Linear fractional transformations look like $$f(x) = \frac{ax+b}{cx+d}$$ for some real constants $$a,b,c,d$$. Perhaps I can convince you they are fractional, but how are they linear? Considering the simplest example, how could something like $$f(x) = 1/x$$ be linear? We’d try to show that to some extent, it is linear. The main idea is that we embed the real line into the cartesian plane and cheat a bit by using projections.

Consider the transformation $$r: (x,y) \to (y,x)$$. This is nothing more but reflecting along $$y=x$$, and it’s certainly linear in the sense that $$r(u+v) = r(u) + r(v)$$ for whatever coordinates $$u, v$$ we throw at $$r$$. We now embed the real line inside the plane as the set $$\{(x, 1) \forall x \in \mathbb{R}\}$$, i.e. the line $$y=1$$. Under the reflection $$r$$, we would map some point $$(c,1)$$ to $$(1,c)$$. Graphically, we are mapping the red points to the blue points as you could see below.

How does this have anything to do with $$f(x) = \frac{1}{x}$$? The trick is that for each blue point $$(1,c)$$, we draw a green line connecting it and the origin, and see where the green line intersects with the red line. We can see that that intersection point is exactly $$(1/c, 1)$$!

Secretly, we’re using two similar triangles. One has vertices $$(0,0), (0,1), (\frac{1}{c}, 1)$$ and the other has vertices $$(0,0), (0,c), (1,c)$$.

As such we have sort of been able to map $$x$$ to $$\frac{1}{x}$$, but we cheated a little by going up one dimension and doing some projection at the end. As an exercise, could we make a similar construction for $$f(x) = 2/x$$ by choosing a different transformation $$r$$? How about $$f(x) = x + 1$$? How about $$f(x) = 1 + \frac{1}{x}$$?

There’s another way of finding inverse points, often used in straighedge and compass construction, that uses circles and a pair of similar right-angled triangles. How is that method different from the above?

To make all of this formal, one needs to study a bit of projective geometry. You can read more about linear fractional transformations from its wiki.

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