Analysis and synthesis of interpolating functions for 3-D objects

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Interpolation technique has been used for generation of three dimensional surfaces. This paper first analyses the objects generated by specified interpolating functions. Both linear and non-linear interpolating functions are used to generate interesting objects. None of these objects can be generated using the technique of surface of revolution. The paper discusses various choices for the interpolating weighting functions and introduces the idea of modifying functions to incorporate second order of non-linearity. Several interesting objects, such as those resembling chess pieces, vase, goblets, etc., are presented. The main feature of the generated objects is that they have a compact mathematical representation. Subsequently, we consider the reverse process where the characteristics of the desired shape are given and the interpolating functions are to be determined. The Fourier series expansion is used to determine an interpolating function which gives rise to the desired object using non-linear interpolation. This synthesis process is illustrated with several examples. CR Categories and Subject Descriptors: G.1.1 [Numerical Analysis]: Interpolation - Linear and Nonlinear, Fourier Series Analysis, I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling - curve, surface, sold and object representation. General Terms: Analysis and Synthesis, Generation of 3-D Objects. Additional Keywords and Phrase: inbetweening, blending and modifying functions.