Coordinate system symmetries

Since computations performed with Cactus are done on a discrete lattice, only a discrete set of coordinate values are used for any coordinate system. The symmetries of how the coordinate values vary on the grid points make coordinates fall into three types: uniform, nonuniform, and warped. (At least these are the three cases that the CoordBase schema considers.) A uniform coordinate varies from each of its neighbors by a constant. i.e. its value can be determined from the index of the grid point from simply an origin and constant `delta'. A nonuniform coordinate has a spatially varying `delta'. For both uniform and nonuniform coordinates, the coordinate values do not vary along the other directions of the grid. (e.g. the $x$ coordinate will be the same regardless of the `j' index of the grid point.) Thus one could completely determine the coordinate values of a 3D system of nonuniform coordinates by providing three 1D arrays. This later assumption is relaxed for a warped coordinate; a warped coordinate will vary across the entire grid. Recall that `coordinate lines' (lines of constant coordinate value) cannot cross (because one n-tuple of coordinate values would specify muliple points in space), so this places a `bound' of sorts on the possible `warping' of the coordinates.

The type of a coordinate system will be the same as that of its coordinates. If there are different types of coordinates within the same system, then the coordinate system is mixed. Note that a warped coordinate system is the most general possible, so any coordinate system could be regarded as warped if one wishes.