Message #1242

From: Roice Nelson <>
Subject: Re: [MC4D] Re: 3D-only rotations in MHT633
Date: Tue, 02 Nov 2010 20:11:39 -0500

Argh! Scratch the "assuming the camera is invisible" comment. I was
letting my rays get ahead of themselves and wrap back around to the viewer.
And profuse apologies for the extra spam.

To try to make this correction a useful post, let me mention that this
particular lapse came from me recalling portions of a very nice discussion
of this exact topic in Thurston’s "Three-Dimensional Geometry and Topology".
See "Example 1.4.2 (the three sphere from the inside)", starting on page

  1. The short two page discussion is visible in google
    and very readable :) I’ll make myself scarce now…


On Tue, Nov 2, 2010 at 7:39 PM, Roice Nelson <> wrote:

> I realized that for a 3^4 "in sphere view", the 1C sticker would always
> block seeing further than the location that is antipodal to the camera in
> S3, regardless of shrink settings. So you could never see beyond that. You
> potentially could see beyond the antipode for the 4^4 though, assuming the
> camera is invisible anyway :) I still think an "in sphere" view might work
> well and would be interesting to play with…
> Roice
> On Tue, Nov 2, 2010 at 7:14 PM, Roice Nelson <> wrote:
>> On Tue, Nov 2, 2010 at 6:15 PM, Andrey <> wrote:
>>> Roice,
>>> I’m not sure what you mean by "3D-only rotations". I see that navigation
>>> in S3 (in MC4D) and in H3 uses only one camera with 6 position/orientation
>>> parameters (+ some zoom parameters) like normal 3D navigation. For example,
>>> in MC4D you can’t freely navigate in projection space (look from center
>>> outside etc.) like you can, say, in MC7D.
>> Sorry that wasn’t clear. By "3D-only rotations", I meant rotations which
>> are done after the H3 -> R3 projection, not before it. They are Euclidean
>> 3D rotations of the projected (3D) object, and are what Melinda was asking
>> for. These rotations would preserve the shape of 3D objects, unlike the
>> left-click dragging currently in MHT633. Does that help clarify?
>> If you implement the ball model with the POV at the origin, your
>> left-click dragging should automatically result in the types of rotations I
>> was intending to describe.
>>> In MC4D we talk about "3D projection" because there are problems with
>>> real view in sphere: all rays from camera meet in opposite point of sphere
>>> and you can’t see farther than that point. It can be solved by a
>>> nonrealistic optics: say, rays are going by circles tangent to camera axis -
>>> or by intermediate projection, as done in MC4D.
>> I think an "in sphere" view could work well for the 3^4. The image of the
>> furthest cube would cover the entire background and would be "inverted".
>> Depending on face/sticker shrink values, you wouldn’t be able to see past it
>> like you describe, but you could still see the entire puzzle and work with
>> it.
>>> Yes, it’s easy to convert view in MHT from real-view to Poincare ball
>>> (centered in POV). The only problem will be with the situation when camera
>>> is outside of the space. In that case we’ll need to recalculate "shift-left
>>> drag" operation (twist of camera or x-y pan): it will become sliding along
>>> some straight line. But I think that it’s not very difficult.
>> Don’s tessellation applet <>had to deal with the same problem of hyperbolic panning outside the model
>> boundary, so you might check out what he did there…
>> seeya,
>> Roice