High level continuity for coordinate generation with precise controls

Coordinate generation techniques with precise local controls have been derived and analyzed for continuity requirements up to both the first and second derivatives, and have been projected to higher level continuity requirements from the established pattern. The desired local control precision was obtained when a family of coordinate surfaces could be uniformly distributed without a consequent creation of flat spots on the coordinate curves transverse to the family. Relative to the uniform distribution, the family could be redistributed from an a priori distribution function or from a solution adaptive approach, both without distortion from the underlying transformation which may be independently chosen to fit a nontrivial geometry and topology.