All mappings from $\mathbb{R}^N$ to $\mathbb{R}^N$ where the image of any line segment is a line segment

Question

I know that a projective transformation maps a line segment to a line segment. Are there mappings other than projective transformations where the image of any line segment is a line segment in $\mathbb{R}^N$?

Thanks a lot!

Answer 1

According to the Wikipedia article on projective transformations, all collineations in real projective spaces are projective transformations. A collineation is a one-to-one mapping of lines.

You could think of a collineation as preserving collinearity of points. I.e. any three collinear points are mapped to three points that are in turn collinear. But any three collinear points sit on a common line segment, so the any map that takes line segments to line segments preserves collinearity and hence is a collineation and in turn a projective transformation.