William F Pokorny <ano### [at] anonymousorg> wrote:
> On 12/2/20 11:53 AM, Mr wrote:
> > Interesting... Does that mean than any shape rendered as a parametric could find
> > an equivalent isosurface? ...But not the other way around...?
> >
>
> :-) Unsure how your mind walked to that question from anything we wrote
> - but it's a good question.

I think I might have implied it in my response.

>  From everything I've read, the parametric to implicit conversion
> ("called implicitization") is always possible, but perhaps not always
> tractable/managable(1).

> As to whether it's always possible... I've seen statements out and about
> that the answer is yes; Bald Eagle's draft bicubic patch paper makes
> this statement.

That bit that I included (mostly as a prompt to explore further reading) was
based upon a number of different search results that I came up with, and largely
paraphrased from

Algebraic Methods for Computer Aided Geometric Design (2002)
[* vide infra]

So, perhaps if you're inclined to get some more details, you could contact the
primary author and see if he has something that you can work with.

Thomas W. Sederberg
tom### [at] csbyuedu


I know that another source explained things in a bit more detail, pointing out
that although it's _possible_, for some conversions, you could run into
equations with zillions of terms and pow(x, 138) + .... etc.
This, in my mind, is somewhat reminiscent of Tupper's Self-Referential Formula.
https://www.popularmechanics.com/science/math/a16593584/tuppers-self-referential-equation-produces-a-graph-of-itself/

So although it's possible, in theory, in practice, it's likely to be anything
but efficient.


*
"Implicitization, Parametrization and Intersection
 For any parametric curve, an implicit polynomial equation exists that describes
exactly the same curve. Likewise, for any parametric surface, there exists an
implicit equation that describes exactly the same surface. The process of
finding the implicit equation of a parametric curve or surface is called
implicitization.
 An inversion formula for a parametric curve can be derived using polynomials.
If the parametrization of a curve is a general one-to-one map between parameter
values and points on the curve, the inversion formula returns the parameter
value t corresponding to a point (x, y) that lies on the curve.
 Algebraic methods also can facilitate the design of algorithms for computing
intersections between curves and surfaces."

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