What is the Hilbert curve's equation?

Question

The Hilbert curve has always bugged me because it had no closed equation or function that I could find. What is its equation or function? For example, if I wanted to find the Hilbert's curve point at 4/7, how would I find it?

Answer 1

As pointed out in this other answer, there is a formula for Hilbert's space filling curve in Space-Filling Curves by Hans Sagan. The formula below appears as formula 2.4.3 on page 18 of the text.

If we first write $t\in[0,1)$ in its base four expansion, $$t=0_{\dot 4}q_1q_2q_3\ldots,$$ then $$h(t) = \sum_{j=1}^{\infty} \frac{(-1)^{e_{0j}}}{2^j}\text{sign}(q_j) \left( \begin{array}{c} (1-d_j)q_j - 1 \\ 1-d_jq_j \end{array} \right), $$ where $e_{kj}$ denotes the number of $k$s preceding $q_j$ and $d_j=e_{0j}+e_{3j} \mod 2$ and $\text{sign}(x)$ is $1$ if $x$ is positive, $0$ if $x$ is zero and $-1$ if $x$ is negative (though $q_j$ is never negative).

While it takes a bit to digest, it's not too outrageously complicated and quite easy to program. If the denominator of $t$ is a power of 2, then the base four expansion will terminate and the computation is exact. For any other rational number, the base four expansion is periodic and the value can still be computed exactly using a geometric series. For irrational numbers, the sum can be truncated to obtain a good decimal approximation.

Consider, for example, your question about $t=4/7$. We have the base four expansion $$\frac{4}{7} = 0_{\dot 4}\overline{210}.$$ Now, $$ h(0_{\dot 4}210) = \left( \begin{array}{c} 1/2 \\ 1/2 \\ \end{array} \right) + \left( \begin{array}{c} 0 \\ 1/4 \\ \end{array} \right) + \left( \begin{array}{c} 0 \\ 0 \\ \end{array} \right) = \left( \begin{array}{c} 1/2 \\ 3/4 \\ \end{array} \right), $$ $$ h(0_{\dot 4}000210) = \left( \begin{array}{c} 1/16 \\ 1/16 \\ \end{array} \right) + \left( \begin{array}{c} 1/32 \\ 0 \\ \end{array} \right) + \left( \begin{array}{c} 0 \\ 0 \\ \end{array} \right) = \left( \begin{array}{c} 3/32 \\ 1/16 \\ \end{array} \right), $$ $$ h(0_{\dot 4}000000210) = \left( \begin{array}{c} 1/128 \\ 1/128 \\ \end{array} \right) + \left( \begin{array}{c} 0 \\ 1/256 \\ \end{array} \right) + \left( \begin{array}{c} 0 \\ 0 \\ \end{array} \right) = \left( \begin{array}{c} 1/128 \\ 3/256 \\ \end{array} \right), $$ and $$ h(0_{\dot 4}000000000210) = \left( \begin{array}{c} 1/1024 \\ 1/1024 \\ \end{array} \right) + \left( \begin{array}{c} 1/2048 \\ 0 \\ \end{array} \right) + \left( \begin{array}{c} 0 \\ 0 \\ \end{array} \right) = \left( \begin{array}{c} 3/2048 \\ 1/1024 \\ \end{array} \right). $$ Looking at the columns, it's not too hard to spot the pattern and we have $$ h(4/7) = \left( \begin{array}{c} 1/2 \\ 1/2 \\ \end{array} \right) \sum _{k=0}^{\infty } \frac{1}{8^k}+\left( \begin{array}{c} 1/32 \\ 1/4 \\ \end{array} \right) \sum _{k=0}^{\infty } \frac{1}{64^k} = \left( \begin{array}{c} 38/63 \\ 52/63 \\ \end{array} \right). $$

Finally, here are some approximations to Hilbert's curve obtained by passing a polygonal curve the points $h(p/2^k)$ for increasing values of $k$.

The code to produce this image can be found in this Observable notebook.

Answer 2

The expression is fairly complicated and the derivation fairly tricky. Look it up in Hans Sagan, Space-filling curves, Springer 1994.