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Conic sections are an important class of curves with numerous scientific applications. A conic section is a curve obtained by intersecting a double-cone (two cones with the same axis and sharing a vertex) with a plane. There are four types of non-degenerate conic sections: circles, ellipses, parabolas and hyperbolas, as well as three degenerate cases: a single point, a line, and two intersecting lines.

Conic sections are represented algebraically by an equation of the form Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, where A, B, C, D, E, and F are real-valued constants. Two more degenerate cases may arise from this description: two parallel lines and the empty set. A proper rotation will eliminate the xy-term. There are two important quantities in this description which are invariant with respect to rotations: the discriminant, B^2 – 4AC, and the trace, A + C. The discriminant tells us what type of conic section is represented by the equation. Assuming the conic section is non-degenerate, if the discriminant is negative, the conic section is a circle or an ellipse, if the discriminant is zero, the conic section is a parabola, and if the discriminant is positive, the conic section is a hyperbola.

All non-degenerate conic sections have one or two foci. Circles and parabolas have a single focus and ellipses and hyperbolas each have two foci. All conic sections may be defined in terms of the foci. For instance, an ellipse is the set of points for which the sum of the distances to each focus is constant, and a hyperbola is the set of points for which the absolute value of the difference of the distances to each focus is constant. A parabola is also defined in terms of a line called the directrix. Specifically, a parabola is the set of points for which the distance to the focus is equal to the minimum distance to the directrix.

One of the hallmarks of Newton’s law of universal gravitation is the prediction of the shape of the orbits of planets. It turns out that their orbits are ellipses with the sun at one of the two foci. Parabolic and hyperbolic orbits are also permitted, though bodies exhibiting these orbits can only pass by the sun once. The orbits of comets are approximately parabolic.

Source by David Terr


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