What fundamental connection between equations and curves did Pierre de Fermat establish?

Pierre de Fermat, alongside René Descartes, established the fundamental principle of analytic geometry, which posits that algebraic equations in two variables can be used to define and represent plane curves.12

By applying algebra to geometry through a coordinate system, Fermat demonstrated that the properties of geometric loci—sets of points with specific characteristics—could be analyzed and solved using algebraic techniques. This breakthrough allowed for the systematic study of curves, such as parabolas and hyperbolas, by expressing them as polynomial equations. This connection between algebra and geometry provided the essential foundation for the later development of calculus, enabling mathematicians to determine tangents, maxima, and minima for a vast class of algebraic curves in a routine manner.123