I have no idea where the idea for this came from. I think I was just thinking about designs with level curves... maybe from watching the Vi Hart on spirals recently.
In this sketch, the blue curve is parameterized by two functions of t. The red curves are spreading out at distance 1 from each previous. n = 4 makes four curves out on both sides.
Some values to try:
x(t) = t, y(t) = cubic.
x(t) = t, y(t) = 2*cos(t)
x(t) = sin(2t), y(t) = cos(t)
x(t) = sin(t), y(t) = 1.5^t
What crazy curves can you make? Are any of the designs pretty?
Note: the sketch can be slow - GeoGebra is doing a lot of computation! If it doesn't work in browser, you might have to download from the "teacher page." Click "Created with GeoGebra" at the bottom of the window.