What are projective transformations?

[A_B]

Hi,

I'm having trouble interpreting projective transformations. Let's confine ourselves to the projective plane $P(\mathbb{R}_0^3)$.

The transformations of the projective plane are $GL(\mathbb{R}, 3) / \sim$. But these include things like reflections in planes and lines through the origin.

I don't see how that corresponds to anything useful. Perhaps it'll help if I explain what I think a projective transformation should do, so you ca correct me.

A scene in $\mathbb{R}^3$ consists of some points $p_1, p_2, ..., p_n$. There is a plane in space, say $\alpha \leftrightarrow x=1$ on which we project the scene from the origin. Now we move the origin, or equivalently the scene and the plane, and do the projection again. The projective points will have changed and a projective transformation describes this change.

Thanks
A_B

 

[A_B]

OK, turns out I was indeed wrong. Here I found a short explanation of what a projective transformation is: http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node16.html.

But as for how the algebra is related to this concept... I'm clueless.

What on Earth is the purpose of a "projective base", and that "extra point", the "unity"?
(In my course material there is a theorem that proves that the extra point is necessary to determine a unique projective transformation, but since I don't know what a projective transformations is, nor what I should imagine when thinking of a projective base, that theorem adds little to my understanding of it all.)

What is the point of calculating the "coördinates of a point relative to a given projective base"?
Why can't we just use a representative (x,y,z) of the point X = [(x, y, z)]? What's the difference?


A_B

 

[saim_]

I don't know anything about projective geometry but your post intrigued me and I think you might have missed the wiki page for an idea why projective transforms are important and what they intuitively "do":
http://en.wikipedia.org/wiki/Homography

 

[A_B]

I have read the wiki page.
I think I roughly understand what a projective transformation does, although the only expamles I can think of right now are perspective transformations and parallel projections (the latter being equivalent to the former, it's just a perspective transformation from a point at infinity). I don't know if there are other examples of projective transformations.

The main problem is making the connection between the formalism (projective base, homogeneous coordinates, projective transformations as equivalence classes in the general linear group etc.) and the intuitive picture .

I have read and understood many arguments that treat the subject synthetically. i.e. they start from incidence axioms and derive the theorems of projective geometry, no problem there. It's the connection between the algebra and the ideas that I can't grasp.

I do understand how P(R^3) is a good model of the projective plane, [(x, y, z)] being points and all that. It's the transformations and projective bases that I don't understand. Why would we even bother with projective transformations?

Does a projective transformation describe a change in the position/orientation of the plane on which the projection is made? Does it describe a movement of the center of the projection? What is the significance of the projective base, and especially of the unity point?
Those are my questions

A_B

 

[A_B]

Hi,

I think I've figured it out (partially, but the rest will follow)
The mistake I made was that the plane on which a projection is made plays no role in projective geometry, because as soon as such a projection is made, you have fixed your "line at infinity" and hence you are dealing with an affine space.

The projective transformations of an n-dimensional projective space act upon the points of this space, which are rays of an n+1 dimensional vector space.

The point of considering these transformations is clear in the context of Klein's Erlangen Program. The invariants under the group of projective transformations correspond exactly to those properties of plane figures that are left invariant under a projection on to another plane.

The thing that isn't entirely clear to me yet is which plane to plane projections are covered by GL(R, n)/~, and which are not. But I suppose that will come with practice.

Also, the meaning of the "unity" in a projective base is still a little vague...


A_B

 

[mathwonk]

they are just represented by invertible linear transformations. But multiplying one by a ≠0 scalar defines the same geometric transformation.