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Thomas de Groot <tho### [at] degroot org> wrote:
> Much better indeed!
Thanks.
> [you could - maybe - light up a tiny bit the publication reference; not
> essential though]
I'll think about it. That text makes the render SLOOOOOW.
While this little project answered a question I had, it raised some others.
First, is it possible to use a polynomial to draw out a sphere sweep?
(I just used an x-y scaled polynomial to highlight the edge)
I may have to try out Mike Williams' parametric isosurface method.
http://www.econym.demon.co.uk/isotut/splines.htm
And second, which I vaguely suspected:
http://tutorial.math.lamar.edu/Classes/Alg/GraphingPolynomials.aspx
"Finally, notice that as we let x get large in both the positive or negative
sense (i.e. at either end of the graph) then the graph will either increase
without bound or decrease without bound. This will ALWAYS happen with EVERY
polynomial...." [emphasis mine]
But that clearly doesn't seem to be the case in this instance!
Sooooo.... maybe someone who does real math[s] can explain what's going on here.
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Am 26.10.2017 um 13:53 schrieb Bald Eagle:
> First, is it possible to use a polynomial to draw out a sphere sweep?
My gut feeling is that it should be possible. You'll need one polynomial
per spline segment though, and you'll need to use CSG to "cut" the
polynomial at the segment borders.
> And second, which I vaguely suspected:
> http://tutorial.math.lamar.edu/Classes/Alg/GraphingPolynomials.aspx
>
> "Finally, notice that as we let x get large in both the positive or negative
> sense (i.e. at either end of the graph) then the graph will either increase
> without bound or decrease without bound. This will ALWAYS happen with EVERY
> polynomial...." [emphasis mine]
>
> But that clearly doesn't seem to be the case in this instance!
>
> Sooooo.... maybe someone who does real math[s] can explain what's going on here.
Note that POV-Ray's `poly` primitive isn't a "graph" of that polynomial.
Instead, it's the set of points where f(x,y,z) is zero.
Look at the 2D plot of a polynomial with one parameter, i.e. y=f(x).
Notice how the set of points where f(x)=0 - the intersection with the X
axis - is typically just a few isolated points (or none at all).
Now imagine a 3D plot of a polynomial with two parameters, i.e.
z=f(x,y): You can plot this polynomial, too, giving you a kind of
"height field" over the XY plane. Notice how this height field will
typically also intersect the XY plane, but now the set of those
intersection points (all the points where f(x,y)=0) will typically form
closed non-intersecting loops (or lines stretching from infinity to
infinity).
If you extend this to a polynomial with three parameters, you can play
the same game in 4D space: The polynomial forms a "hyper-heightfield"
over the XYZ "hyperplane" (aka volume), and the points with f(x,y,z)=0
typically form a closed surface in 3D space. That set of points is what
POV-Ray's `poly` primitive represents.
In other words, in POV-Ray's `poly` statement the function result
t=f(x,y,z) does /not/ correspond to any spatial dimension; instead, it
can be thought of as a potential field (e.g. temperature), and the shape
is defined as the region where that potential field is below zero (think
water and ice).
So the statement "as we let x get large [...] then the graph will either
increase without bound or decrease without bound", might be translated
to the `poly` situation as follows:
"as we let x, y and z get large [...] then the potential field value
will either increase without bound or decrease without bound"
... except that this statement is actually not generally true for
polynomials with 2 or more parameters.
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