Hilbert Curves, Part II

Last month, I blogged the following:

One standard to change is that of using scan lines in TV. If the electron guns scanned the tube in a Hilbert curve, either the broadcaster or the receiver could double the potential resolution while remaining compatible with the old resolution at the other end.

Since that isn't very understandable to non-mathematicians, I'll try to translate it into English.

In the normal scan-line method, the electron beam will go to the end of a line before starting on the next. As a result, if the transmitter doubles its resolution, it will go through each line in half the time. That means the transmitter will broadcast the far edge of screen while the receiver is ready for middle. The receiver will show two copies of the screen, each half width.

If the screen is scanned by a Hilbert curve, the transmitter will broadcast a pixel from near the point the receiver is prepared for even if the resolution doubles. Doubling the resolution will not produce any large-scale distortion.

When the resolution is doubled, the order of the curve increases by one. Each order Hilbert curve (the top illustration here shows the first few orders of the Hilbert curve) has double the resolution of the preceding one. The curves from the second order onward will go through each quarter of the square in the same order; the curves from the third order onward will go through each sixteenth of the square in the same order; etc.