With what analytic functions can one construct the $(x,y)$ coordinate axes using a straightedge and a compass?

Given the graph of $y = \frac{1}{x}$, construct the $(x,y)$ coordinate axes using a straight edge and a compass.

The solution to this problem is known (mouse over the spoiler text below for a hint).

Straight edge and compass as in classic Euclidian geometry. There is no way to measure distance. The $y = \frac{1}{x}$ function has an interesting property: if you choose one part of the graph (say $x>0$) and have two distinct parallel segments on it, the line that goes through the midpoints of those segments has to pass through the origin. After you’re able to show that, it’s not hard to figure out the rest.

What other analytic functions can one substitute for $y = \frac{1}{x}$, and still be able to do so?

Solution 1:

Draw a chord between any two points on one branch of the hyperbola. Choose another point on that branch and draw a chord parallel to the first chord. Bisect each of these chords. The line through these midpoints passes through the origin of the coordinate system. For a proof, see:

P. Pinkerton, "The Asymptotes of the Hyperbola," Mathematical Notes, Volume 6, December 1910, pp 63-65 Link to this article: http://journals.cambridge.org/abstract_S1757748900000761.

This line cuts both branches of the hyperbola, say at $B_1$ and $B_2$. Now bisect the line connecting these cut points, i.e., $B_1B_2$, at $O$. This is the center of the coordinate system.

Through this point $O$ draw a perpendicular line. Bisect the angle between line $B_1B_2$ and the perpendicular. This is one axis of the hyperbola. The other is perpendicular to it.