**ABSTRACT: Computer-Aided Geometric Modeling of Geologic Surfaces**

**H. Schaeben, S. Auerbach**

Polynomial B-splines associated with irregular triangles provide a method combining all the properties required by geometric modeling of geologic surfaces and bodies: (1) triangles are adjustable to the spatial distribution of data with respect to their abscissae; (2) data providing positional ( x, y, z) information can be processed; (3) data providing information on any directional derivatives can be processed when available or accessible (however, they are not necessarily required); (4) data providing information on the geometry to be recovered/modeled that can be formalized into discontinuities of the function itself or any of its directional derivatives can be processed; (5) given a topology consistent with the data, i.e., an approximate parameterization of the geomet y to be recovered/modeled, geometries can be represented which cannot be represented by functions; and (6) local control of the geometry is provided by corresponding control points establishing the convex hull, making interactive computer aided geometric modeling possible.

Therefore, bivariate quadratic simplex B-splines defined by their corresponding set of knots derived from a constrained suboptimal Delaunay triangulation of the domain are employed to obtain a C^{1} smooth surface. The generation of triangle vertices is adjusted to the areal distribution of the data in the domain. We emphasize here that the vertices of the triangles initially define the knots of the B-splines and generally do not coincide with the abscissae of the data.

Applications of this approach to least squares approximation of geoscientific data will be presented as examples.

AAPG Search and Discovery Article #91003©1990 AAPG Annual Convention, San Francisco, California, June 3-6, 1990