From drorbn@math.toronto.edu Fri Sep 21 12:10:31 2007
Date: Thu, 30 Aug 2007 16:36:42 -0400 (EDT)
To: Robert Palais
Cc: Song Matthew
Subject: Re: Belt trick
Bob,
You are absolutely right! Indeed the parameterization that I sent you
(which I believe is right) is different from the parameterization on the
web site. I have no idea how the error occurred. It is amazing that nobody
noticed the error so far! It seems that the wrong parameterization now on
the site is so close to the right one that the difference is quite hard to
tell visually. With effort I found one specific set of parameters for
which the web-picture is clearly wrong - try t=Pi/4=0.785398 and
half-width=1.
[...]
Best,
Dror.
[...]
On Thu, 23 Aug 2007, Robert Palais wrote:
> Thanks Dror!
>
> [...]
>
> So my next question is that, to the naked eye, the
> parametrization you replied with in mathematica appears
> different than the one on the website - for instance,
> your y component is:
>
> -Cos[v] Sin[2 t] + u Cos[t]^2 Sin[2 v]
>
> but the one I used from the webpage had y component
>
> -(-1 + cos(v)) sin(2t) + u [-cos2(t){2 + cos(v - 2 t) - 2 cos(2t) + cos(v
> + 2t)} sin(v)]
>
> Are these actually the same?
>
> Thanks again,
>
> Bob
>
> [...]
>
> On Thu, 23 Aug 2007, Dror Bar-Natan wrote:
>
> > Dear Bob,
> >
> > Here's the parametrization and a short mathematica computation to show
> > that the two frame vectors are indeed orthogonal:
> >
> > In[1]:= sigma = {x, y, z} = {
> > u (-Cos[2 t] Sin[t]^2 + Cos[t]^2 (Cos[2 t] Cos[2 v] + 4 Cos[v] Sin[t]^2)) +
> > 1/2 Sin[4 t] (-v + Sin[v]),
> > -Cos[v] Sin[2 t] + u Cos[t]^2 Sin[2 v],
> > 1/2 (v - 8 u Cos[t] (1 + 2 Cos[t]^2 Cos[v]) Sin[t] Sin[v/2]^2 +
> > Cos[4 t] (v - Sin[v]) + Sin[v])
> > };
> >
> > In[2]:= B = D[sigma, u] /. u -> 0;
> >
> > In[3]:= T = D[sigma, v] /. u -> 0;
> >
> > In[4]:= B.T // Simplify
> >
> > Out[4]= 0
> >
> > Best,
> >
> > Dror.
> >
> > On Fri, 17 Aug 2007, Robert Palais wrote:
> >>
> >> Dear Dror,
> >>
> >> I checked both numerically and with Mathematica, and
> >> I believe they are non-orthogonal except at endpoints?
> >> Of course you could project on the orthogonal complement,
> >> but my feeling was that the belt should correspond more
> >> directly with an orthogonal frame?
> >>
> >> I will try to go back and take a specific parameter
> >> where the inner product can be evaluated explicitly,
> >> or I could send you my numerical and Mathematica
> >> analyses.
> >>
> >> Best regards,
> >>
> >> Bob.
> >>
> >> On Fri, 17 Aug 2007, Dror Bar-Natan wrote:
> >>
> >>> Dear Bob Palais,
> >>>
> >>> [...]
> >>>
> >>> I believe the belt-width direction and the belt-length direction are
> >>> orthogonal, aren't they? Do you know specific values for the parameters
> >>> for which they are visibly not orthogonal?
> >>>
> >>> Best,
> >>>
> >>> Dror.
> >>>
> >>> On Mon, 13 Aug 2007, Robert Palais wrote:
> >>>
> >>>> Dear Prof. Bar-Natan,
> >>>>
> >>>> I enjoyed your homepage and several of your students' visualizations.
> >>>> I am curious about in particular depicting the belt trick. I would have
> >>>> expected that the belt-width direction dR/du |_{u=0} would be orthogonal
> >>>> to the belt-axis direction dR/dv |_{u=0} but it is not.
> >>>> I thought that these unit vectors (which they are) would give the frame
> >>>> that gives an element of SO(3) for each t and v.
> >>>>
> >>>> Thanks!
> >>>>
> >>>> Bob Palais
> >>>>
> >>>> You might also enjoy a 3D visualization program that allows you
> >>>> to see and rotate objects including many knots with anaglyph or
> >>>> color parallel stereo vision - glasses are inexpensive:
> >>>>
> >>>> http://3d-xplormath.org/
> >>>>
> >>>> It used to be primarily a Mac program, but now is platform
> >>>> independent with Java, developed by my dad. It also exports
> >>>> to Live 3D and other formats.
> >>>>
> >>>> BP