is the set of zeros of the function
(
) =+- ()
222
Fxyz
,,
y
z
f x
.
(12.10)
12.3.2 Example. We rework Example 12.2.4 using this approach. Using equation
(12.10), the surface is defined by the equation
(
) =+-=
22
Fxyz
,,
y
z
x
0
.
But —F(x,y,z) = (-1,2y,2z) and —F(4,0,2) = (-1,0,4). The latter is, up to a scalar
multiple, the same normal as the one we got before.
Finally, we mention Barr's ([Barr81]) superquadric surfaces. These are the surface
analogs of superellipses and are useful in representing shapes such as rounded blocks
or rounded square toroids. They are defined by trigonometric functions raised to expo-
nents. [Barr92] presents expressions for their volume, center of mass, and rotational
inertia tensor.
12.4
Ruled Surfaces
Ruled surfaces are probably the next simplest surfaces after planes. Special cases of
these are extrusions. These are surfaces obtained by sweeping a vector along a curve.
Given a curve f : [a,b] Æ R 3 and a vector v Œ R 3 , the parametric surface
Definition.
[
] ¥ [ Æ
3
pab
:,
01
,
R
defined by
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