VSFCF  CVM 1.1 
Figure 12. Curves of dimension two increasing in area from zero to two as they converge to the Hilbert curve. Also available are an animation (121kb) and a speed controlled version. Use the browser back button to return. Source code is available.
One goal of this document is to try to better visualize spacefilling 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 spacefilling curve maps a subset of R^{1} to a subset of R^{2} we can naturally view the graph as a subset of R^{3}.
In Figure 13 we look at such a graph from several vantage points. The usual xaxis is taken to be the independent axis, while the yzplane 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 xaxis piercing the surface at latitude 0 longitude 0, the positive yaxis piercing at latitude 0 longitude 90, and the positive zaxis 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 3d 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 R^{3} that projects to a space filling curve (in particular, the image contains a disk) in one direction. In fact, there are simple curves in R^{3} whose projections contain disks in all directions (see [Me, Corollary 4]).

