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Am 10/21/2017 um 3:51 schrieb Bald Eagle:
> Mostly because they didn't have a picture to go with the Wikipedia article...
>
> https://en.wikipedia.org/wiki/Curve_of_constant_width
>
>
> So here it is folks, the eighth-order polynomial defined by:
>
> (x^2+y^2)^4 - 45(x^2+y^2)^3 - 41283(x^2+y^2)^2 + 7950960(x^2+y^2) +
> 16(x^2-3y^2)^3 + 48(x^2+y^2)(x^2-3y^2)^2 + (x^2-3y^2)x[16(x^2+y^2)^2 -
> 5544(x^2+y^2)+266382] - 720^3
>
>
> [also see https://arxiv.org/pdf/1504.06733.pdf pg. 21]
>
Please do not use this 3d-look text (that even throws shadows) with this
image. Besides that it IMHO does not look good it is more importantly
very hard to read.
And personally - e.g. on a wiki-page - I hate text information where
copy and paste doesn't work ;)
-Ive
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On 25/10/2017 09:29, Ive wrote:
> Am 10/21/2017 um 3:51 schrieb Bald Eagle:
>> Mostly because they didn't have a picture to go with the Wikipedia
>> article...
>>
>> https://en.wikipedia.org/wiki/Curve_of_constant_width
>>
>>
>> So here it is folks, the eighth-order polynomial defined by:
>>
>> (x^2+y^2)^4 - 45(x^2+y^2)^3 - 41283(x^2+y^2)^2 + 7950960(x^2+y^2) +
>> 16(x^2-3y^2)^3 + 48(x^2+y^2)(x^2-3y^2)^2 + (x^2-3y^2)x[16(x^2+y^2)^2 -
>> 5544(x^2+y^2)+266382] - 720^3
>>
>>
>> [also see https://arxiv.org/pdf/1504.06733.pdf pg. 21]
>>
>
> Please do not use this 3d-look text (that even throws shadows) with this
> image. Besides that it IMHO does not look good it is more importantly
> very hard to read.
Oh! I liked it.
> And personally - e.g. on a wiki-page - I hate text information where
> copy and paste doesn't work ;)
>
In that case you might like capture2text. It is a utility that takes a
screenshot of part of your screen then OCR's it and puts the text in the
clipboard. I've been using it for a couple of years.
Although it doesn't work too well with Bald Eagle's 3D text.
It can also use Google Translate.
http://capture2text.sourceforge.net/
--
Regards
Stephen
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On 25-10-2017 11:29, Stephen wrote:
> On 25/10/2017 09:29, Ive wrote:
>> Am 10/21/2017 um 3:51 schrieb Bald Eagle:
>>> Mostly because they didn't have a picture to go with the Wikipedia
>>> article...
>>>
>>> https://en.wikipedia.org/wiki/Curve_of_constant_width
>>>
>>>
>>> So here it is folks, the eighth-order polynomial defined by:
>>>
>>> (x^2+y^2)^4 - 45(x^2+y^2)^3 - 41283(x^2+y^2)^2 + 7950960(x^2+y^2) +
>>> 16(x^2-3y^2)^3 + 48(x^2+y^2)(x^2-3y^2)^2 + (x^2-3y^2)x[16(x^2+y^2)^2 -
>>> 5544(x^2+y^2)+266382] - 720^3
>>>
>>>
>>> [also see https://arxiv.org/pdf/1504.06733.pdf pg. 21]
>>>
>>
>> Please do not use this 3d-look text (that even throws shadows) with
>> this image. Besides that it IMHO does not look good it is more
>> importantly very hard to read.
>
> Oh! I liked it.
Me too, but I must agree with Ive here. I would also add (sorry for
this) that the object texture is misleading: it looks like a 3D surface,
curved towards the viewer, while it is flat in reality. It took me a
while before I understood what I was looking at.
--
Thomas
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Ive <ive### [at] lilysoft org> wrote:
> Please do not use this 3d-look text (that even throws shadows) with this
> image. Besides that it IMHO does not look good it is more importantly
> very hard to read.
I usually use pretty flat, functional text, and I did think about the look and
readability. I didn't think it looked TOO bad, but perhaps for a wikipedia
page, it out to be a little less artistic.
> And personally - e.g. on a wiki-page - I hate text information where
> copy and paste doesn't work ;)
Perhaps I can add the text info as plain text or formatted text or MathJax, but
I wanted to have the info in the graphic too - for inseparability purposes.
Along those lines, I wouldn't mind embedding the full POV-Ray scene used to
generate the graphic in the metadata header section of the file - I just have to
puzzle out _exactly_ how.
Thanks for the constructive feedback!
> -Ive
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Thomas de Groot <tho### [at] degrootorg> wrote:
> Me too, but I must agree with Ive here. I would also add (sorry for
> this) that the object texture is misleading: it looks like a 3D surface,
> curved towards the viewer, while it is flat in reality. It took me a
> while before I understood what I was looking at.
>
> --
> Thomas
It's not really textured - I have it lit with 4 lights - white, magenta, yellow,
and blue.
It IS 3D - just not concave or convex, since the z components are all zero, and
that winds up giving a polynomial object that is infinitely scaled in the
z-direction. So I intersected it with a thin box.
But I take your meaning.
This leads into a topic about the Documentation that I will post in a fresh
thread.
No need for apologies: it's constructive, and honest feedback about how the
render comes out.
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Simpler version
Post a reply to this message
Attachments:
Download 'polynomial1.png' (224 KB)
Preview of image 'polynomial1.png'
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On 26-10-2017 12:42, Bald Eagle wrote:
> Simpler version
>
Much better indeed!
[you could - maybe - light up a tiny bit the publication reference; not
essential though]
--
Thomas
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Thomas de Groot <tho### [at] degrootorg> 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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On 26/10/2017 12:53, Bald Eagle wrote:
> who does real math[s]
+1 for diversity. :-)
--
Regards
Stephen
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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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