Investigation of Planar Algebraic Curves using Projective Geometry and Computer Algebra Methods

Abstract

This study investigates planar algebraic curves using a combined framework of projective geometry and computer algebra methods to address persistent theoretical and computational challenges. Although algebraic curves—defined by polynomial equations in two variables—are foundational to classical geometry, there remains a significant knowledge gap in integrating symbolic computation with geometric intuition, particularly in the context of high-degree curves and singularities. By transforming affine equations f(x,y)=0 into their projective counterparts F(x,y,z)=0, the study enables the analysis of curves at infinity and resolution of singularities through blow-up techniques and intersection theory. Symbolic tools, including Gröbner bases, discriminants, Hessians, and resultants, are employed using systems like Mathematica and Singular to identify singular points, compute multiplicities, and classify curve types. Results confirm the effectiveness of this approach, such as successfully converting rational parametric curves into implicit forms and resolving complex intersection behavior. Findings highlight the computational benefits and theoretical clarity offered by projective embeddings and symbolic algorithms. The implications extend to geometric modeling, robotics, and computer-aided design, while future research is encouraged in birational classifications, moduli spaces, and hybrid symbolic-numerical methods. This research contributes to bridging the gap between abstract algebraic theory and practical algorithmic implementations.