We start with a rational polynomial parametric surface to implicit
algebraic surface intersection problem defined as:
(5.77)
This leads to four algebraic equations in five unknowns
(underconstrained system). For the usual low degree surfaces
and low degree patches
, we can
substitute
into
to obtain an
implicit algebraic curve in
[124,333,211,302,212].
Examples of low order implicit algebraic surfaces in practical use are
planes (degree 1), the natural quadrics (cylinder, sphere, cone)
(degree 2), and torii (degree 4). In fact in a survey of mechanical
parts (mechanical elements), over
of all surfaces involved are
of these types
[149]. It is also well known that low order implicit
algebraic surfaces have a low degree rational polynomial parametric
representation (which can be easily obtained
[314]), so that when two such low order
implicit algebraic surfaces are intersected, the methods of this
section may be also used.