Transformations taking straight lines to straight lines

1. Oct 28, 2006

Palindrom

On R^n, I'd say the only smooth transformations taking straight lines to straight lines are the affine transformations.

Would I be right saying that?

How would one go about proving that?

2. Oct 28, 2006

pmb_phy

I believe so.

Best wishes

Pete

3. Oct 28, 2006

robphy

If I'm not mistaken, Affine Transformations require that parallelism be preserved... however, Projective Transformations also take straight lines to straight lines without requiring parallelism.

4. Oct 28, 2006

quasar987

I'm not familiar with the terminology used here, but surely if a map takes a straight line to another straight line, it is made of a translation + a rotation. So something like

5. Oct 29, 2006

Doodle Bob

The problem here then is that on R^n (as opposed to R^n unioned with an (n-1)-sphere at infinity) the transformation wouldn't be onto. In other words, if a transformation is a bijection on R^n and maps lines to lines, it must needs to map parallel lines to parallel lines.

If we're talking about non-bijections as well, then the projective transformations might be allowable.

6. Oct 29, 2006

Office_Shredder

Staff Emeritus
You can also do a reflection