My pet theory about linear transformations

[quickfur]

Here's a pet theory i have about linear transformations. Linear transformations are essentially the same as orientation in n-space.

Take 2D transformations, for example. Say you have a hexagon centered on the origin, and you want to scale it so that its height becomes halved while retaining its width. Obviously, this is a straightforward scaling by (1/2,1). But you can also achieve the same effect if you rotate the hexagon into 3-space and project it back to 2-space. To be precise, you rotate it in the YZ plane by 60 degrees, and then project (parallel) the result back to 3-space.

What about shearing? Shearing is essentially rotation in a higher dimension (think about what a rotating square looks like, when viewed from an angle).

What about scaling? Scaling is essentially orientation in 2n dimensions. (Think about how a polygonal face on a 4-polytope's cells can appear reduced in size when the cell is rotated into 4-space.)

What about linear combinations of transformations? They are essentially performing the equivalent orientation operations in distinct additional dimensions. I.e., they are equivalent to orientation in (n+m)-space.

In other words, all linear transformations can be thought of as taking the target object, orienting it appropriately in up to (n+m)-space, and projecting it back to n-space.

What does everyone think about this theory? :-)

If this equivalence is valid, it certainly has very interesting consequences. For example, we would be able to represent linear transformations as a sequence of angles defining the equivalent orientation in (n+m)-space. Conversely, to visualize orientation in (n+m)-space, we can simply reduce it to a linear transformation in n-space.

Does any of this make any sense? :-)

[Marek14]

How about scaling which increases some of the dimensions?

I guess the whole idea is to replace general linear transformation in n dimension by more restricted transformation in >n dimensions. It might have something to do with transformation matrices.

[quickfur]

How about scaling which increases some of the dimensions?

I thought about that, and believe that perspective projection could solve the problem. To scale an object by a factor of N, you could place it at a distance of 1/N in the (n+1)'th dimension, and do a perspective projection back to n-space.

I'm not sure if perspective projection will introduce unwanted distortions, though.

I guess the whole idea is to replace general linear transformation in n dimension by more restricted transformation in >n dimensions. It might have something to do with transformation matrices.

What started me off on this whole thing was noticing how objects appear to undergo all manner of linear transformations when projected from 4-space into 3-space. I thought it gave fresh insights into the nature of linear transformations.

[pat]

The theory suggests that the zero-transform should be invertible, no?

Rotate an object to produce a zero-projection. Now, rotate it some more and you've inverted zero.

[wendy]

Any tilting of the object, viewed far off, results in a compression in some dimensions. Couple this with scaling, and you can get 1:1:2 by first viewing it at 0:60 , 0:60, 1:0 and then doubling it.

To handle multiple dimensions, one imagines this, eg as a 3d object (ie the demension of the solid), living in 3+2 dimensions (2 = number of dimensions to be reduced).

One can reduce polyhedra something feice in 5d.

W

[quickfur]

The theory suggests that the zero-transform should be invertible, no?

Rotate an object to produce a zero-projection. Now, rotate it some more and you've inverted zero.

Not really. The "zero transform" only applied to that particular object in a particular orientation. If you apply it to another object, it may not result in a zero transform (e.g. if the other object has non-zero orientation in the extra dimensions). A "real" zero transform should reduce every object to 0.

OTOH, though, this "virtual" inversion of zero could result in rather nice algebraic properties, sorta like the analog of "0 * x/0 = x" for suitable definitions of *, /, and 0. :-) Sorta reminiscient of extensions to the real numbers which deal with negative infinity and invertible divisions by zero. (I forget the reference to this, though. You end up with pretty weird algebraic systems which have unexpected nice properties.)

[mr_e_man]

How about scaling which increases some of the dimensions?

I thought about that, and believe that perspective projection could solve the problem. To scale an object by a factor of N, you could place it at a distance of 1/N in the (n+1)'th dimension, and do a perspective projection back to n-space.

I'm not sure if perspective projection will introduce unwanted distortions, though.

Perspective projection is not linear, unless you use projective space. Then the projection sending (x,y,z) to (x/z,y/z,1) is represented as the matrix product [1,0,0,0; 0,1,0,0; 0,0,1,0; 0,0,1,0] [x; y; z; 1] = [x; y; z; z] = z [x/z; y/z; 1; 1].

In Euclidean space, rotations shorten the length of the orthogonal projection (by cos t ). If we leave Euclidean space, and use pseudo-Euclidean space, then a hyperbolic rotation can increase the projected length (by cosh t ).

I guess the whole idea is to replace general linear transformation in n dimension by more restricted transformation in >n dimensions. It might have something to do with transformation matrices.

What started me off on this whole thing was noticing how objects appear to undergo all manner of linear transformations when projected from 4-space into 3-space. I thought it gave fresh insights into the nature of linear transformations.

This looks like something I've been studying recently: using Clifford algebra to represent linear transformations. It is known that the "sandwich product" aba^-1 can produce ordinary rotations and reflections, but what about general linear transformations?

An n-dimensional space can be embedded in the null cone of a 2n-dimensional psEuc space. Then any linear transformation (with positive determinant) is represented as a rotation in psEuc space. A transformation with negative determinant is represented as a more general "indefinite special orthogonal transformation". These are all produced by sandwich products with multivectors.