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How did Euler and other mathematicians develop analytic geometry in three dimensions?

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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).

Although René Descartes and Pierre de Fermat initially proposed using three coordinates to study spatial geometry, the development of three-dimensional analytic geometry remained limited until approximately 1730.1

During this period, Leonhard Euler, Jakob Hermann, and Alexis Clairaut began establishing general equations for surfaces such as cylinders, cones, and surfaces of revolution. For instance, Euler and Hermann demonstrated that the equation $f (z) = x^{2} + y^{2}$ describes a surface generated by revolving a curve about the z-axis.1

In 1748, Euler further advanced the field by utilizing rotation and translation equations to simplify the general quadric surface equation, aligning its principal axes with the coordinate axes. Alongside Joseph-Louis Lagrange and Gaspard Monge, Euler was instrumental in making analytic geometry a field independent of synthetic geometry. Additionally, Clairaut and François Nicole proved that all plane cubics could be derived as sections of specific cubical cones, bridging the gap between algebraic forms and spatial geometry.1