Reciprocal System Research Society

Somewhere it was said that in Projective Geometry there are only cross-ratios (ratios of ratios) and they have 15-degrees of freedom, which are represented by a 4x4 matrix (with 1 element fixed at unity).

Since a cross-ratio is simply (a/b)/(c/d) which is an expression of 4 variables, shouldn't the cross-ratio have only 4 degrees of freedom ?

Where do the remaining 11 come from ?

If I understand right mathematically, cross ratio isn't the only invariant... other things like incidence, tangency, collinearity, and intersection are also unchanged under projective transformation.

Still trying to work out the relations, I'll get back with it in a while...

Alas, friends-- a few of us-- the crash test dummies--- are still struggling, yet intrigued with the definitions and relationships of the first two words: "time" and "space." At least for one crash test dummy, the words "cross ratio, incidence, tangency, collinearity, intersection" and "projective transformation" are so far beyond our current level that we fear you may grow impatient with any response-- including basic questions---we might have. So forgive us our silences (and stillnesses) .... peace. Bear

Bear wrote:

... are so far beyond our current level that we fear you may grow impatient with any response-- including basic questions---we might have. So forgive us our silences (and stillnesses) .... peace. Bear

Oh, I'm sorry for just throwing in that jargon! Those are some terms used in geometry... the difficulty is still in the basics: how to connect that to motion. So your questions would be the most relevant of all... please post anything you feel like, we are all searching! The more 'basic' the better.

Coming back to Horace's point, I think it could be this way... The cross ratio has four variables to it, that is fine. But the cross ratio doesn't determine the geometry, how it transforms does. And to represent any general transformation on a four-variable, we need 16 (4X4) variables, minus one for scale [can't have ALL the variables going to zero].

Cheers,

Gopi

Tomorrow's Thanksgiving, so I grateful that I think I'm starting to understand. Do I have it right-- that if you divide the circumference of a pumpkin by its diameter, you end up with pumpkin pi? Just checking-- Bear

Bear,

You know, pumpkins have the same geometry as the earth—same number of segments. Check out the little ones. You can see the ten primary meridians (stem is the north pole),

My pi was a bit abstract and dry, I prefer it with a bit more sugar and substance.

Rick

Hmmm.. interesting. I will check it out. (I happen to have a pumpkin sitting on my front porch.) Makes sense that it should be so. "As within, so without, as above, so below." As goes the pumpkin, so goes the.... (?) My tummy's starting to look like a pumpkin. See you soon----bear.

I must have a Dewey Larson pumpkin--- it has thirteen meridians!

Horace wrote:

Somewhere it was said that in Projective Geometry there are only cross-ratios (ratios of ratios) and they have 15-degrees of freedom, which are represented by a 4x4 matrix (with 1 element fixed at unity).

Since a cross-ratio is simply (a/b)/(c/d) which is an expression of 4 variables, shouldn't the cross-ratio have only 4 degrees of freedom ?

Where do the remaining 11 come from ?

Consider the cross-ratio as a type of homogeneous coordinate: [a b c d]. I did not use [x y z 1] because it does not represent a point or plane, but represents a dimension of motion, itself.

The Universe is composed of 3 dimensions: 4 variables x 3 dimensions = 12 DOF:

[a b c d]

[e f g h]



This leaves three DOF, and one constant: . This is the "dimension of perspective"--the "camera" or observer of the other three dimensions. It is in this dimension that the assumptions are placed from the other strata of geometry, such as the plane at infinity. basically says that the observer's scale is fixed in the system at unity, yet all the other parameters are still variable. In other words, "we're still" and the rest of the Universe is moving, in a scalar fashion.

Total: 3 dimensions x 4 variables = 12 DOF + 3 perspective DOF = 15 DOF for the basic projective transform.