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On 12/2/20 11:53 AM, Mr wrote:
> William F Pokorny <ano### [at] anonymous org> 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.
From everything I've read, the parametric to implicit conversion
("called implicitization") is always possible, but perhaps not always
tractable/managable(1).
The implicit to parametric conversion ("called parameterization") in
practice I think mostly can work.
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.
I've also see statements parameterization is sometimes impossible. In
Jules Bloomenthal's introduction to the book "Introduction to Implicit
Surfaces" he writes:
"Parameterization is not always possible, however; for example, implicit
surfaces that are defined by certain polynomials of fourth or higher
degree cannot be parameterized by rational functions [Salmon 1914]."
(1) - Always possible is not the same thing as saying the converted
parametric -> "implicit equation" is usable in an isosurface object.
Bill P.
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