Surfaces in computer aided geometric design: a survey with new results
Computer Aided Geometric Design
C2 interpolation over hypercubes
Computer Aided Geometric Design
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Multivariate interpolation problems, which occur often in scientific research, can sometimes be approached using Coons' patches. Coons' patches are smooth, local interpolants to lower dimensional data sets. (For example, they interpolate curves of data when considering 3-dimensional problems.) However, they do not interpolate as desired unless all the mixed partial derivatives, the 'twists', are equal. The twists are not equal in many cases of practical importance, such as for wire frame data. Gregory (1983) has developed a compatibly corrected Coons' patch for 3-dimensional surfaces. We generalize this interpolant, 'Gregory's square', to the case of functions of n variables. The interpolant we propose is built up inductively from one to n-dimensions, requiring, at each step, only one additional term to be defined. This is the key to the whole process and involves the definition of a general 'twist operator'.