Message #4214

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Spherical distortion on 4D to 3D cameras
Date: Wed, 07 Aug 2019 18:34:02 -0500

Cool! I’ve wanted to see MC4D on the hypersphere for a long time :)

I had a couple thoughts catching up on this thread…

• Perhaps the simplest way to map points (from a square, cube,
  hypercube, or any shape really) to a sphere is simply to normalize the
  points. This makes them unit length, i.e. radially projects them to the
  sphere. I’m not sure how your functions compare to radial projection.

• A natural choice for the 4D camera point ‘c’ is the north pole of the
  sphere <0,0,0,1>, which makes the projection of the sphere "stereographic
  projection <https://en.wikipedia.org/wiki/Stereographic_projection>".
  This projection has the nice property that it is "bijection" with Euclidean
  space plus a "point at infinity" (meaning it is both one-to-one and onto -
  in short, no projected points can crash into each other). Stereographic
  projection has many other nice properties as well. I would like to see one
  of your Blender images with the camera placed at the north pole. It looks
  like you might have picked a 4D projection point off the sphere (?)

Cheers,
Roice

On Wed, Aug 7, 2019 at 6:04 PM programagor@gmail.com [4D_Cubing] <
4D_Cubing@yahoogroups.com> wrote:

>
>
> Greetings again
>
> The whole thing is done in Blender 2.80, which has excellent Python
> integration. The rendering is done using a real-time approximating
> raytracer Eevee, which works really smoothly, but the shadows are not very
> realistic. The bloom is also turned on, which in combination with the
> highlighted outlines made for the cartoon look, I think.
>
> The W- face does experience a little bit of distortion as well, just not
> as much since it is further from the surface of the unit sphere.
>
> Also, I uploaded all related files here:
> https://git.mckay-bednar.net/jiri/mc4d-hw
>
> And if anyone wants to help out with an electronic version of the MC4D,
> here’s a little challenge::
> https://math.stackexchange.com/questions/3316660/how-to-reliably-lay-out-continuous-unfolded-diagrams-of-3d-shapes
>
>
> —In 4D_Cubing@yahoogroups.com, <marnix.lenoble@…> wrote :
>
> I love this representation! Would be nice to have it in the software as
> well.
>
> On Tue, 6 Aug 2019, 09:50 Melinda Green melinda@… [4D_Cubing], <
> 4D_Cubing@yahoogroups.com> wrote:
>
>
>
> On 8/5/2019 4:49 PM, mananself@… [4D_Cubing] wrote:
>
> […] Somehow the rendering of the image has a cartoon-ish style. I think
> it has to do with the curvy surfaces. Is there anything special about the
> colors? What did you use to rendered after getting the coordinates?
>
>
> It looks to me like the squares are subdivided into 11x11 grids and lit
> individually. Part of the effect might simply be self-shadows which we’re
> not used to seeing.
>
> -Melinda
>
>
>
>