Sturmian word

Sturmian word
You are encouraged to solve this task according to the task description, using any language you may know.

A Sturmian word is a binary sequence, finite or infinite, that makes up the cutting sequence for a positive real number x, as shown in the picture.

Example Sturmian word when x = 0.618..., the golden ratio.

The Sturmian word can be computed thus as an algorithm:

• If $x>1$, then it is the inverse of the Sturmian word for $1/x$. So we have reduced to the case of $0<x\leq 1$.
• Iterate over $floor(1x),floor(2x),floor(3x),\dots$
• If $kx$ is an integer, then the program terminates. Else, if $floor((k-1)x)=floor(kx)$, then the program outputs 0, else, it outputs 10.

The problem:

• Given a positive rational number ${\frac {m}{n}}$, specified by two positive integers $m,n$, output its entire Sturmian word.
• Given a quadratic real number ${\frac {{\sqrt {a}}+m}{n}}$, specified by three positive integers $a,m,n$, where $a$ is not a perfect square, output the first $k$ letters of its Sturmian word when given a positive number $k$.

(If the programming language can represent infinite data structures, then that works too.)

Stretch goal: calculate the Sturmian word for other kinds of definable real numbers, such as cubic roots.