I am delighted to announce that in 2022, Springer accepted my manuscript, Linear Fractional Transformations: An Illustrated Introduction, for publication in the Undergraduate Texts in Mathematics. An excerpt from the preprint of this book can be found here: LFT Chapter 2. Further details/links will be added here once they are available.
Linear Fractional Transformations is aimed primarily at an undergraduate audience, such as first-year mathematics students who wish to see some beautiful mathematics beyond the calculus/linear algebra curriculum. It is lovingly illustrated: in the final ~250-page manuscript, there are literally hundreds of figures. There were, in fact, too many illustrations and too much material in the original preprint, and almost half of it had to be cut. I am confident, however, that this material will eventually make it into future books.