Transformations taking straight lines to straight lines

[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?

 

[pmb_phy]

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?

I believe so.

Best wishes

Pete

 

[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.

 

[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

 

[Doodle Bob]

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.

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.

 

[Office_Shredder]

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

You can also do a reflection