POV-Ray : Newsgroups : povray.advanced-users : Shape Request Server Time: 18 Jun 2019 16:38:55 GMT
 Shape Request (Message 1 to 5 of 5)
 From: Bald Eagle Subject: Shape Request Date: 15 May 2017 16:35:00 Message:
```I have looked into doing some 4D projections into 3D space, but alas, the math

With regard to constant-perceived-width objects, specifically in the present
case of planetary and satellite orbits, the projection of a torus would be a
useful little object to be able to have on-hand.

http://mathforum.org/mathimages/index.php/Projection_of_a_Torus

I'm envisioning 3 parameters for this:  major radius, and 2 minor radii - the
max and the min.

Thanks to anyone taking an interest!  :)
```
 From: Le Forgeron Subject: Re: Shape Request Date: 15 May 2017 18:13:36 Message: <5919efd0\$1@news.povray.org>
```Le 15/05/2017 Ã  18:30, Bald Eagle a Ã©crit :
> I have looked into doing some 4D projections into 3D space, but alas, the math
> for the moment evades me.
>
> With regard to constant-perceived-width objects, specifically in the present
> case of planetary and satellite orbits, the projection of a torus would be a
> useful little object to be able to have on-hand.
>
> http://mathforum.org/mathimages/index.php/Projection_of_a_Torus
>
> I'm envisioning 3 parameters for this:  major radius, and 2 minor radii - the
> max and the min.
>
> Thanks to anyone taking an interest!  :)
>
>

A quick search for 4D torus found me that page:

http://www.dr-mikes-maths.com/4d-torus.html

did not get the notion of 4D.

Now, if all you need is a classical 3D-donut shape with a hole not at
the center and yet a regular circle as bounding on the equator plane, it
might just be a matter of playing with fourth order polynomial.

So, can you clarify what you want ?
```
 From: clipka Subject: Re: Shape Request Date: 15 May 2017 18:35:54 Message: <5919f50a\$1@news.povray.org>
```Am 15.05.2017 um 18:30 schrieb Bald Eagle:
> I have looked into doing some 4D projections into 3D space, but alas, the math
> for the moment evades me.
>
> With regard to constant-perceived-width objects, specifically in the present
> case of planetary and satellite orbits, the projection of a torus would be a
> useful little object to be able to have on-hand.
>
> http://mathforum.org/mathimages/index.php/Projection_of_a_Torus
>
> I'm envisioning 3 parameters for this:  major radius, and 2 minor radii - the
> max and the min.
>
> Thanks to anyone taking an interest!  :)

I suspect that such a torus might not have the necessary numeric
stability to form artifact-free thin lines.

Personally, I'd try a sphere sweep based approximation instead --
provided the newest experimental version turns out to work as I'm hoping.
```
 From: Bald Eagle Subject: Re: Shape Request Date: 15 May 2017 19:20:01 Message:
```To clarify,
I'd just like to have a variable-minimum-radius torus.
Small at "apogee", and wide at "perigee".

Here, the Dupin Cylide - it's exactly what I was thinking of.
https://en.wikipedia.org/wiki/Dupin_cyclide

clipka <ano### [at] anonymousorg> wrote:

> I suspect that such a torus might not have the necessary numeric
> stability to form artifact-free thin lines.

No idea, but good observation.
I know it's possible to do in POV-Ray (the Dupin Cyclide), because it was done
here:

http://www.dimensions-math.org/Dim_CH7_E.htm
http://www.geometrie.tuwien.ac.at/vis/vis036.html
http://www.geometrie.tuwien.ac.at/vis/vis037.html

> Personally, I'd try a sphere sweep based approximation instead --
> provided the newest experimental version turns out to work as I'm hoping.

That was my immediate idea, but IIRC from my investigations into Bezier splines,
you can't make a circle with a single spline, so it would have to be at least 2
pieces.
```
 From: Le Forgeron Subject: Re: Shape Request Date: 15 May 2017 20:35:01 Message: <591a10f5@news.povray.org>
```Le 15/05/2017 Ã  21:17, Bald Eagle a Ã©crit :
> To clarify,
> I'd just like to have a variable-minimum-radius torus.
> Small at "apogee", and wide at "perigee".
>
> Here, the Dupin Cylide - it's exactly what I was thinking of.
> https://en.wikipedia.org/wiki/Dupin_cyclide

So, you even found a page with all that is needed:
* for a parametric
* for a polynomial (quartic)

Per your specifications, it's an elliptic cyclide and we have c < d
(ring cyclide).

Remains to be done: mapping your input (3 radii) into a,b,c & d.
```