In a message dated 5/16/2010 5:28:36 PM Eastern Daylight Time, jsd@av8n.com
writes:
I recently stumbled across something interesting:
Conformal Geometric Algebra (CGA).
By way of analogy, consider how special relativity requires
rethinking basic notions of space and time. The old physics
can be understood in terms of the new physics, but not vice
versa. The old physics is recovered by considering the
non-relativistic limit. Similarly classical mechanics can
be understood in terms of QM but not vice versa. The old
physics is recovered by considering the classical limit.
In this way, CGA is a significant upgrade to ordinary old-
style geometric algebra. It requires rethinking basic
notions of "what is a point" and "what is a vector". The
old GA is readily understood in terms of CGA but not vice
versa.
In particular, CGA provides the answers to some questions
that have been bugging me for decades. For starters,
consider the fundamental principle that physics is
independent of absolute location. We are free to choose
an origin of coordinates, but somebody else might choose
differently, and all the fundamental equations of physics
really ought to come out the same. This is an example of
gauge freedom. This independence has been known for a
long time; it predates Galileo's principle of relativity.
Question for the group: Does this principle have a
fancy name? I reckon it doesn't really need a name,
but if it does have a convenient name it would be nice
to know.
To say the same thing another way, I think Descartes did
us no favor by emphasizing the correspondence between points
and displacement vectors relative to "the" origin.
CGA tells us that points are not vectors! The displacement
vector from one point to another is a vector, but each point
by itself is not a vector. We can choose an origin and
speak about the displacement from the origin to each point
... but what if we don't choose an origin? What happens if
we want to shift the origin?
As an interesting specific situation, consider the proverbial
assertion that three points determine a plane. That's true,
but there's more to the story. In the plane, three points
(in general position) determine a circle. That's interesting,
because a circle is finite and has a definite location in the
plane, whereas the plane itself is infinite and is invariant
with respect to shifts in directions parallel to the plane.
Going up a dimension, four points determine a sphere.
Things get interesting when we consider that a sphere turns
into a plane, if the sphere is centered at infinity. Similarly
a circle turns into a straight line, if the circle is centered
at infinity. So if we construct an algebra of circles and
spheres, we get lines and planes for free.
This is apparently very trendy in the computer graphics
business. It is beginning to trickle into physics.
I am definitely not an expert on this. I'm still stuck in
Plato's cave, looking at the shadow of a chair. I've not
yet seen the chair, but I'm pretty much convinced that the
chair exists, and that it is an interesting and useful
chair. I reckon we are going to hear lots more about this.
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There is a nifty interactive computer-graphics system that
allows you to visualize various CGA concepts. And there is
a tutorial that walks you through a bunch of examples.