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How did Pierre de Fermat contribute to analytic geometry?

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Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated. — analytic geometry The conic sections result from intersecting a plane with a double cone, as shown in the figure. There are three distinct families of conic sections: the ellipse (including the circle), the parabola (with one branch), and the hyperbola (with two branches).

Pierre de Fermat is recognized as one of the two primary founders of analytic geometry, having discovered its fundamental principles independently of René Descartes.12

Fermat’s contribution was rooted in his efforts to reconstruct the lost Plane Loci of Apollonius. By 1636, in his paper Ad Locos Planos et Solidos Isagoge ("Introduction to Plane and Solid Loci"), he demonstrated that the study of loci—sets of points with specific characteristics—could be facilitated by applying algebra to geometry through a coordinate system.23

His work established the fundamental principle of analytic geometry: that an equation involving two variable quantities defines a specific curve or locus in space. Specifically, Fermat showed that quadratic equations correspond to conic sections, such as the ellipse, parabola, or hyperbola. Additionally, his analytical methods for determining tangents to curves and identifying their maximum and minimum points were significant precursors to the development of differential calculus.234