Graphics Reference
In-Depth Information
13.5.5
Surface Algebraic Methods
When one tries to use algebraic methods to solve intersection problems, one looks
first for special cases that can be handled by special techniques. For example,
plane/plane or plane/quadric intersections produce lines and conics, respectively, and
can be solved for explicitly. In general, because the implicit/parametric case is rela-
tively easy, attempts have been made to reduce other cases to this one.
An implicit surface
S
can be parameterized, but not necessarily by rational poly-
nomial functions. If
S
is defined by linear or quadratic polynomials, then
S
can be
parameterized by rational polynomial functions. If
S
is defined by higher-degree
polynomials, then it may not admit such a parameterization. See Section 10.15 in
[AgoM05] for more details. Parameterized surfaces, on the other hand, can always be
represented implicitly by rational polynomial functions if the parameterization was
also of that form. The only problem is that the standard implicitization algorithms
may produce very complicated equations. See Sections 10.9 and 10.10 in [AgoM05].
For example, it can be shown that a bicubic patch is equivalent to an algebraic surface
of degree 18 whose equation contains 1330 terms. For these reasons, algebraic geom-
etry methods seem to be impractical currently, but there is a lot of ongoing research.
At any rate, because it is known that every algebraic curve in
R
3
can be mapped to
an algebraic curve in
R
2
(although the latter may be more complicated than the
former), one general algebraic approach to solving the surface intersection problem is:
(1) Map the intersection curve in
R
3
to a plane curve defined by an equation
(
)
= 0
huv
,
.
(13.23)
(2) Solve equation (13.23).
(3) Map the solution back to
R
3
.
Step (2) is considered in more detail in Sections 14.5.1 and 14.6. Here we describe
two approaches to (1) and (3). One is based on substitutions and the other, on pro-
jections. Let
S
1
and
S
2
be surfaces in
R
3
.
The Substitution Approach.
Suppose that surface
S
1
is defined implicitly by an
equation
(
)
= 0
fxyz
,,
(13.24)
and surface
S
2
is defined via a parameterization
(
)
=
(
(
)
(
)
(
)
)
guv
,
g uv g uv g uv
,
,
,
,
,
.
1
2
3
Substituting into equation (13.24) gives
(
)
=
(
(
)
(
)
(
)
)
=
huv
,
fg uv g uv g uv
,
,
,
,
,
0
.
(13.25)
1
2
3
If we solve equation (13.25) in the u-v plane, then we can map the solution back to
R
3
using g.
