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When Nine Points Are Worth But Eight: Euler’s Resolution of Cramer’s Paradox - The Conic Sections

Robert E. Bradley (Adelphi University) and Lee Stemkoski (Adelphi University)

The two curves we have just considered – the circle and the parabola – are special cases of the conic sections.

A conic section is a curve obtained by the intersection of a plane with the surface of a (double-napped) cone, as shown in Figure 4. When the plane is parallel to the edge of one cone, the intersection is a parabola. When the plane and cone intersect in a closed curve, the result is an ellipse; in the special case where the plane is perpendicular to the axis of symmetry of the cone, the ellipse is actually a circle. When the plane and cone intersect in two curves, the result is a hyperbola.

Figure 4. The Conic Sections: (1) Parabola, (2) Ellipse and Circle, (3) Hyperbola. (This image is in the public domain.)

If the plane intersects the vertex of the cone, then the result will be one of three "degenerate" cases: a pair of intersecting lines, a single line, or a single point.

The conic sections were first studied by mathematicians of ancient Greece. Pappus credited Euclid with writing four volumes on conic sections (which have been lost to history), and Apollonius with completing these volumes and writing an additional four. In Conics, Apollonius proved that two (distinct, nondegenerate) conic sections intersect in at most four points [Katz 2008, p. 122]. Therefore, four points do not uniquely determine a conic section, as illustrated in the applet in Figure 5.

Figure 5.  Multiple conics passing through four points. Move points \(A,B,C,\) and \(D\) to explore the possibilities. (Interactive applet created using GeoGebra.)