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How did Johannes Kepler use an ellipse to describe planetary motion?

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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).

Johannes Kepler used the ellipse to describe planetary motion by empirically analyzing the precise observational data of Mars recorded by Tycho Brahe.12

In his first law of planetary motion, published in Astronomia Nova (1609), Kepler concluded that planets do not travel in perfect circles, as previously theorized, but in elliptical orbits with the Sun positioned at one of the two foci.132

Mathematically, an ellipse is defined as the locus of points where the sum of the distances to two fixed points (the foci) remains constant. By applying this geometry, Kepler accounted for the varying distances between a planet and the Sun throughout its orbit. This model, characterized by the eccentricity of the orbit, allowed him to accurately match Tycho’s observations, which circular models could not explain. This shift from circular to elliptical paths was a foundational development in celestial mechanics, later providing the basis for Isaac Newton’s law of gravitation.24135