A simpler approach to working with affine spaces

This is a much simpler approach to handling affine spaces. It's
actually the original thing I was doing before I got mixed up with
homogeneous coordinates. This is both topologically simpler than the
representation I was talking about before and lets you do more stuff
that's related to regular temperaments than homogeneous coordinates
do.

If A^n_R is an affine space defined over R, then we can embed A^n into
R^(n+1). The embedding is given by (a,b,c,...) |---> |a b c ... 1>,
where the former is a point and the latter is a vector. We're not
considering the points in this latter space to be homogeneous - we're
just embedding A^n into the affine subspace of R^(n+1) indexed by |a b
c ... 1>. I'll call the last coordinate "w" to keep up with the usual
convention for homogeneous coordinates.

In general, if you subtract two vectors representing points, you get
something of the form |a b c ... 0>. As vectors of the form |a b c ...
1> represent points, so do vectors of the form |a b c ... 0> represent
the difference of points. These vectors make both intuitive and
algebraic sense as the representation of vectors existing as the
difference between affine points in the original space.

This lets us represent affine transformations on A^n as linear
transformations on R^(n+1) in exactly the same way that they're
represented with homogeneous coordinates. The only real difference is
that, if you want to sum points together, you have to be careful to
explicitly make sure that the sum you're taking is actually an affine
combination of the points. If you don't this, you'll end up with
something that's off of the embedded representation of A^n and hence
which doesn't represent a point. That's not to say that vectors off of
the representation of A^n can't be very useful; you just need to
remember that they aren't representing points.

Additionally, and more relevant to our purposes, this approach allows
us to represent affine spans in A^n as linear spans in R^(n+1), or
more precisely the intersection of a linear span in R^(n+1) and the
subset of R^(n+1) in which A^n is embedded. You can do so by making a
matrix like this, which represents the affine hull of 3/2 and 40/27

[-1 1 0 1]
[3 -3 1 1]

If we put this matrix in column-reversed hermite form, defined as
fliplr(hermiteForm(fliplr(matrix))), we get

[-1 1 0 1]
[4 -4 1 0]

Note that the bottom row has w=0, which means it's a vector. This
makes it easy to see that the affine hull of 3/2 and 40/27 is the
subspace spanned by 81/80, translated from the origin to 3/2. In
general, it becomes trivially easy to represent an affine span as the
translation of a linear span this way.

In fact, not only does this let us represent an affine span, but it
lets us represent an affine lattice coset as well, if we consider the
matrix to represent the set of all Z-linear combinations of its rows,
and hence the set of all Z-affine combinations of its represented
points.

You'll note that things like [-2 2 0 2], which aren't points, are in
the row space of the above matrix. What we really care about are the
set of all vectors of the form [x y z 1]. Again, we obtain this set by
taking the intersection of the linear span and the affine subspace in
which A^n is embedded. This serves as a very useful alternative to the
more typical operation of dividing through homogeneously and simply
renormalizing so that w=1 - considering all scalar multiples of
vectors to be equal in this way destroys the nice representation of
vectors that we have above.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> This is a much simpler approach to handling affine spaces. It's
> actually the original thing I was doing before I got mixed up with
> homogeneous coordinates.

Congratulations!

On Fri, May 18, 2012 at 8:15 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
> >
> > This is a much simpler approach to handling affine spaces. It's
> > actually the original thing I was doing before I got mixed up with
> > homogeneous coordinates.
>
> Congratulations!

OK! We're on the same page then? No more ambiguity? I can get back to
Bosanquet positive temperaments and such?

-Mike

LOL, still cracking up at this response, btw. :)

-Mike

On Fri, May 18, 2012 at 8:15 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
> >
> > This is a much simpler approach to handling affine spaces. It's
> > actually the original thing I was doing before I got mixed up with
> > homogeneous coordinates.
>
> Congratulations!

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> OK! We're on the same page then? No more ambiguity? I can get back to
> Bosanquet positive temperaments and such?

Yes, you need to explain what the point of finding an affine hull is.

On Fri, May 18, 2012 at 11:53 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
>
> > OK! We're on the same page then? No more ambiguity? I can get back to
> > Bosanquet positive temperaments and such?
>
> Yes, you need to explain what the point of finding an affine hull is.

This question makes no sense to me at all. This is like asking what
the point of finding a subspace is. There are obviously many points.

-Mike

Are you saying the music-theoretic applications of finding affine
hulls of things? What was wrong with the last application I mentioned,
finding generators for temperaments?

-Mike

On Sat, May 19, 2012 at 12:01 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
> On Fri, May 18, 2012 at 11:53 PM, genewardsmith
> <genewardsmith@sbcglobal.net> wrote:
>>
>> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
>> wrote:
>>
>> > OK! We're on the same page then? No more ambiguity? I can get back to
>> > Bosanquet positive temperaments and such?
>>
>> Yes, you need to explain what the point of finding an affine hull is.
>
> This question makes no sense to me at all. This is like asking what
> the point of finding a subspace is. There are obviously many points.

You know, I've come to the conclusion that there's basically no
possible way to "win" this latest challenge to be "mathematically
rigorous" or what have you. If I post the idea first, you complain
that I didn't take time to mathematically define everything
rigorously. If I then take the time to actually do that, you complain
that I'm now not posting something that's related to music.

This is apparently even true retroactively, even though I'd already
posted the conceptual idea first (and explained it on IRC, and posted
three applications and two coordinate systems).

If there's any part of me that is a mathematician, it's telling me
that there's no possible way to win. I'm just going to plow ahead
anyway and post things in an order that seems sensible.

-Mike

On Sat, May 19, 2012 at 1:08 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
> Are you saying the music-theoretic applications of finding affine
> hulls of things? What was wrong with the last application I mentioned,
> finding generators for temperaments?
>
> -Mike
>
>
> On Sat, May 19, 2012 at 12:01 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
>> On Fri, May 18, 2012 at 11:53 PM, genewardsmith
>> <genewardsmith@sbcglobal.net> wrote:
>>>
>>> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
>>> wrote:
>>>
>>> > OK! We're on the same page then? No more ambiguity? I can get back to
>>> > Bosanquet positive temperaments and such?
>>>
>>> Yes, you need to explain what the point of finding an affine hull is.
>>
>> This question makes no sense to me at all. This is like asking what
>> the point of finding a subspace is. There are obviously many points.

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Are you saying the music-theoretic applications of finding affine
> hulls of things? What was wrong with the last application I mentioned,
> finding generators for temperaments?

Describe it, using your new approach.

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> You know, I've come to the conclusion that there's basically no
> possible way to "win" this latest challenge to be "mathematically
> rigorous" or what have you.

In your lexicon "congratulations" is an indication of disapproval? You've now explained what you are doing in a way which makes sense. Having done that, you can explain what to do with it in a way which makes sense. Why is that a problem?