Orthogonal grids on meander-like regions: A database approach

We deal with the automatic construction of orthogonal grids on meander-like regions. These are similar to riverbeds and appear naturally in geography (rivers, regions between two fixed heights in a hilly landscape) and medicine (longitudinal slices of organs such as veins and arteries). We use pairs of lemniscates of two foci and pairs of ellipses to approximate subregions of the meander-like region. A subregion determines an approximating lemniscatic or elliptic sector given by two confocal lemniscates or two ellipses from a certain family and, in both cases, two orthogonal hyperbolas. We use the fact that a sector is conformally equivalent to a region bounded by two concentric circles and two rays, to map the orthogonal grid on this disc domain, given by polar coordinates, into a grid in the approximating sector. Finally the lemniscatic and elliptic sectors that approximate the subregions are assembled to conform an orthogonal grid on the meander-like region.