Abyss and Game of Polynomials and Coefficients : Algebraic Cubic And Elliptic Curves

In this paper, an important novel study with interesting conclusions on algebraic cubic curves and elliptic curves is being presented. Various popular algebraic cubic curves and elliptic curves viz., Ochoa curve, Mordell curve, projection, birational transformation, Conchoid of de Sluze, Maltese cross curve, Semicubical Parabola, Tschirnhausen Cubic, Right Strophoid, Strophoid, have been studied to find exhaustively which could resemble with the same shape but with a different degree of polynomial and coefficient. The affect of higher order polynomials and the accuracy obtained with RMSE and interpolation has also been studied and in total 392 plots are submitted with equations. 

The manuscript has three novelties - different curves obtained by changing coefficients, the solutions in both x and y are given and the roots which are obtained from the algebraic curve equation, are again interpolated with different degrees viz., degree 2, 4, 6 and 10, with root mean square error.