In structurally deformed areas, fold geometry usually varies from point to point ranging from parallel folds to those with thickened hinges. The latter type of folds (i.e. thickened, or T-folds) are believed to form by the superimposition of homogeneous ductile strain during progressive deformation. The geometry of the various folds thus formed appears to be the result of the cumulative effect of, amongst others, both ductile strain and the rheology of the folded material. Understanding of the role of ductile strain and rheology on fold geometry is rather easier for experimentally deformed folds wherein we can “see”all the stages of fold development. In naturally deformed folds, on the other hand, this is not possible and one has to apply mathematical treatment/analysis of the data on fold geometry. Thus, in order to understand how ductile strain and rheology control the shapes of natural folds, a mathematical treatment of a polynomial equation for natural folds has been done in the paper. Since for a polynomial equation, it is very difficult, at times even practically impossible, to understand the functional relations properly, it would be easier to do so if the polynomial is transformed to some linear form. This has been done in the paper by applying suitable transformations to a polynomialy=1+AB^x/x
such that it has been brought down to some simpler derivative/equivalent in a linear form of the type y′=a+px′
The rheological implications of the linear relation, vis-a-vis its polynomial parent, have been interpreted and the role of ductile strain and rheology on the overall shape/geometry of the natural folds during various stages of folding has been discussed.

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The mathematical function for quantiative description of zircon crystals was incorporated into the Computerized Image-Prossesing System for Fission Track Dating (LIPS-FTD) . This function enables the crystal form measurement and description of the zircon morphology and is useful in fission track dating. The mathemitical function was used to measure the zircon crystal form and the Pyramid and Prism Indices were compared with the values of the zircon morphological classification diagram by Pupin. Further, the Pyramid and Prism indices of zircon samples from the Greenhorn Formation in Utha, USA, were determined and compared with INDICES A and T determined from the Pupin's diagram. Based on the enhancement of the CIPS-FTD, detrital zircon in acidic tuff beds could be identified.

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Discontinuity is considered in the gridding algorithm of geologic surfaces by breaking the linkage between grid nodes which cross the line of discontinuity in the similar manner to Bolondi et al. (1976) . Shiono et al. (1986, 1987) proposed a gridding algorithm for numerical determination of the optimal continuous surface: z=f (x, y) .In this algorithm, the smoothest function is chosen among many feasible functions which satisfy elevation data as well as dip and strike-data, using a functional:
J (f) =m₁J₁ (f) +m₂J₂ (f)
as a measure of smoothness of the function f (x, y) . Applying Bolondiet al. (1976) 's idea to the algorithm, Noto et al. (1988) developed a N₈₈-BASIC program for gridding of a faulted surface, in which discontinuity of surface is realized by neglecting terms of J₁ (f) and/orJ₂ (f) related to neighboring nodes located on both sides of the line of discontinuity. However, the program has some limitations due to its restricted use of computer memory and low processing speed. Expanding the algorithm proposed by Noto et al. (1988), we deveolped a FORTRAN program to determine optimal shapes of not only discontinuous surfaces but also surfaces including discontinuity of partial derivative of first order. This program realizes both rapid processing and expansion of gridding area.

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