Short answers, pulled from the story.
René Descartes published the essay La Géométrie in the year 1637 as an appendix to a philosophical treatise on method. The work was initially rejected by the academic community for its gaps in logic and its use of the French language rather than Latin. It took over a decade and a translation by Frans van Schooten in 1649 before the world recognized the Cartesian coordinate system.
The Greek mathematician Menaechmus active in the 4th century before the common era solved problems using a method resembling coordinate geometry. Apollonius of Perga developed a system in his work On Conics that utilized reference lines and tangents to describe curves. The 11th-century Persian mathematician Omar Khayyam identified the foundations of algebraic geometry and laid down principles transmitted to Europe.
Pierre de Fermat began with an algebraic equation and then described the geometric curve that satisfied it. René Descartes began with geometric curves and produced their equations as properties of those curves. Fermat's work was not published in his lifetime and it was only after his death that his contributions were recognized.
The Cartesian coordinate system is a grid where every point in a plane is defined by an ordered pair of real numbers. In two dimensions a point is represented as x and y coordinates while three-dimensional space adds a z-coordinate to describe height. This system allows mathematicians to translate geometric shapes into algebraic equations and calculate distance and angle using formulas derived from the Pythagorean theorem.
A linear equation like y equals x describes a straight line while quadratic equations define conic sections such as ellipses parabolas and hyperbolas. In three dimensions a single equation typically defines a surface such as an ellipsoid paraboloid or hyperboloid. Quadric surfaces are defined as the locus of zeros of a quadratic polynomial in three variables and include cylinders cones and planes.
Analytic geometry is the foundation of most modern fields of geometry including algebraic differential discrete and computational geometry. It is indispensable in physics aviation and spaceflight for precise calculations required for global positioning systems and computer graphics. The legacy of René Descartes and Pierre de Fermat lives on in every equation that describes the shape of the universe from the orbit of a planet to the curve of a satellite dish.