Do projections of lines which are not perpendicular correspond to FLTs?

[imurme8]

A math question about projections of lines: Say we have two straight lines which we consider as number lines ($\mathbb{R}$). I've learned that a projection of one line onto another is of the form $ax + b$ for $a,b\in \mathbb{R}$ when the two lines are parallel. If we allow the possibility that the lines are perpendicular, we have a fractional linear transformation of the form $\frac{ax+b}{cx+d}$.

Now if the lines are neither perpendicular nor parallel, do we still have a fractional linear transformation? I've been trying to build such a projection out of projections I know but with no success.

 

[pat_connell]

The answer is no, we do not have a fractional linear transformation. We only have a fractional linear transformation when the lines are either perpendicular or parallel. When the lines are neither perpendicular nor parallel, we have an affine transformation instead, which is of the form ax + by + c for a,b,c \in \mathbb{R}.