Rotating a Hyperbola

The equation of a hyperbola gives it some very interesting properties under a series of stretches. Consider the hyperbola $xy=1$: If we stretch the hyperbola horizontally by a factor of 2 about the x axis — replacing $x$ in the equation $xy=1$ with $\frac{x}{2}$, we have the equation $xy=2$.

Now, if we compress the resulting hyperbola vertically by a factor of 2 — replacing $y$ in the equation $xy=2$ with $2y$ and simplifying, we get the original hyperbola, $xy=1$. So by a horizontal stretch followed by a vertical compression, we get the original hyperbola.

However, consider a point on the hyperbola $xy=1$, say $(x,y)$. Stretching the hyperbola horizontally by a factor of 2, this point gets transformed to $(2x,\frac{y}{2})$. Even though each point has a different image point, however, the graph as a whole remains identical.

More generally, if we stretch by a factor of $k$ instead of 2, where $k$ is any positive number, we still obtain the original hyperbola after the two stretches. The point $(x,y)$ gets transformed to the point $(kx,\frac{y}{k})$. We call this combinations of transformations a hyperbolic rotation. Intuitively, the points on the hyperbola are ‘rotated’ downwards.

There are some interesting things about the hyperbolic rotation. For one, by choosing the right $k$ for the transformation, it is possible to transform any point on the hyperbola into any other point on the same arm of the hyperbola (by allowing $k$ to be smaller than 1). Also, since the hyperbolic rotation is composed of two simple stretches, parallel lines as well as ratios of a line segments are preserved. The area of a figure is also preserved since one stretch reduces the area, while the other multiplies the area by the same amount.

With these facts, we can prove things about the hyperbola itself:

Theorem: Let P be a point on the hyperbola $xy=1$. Let $l$ be the line passing through P tangent to the hyperbola. Let X be the intersection of $l$ and the x-axis, and Y be the intersection of $l$ with the y-axis. Prove that P is the midpoint of XY. The proof is very simple with the hyperbolic rotation. Consider when $P = (1,1)$. The theorem obviously holds here, because of symmetry. But by applying a hyperbolic rotation, we can transform the point $(1,1)$ into any other point on the hyperbola. Since ratios between line segments don’t change with a hyperbolic rotation, P is always the midpoint, and we are done.