I am a 2nd year undergraduate doing a BS-MS in an Indian Institute and am taking a course in Elementary Differential Geometry (studied till the basics of surfaces). Our instructor has assigned a term-project on fractals and space filling curves. One of the things he has asked us is, to prove that the Hilbert's curve is parametric and write down it's parametric equations.
I tried to google this but got no information on this topic. So could anybody give some idea as to how to do this not only for the Hilbert's curve but for any fractal or space filling curve?
Your help will be highly appreciated.
Hint Construct the Hilbert $H(t)$ curve piecewise out of smaller copies of the Hilbert curve:
$$ H(t) = \begin{cases} f_1(H(4t)) & 0 \le t \le \frac{1}{4} \\ f_2(H(4t-1)) & \frac{1}{4} \le t \le \frac{1}{2} \\ f_3(H(4t-2)) & \frac{1}{2} \le t \le \frac{3}{4} \\ f_4(H(4t-3)) & \frac{3}{4} \le t \le 1 \end{cases} $$
Geometric transformations $f_k$ are left as an exercise. You may need to prove continuity at the transition points.