The main idea of this paper is to provide an insight to the reader and create awareness on some of the basic Curve Fitting techniques that have evolved and existed over the past few decades. A critical survey has been done on the various Curve Fitting methodologies proposed by various Mathematicians and Researchers who had been working in the fields of Computer Graphics, Image Processing and Pattern Recognition etc. to name a few over the past. Most of the methodologies that have evolved over time have their core structure derived from few significantly basic concepts and it is these basic concepts that have been tried to grasp through this survey and its tried to present the same as the essence of the survey. Note that the same physical variables may be named differently under different methodologies. The purpose was to maintain the originality of the work published by the original authors and this survey to only report their work as it is.