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How did Fermat show that an equation involving two variables can determine a curve?

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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 established the fundamental principle of analytic geometry by demonstrating that an equation involving two variables defines a geometric locus, or curve.12

In his work Ad Locos Planos et Solidos Isagoge (1636), Fermat utilized a coordinate-like approach to bridge algebra and geometry. He considered two variable quantities: one representing a line measured horizontally from a fixed initial point, and a second line positioned at the end of the first, inclined at a constant angle. As the first variable changed in magnitude, the second variable took on a value determined by the algebraic equation, causing the endpoint of the second line to trace a specific curve in space.2

Through this construction, Fermat formulated the principle that whenever two unknown quantities are found in a final equality, their relationship determines a fixed locus, with the endpoint of one variable describing a straight line or a curve. He further showed that quadratic equations correspond to conic sections—such as the ellipse, parabola, or hyperbola—and that these curves could be simplified through a change of variables.2