On 1/19/07, Jason Holt <jason@lunkwill.org> wrote:
Or say you're mapping out things you find at an archaeological dig. Create an X and Y axis at your site, and when you want to find the placement of an item, have a friend walk along each axis holding ends of a tape measure, with you holding it in the middle. When you read half the value seen by the friend holding the other end, they record their positions on the axes. (Or, in fact you could also record [a,b,r] for any line passing through the item, (a,0), (0,y) and showing a distance of r on the tape).
I don't think I'm deliberately trying to hijack this discussion, to go banging on again about one of my hobby-horses ... But surely a better solution to this engineering problem would be to use distance geometry. Establish base points B_i for i = 1,2,3,4 at known locations on your site --- not concyclic and no three collinear --- for example, at the corners of a square enclosing it. To assign coordinates to a point X, measure the distances d(X, B_i) to the base points. The vector of squared distances [d(X, B_i)^2] will be a linear combination of the four vectors [[d(B_k, B_i)^2], for k = 1,2,3,4. The coefficients of this combination, obtained by multiplication by the inverse of this last matrix, give you coordinates for the point. These coordinates are a special case of classical (general) plane tetracyclic coordinates: a second linear transformation recovers the special tetracyclic form more usually employed, whose first two components are (ta-DAH) the Cartesian coordinates of the point. Under it, coordinates may be assigned to circles and lines as well as points; the linear transformations include the conformal Moebius group. However, there is now only a single inversive element at infinity; projective transformations other than similarities are no longer available. Finally, adding a fifth component --- the radius --- yields the still more general Lie system, the linear transformations of which include "equilong" Laguerre transformations as well as Moebius and Euclidean. Because of the redundancy, these systems are more robust than the standard Cartesian, special projective homogeneous, or barycentric options: for instance, small errors in pacing out the distances to your base points would result instead in a circle, whose radius estimates the uncertainty. These coordinate systems and their associated symmetry groups deserve to be better known. For instance, a spectacular member of the 3-D Laguerre group is the "offset" transformation computing the path of a milling machine cutter required to form a given contour. Fred Lunnon