Interpolation is a fundamental process in scientific visualization.
It is the process by which we create an empirical model of the phenomenon we wish to visualize, from a set of data samples. This thesis looks at ways by which the interpolant can achieve certain properties. The properties considered include satisfying simple bounds - such as positivity continuity of derivatives interpolation of derivative values concentrating mainly on the first property. The thesis concentrates on gridded data for surfaces and volumes, but 2D scattered data is briefly discussed in the conclusion. In the polynomial methods we study the linear and cubic approached.