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Figure 12. Curves of dimension two
increasing in area from zero to two as they converge to the Hilbert curve.
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One goal of this document is to try to better visualize space-filling curves by "spreading out" the curve using approximations. Another approach is to spread them out using a third dimension to actually draw the graph. Since a space-filling curve maps a subset of R1 to a subset of R2 we can naturally view the graph as a subset of R3.
In Figure 13 we look at such a graph from several vantage points. The usual x-axis is taken to be the independent axis, while the yz-plane represents the dependent coordinates. To get a better view of the curve it is stretched by a factor of four in the x direction. So we have x values from 0 to 4 while the y and z values go from 0 to 1. Consider then this graph lying at the center of the earth (all angle measurements will be in degrees) with the positive x-axis piercing the surface at latitude 0 longitude 0, the positive y-axis piercing at latitude 0 longitude 90, and the positive z-axis piercing at the north pole. We then take a long walk looking at the fixed curve from different perspectives. (Alternately, one could consider this as rotating the curve.) Our walk has four parts:
Figure 13. Walking around the 3-d graph of the classical Hilbert curve. First is a north pole view, and second is a view from latitude 30 longitude 120. Several other viewing options are available: an animated version, a speed controlled version, a low resolution VRML version (600kb), and a high resolution VRML version (2500kb). Use the browser back button to return. Not all browsers will support VRML. Source code is available.
Note that we have a simple curve in R3 that projects to a space filling curve (in particular, the image contains a disk) in one direction. In fact, there are simple curves in R3 whose projections contain disks in all directions (see [Me, Corollary 4]).