The implicit
representation for a space curve can be expressed as an
intersection curve between two implicit surfaces
(1.8)
or parametric and implicit surfaces
(1.9)
or two parametric surfaces
(1.10)
The differential geometry properties of the intersection curves
between implicit surfaces are discussed in Sects.
2.2
and 2.3 as well as in Chap. 6
together with the intersection curves between parametric and implicit,
and two parametric surfaces. In Sect. 5.8
algorithms for computing the intersections (1.8),
(1.9) and (1.10) are discussed.
If
can be expressed as a function of
,
, or
, we can eliminate
from the parametric form (1.7) to generate
the explicit form. Let us assume
is a function of
, then we have
(1.11)
This is always possible at least locally when
[412]. Also if the two implicit equations
and
can be solved for two of the variables in terms of the
third, for example
and
in terms of
, we obtain the explicit
form (1.11). This is always possible at least
locally when
[412]. Therefore the explicit equation for the space curve
can be expressed as an intersection curve of two cylinders projecting
the curve onto
and
planes.
Next: 1.2 Analytic representation of
Up: 1.1 Analytic representation of
Previous: 1.1.1 Plane curves
Contents Index
December 2009