In fact, a Moore curve is a continuous fractal space-filling curve which is a variant of the Hilbert curve. Precisely, it is the loop version of the Hilbert curve, and it may be thought as the union of four copies of the Hilbert curves combined in such a way to make the endpoints coincide [Wikipedia]. Since, Moore Curve is space-filling curve therefore dimensionally it is Hausdorff dimension.

If your surface satisfies the triangle inequality, i.e. is a Euclidean space the best algorithm to record a spatial index is a space filling curve or a monster curve. It reduces the 2d problem to a 1d problem and discretised and solves the address of the space. The lookup up for similar locations is something like O(log(n)). You can find a good article at http://blog.notdot.net/2009/11/Damn-Cool-Algorithms-Spatial-indexing-with-Quadtrees-and-Hilbert-Curves

http://architects.dzone.com/articles/algorithm-week-spatial

You can do it with Moore Curve. You can also do it using Wolfam math as link given below:

http://demonstrations.wolfram.com/HilbertAndMooreFractalCurves/

In addition to  Prof. Khan's incisive observations about Moore curves, see

J. Lawder,  The Application of Space-Filling Curves to the Storage and Retrieval of Mullti-Dimensional Data, Ph.D. thesis, University of London, 1999:

http://www.dcs.bbk.ac.uk/~jkl/thesis.pdf

For a discussion about Moore curves (section4.3.5.2, starting on page 68) as well as Hilbert curves (section 4.3.7.2, starting on  page 74), see Chapter 4 of the thesis.

Mapping algorithms are given in great detail in chapter 5, starting on page 81.

Another good place to check in this thesis is Appendix B (The Hilbert curve, starting on page 180) and Appendix C (Moore's Curve: Our Variation), starting on page 194.

Thank you James.

Thank you i will check it out.