The Theory of Conics

Conics are curves obtained when a cone is intersected by a plane in a certain way.  They include ellipses, circles, parabolas and hyperbolas. 

Fig. 26 – An ellipse is formed when the cone is intersected by a plane.

Conics belonged almost exclusively to the world of learned mathematics, and treatises before the sixteenth century rarely mentioned them as they were considered difficult subjects.  A rare treatise on conics was written by Albrecht Durer, entitled “Treatise on measurement with compasses and straightedge (Underweysung der Messung mit dem Zirkel und Richtscheyt, Nuremberg, 1525).

Johannes Kepler (1571 – 1630) was one of the first to give a greater degree of unity to the theory of conics.  He developed the idea that the properties of conics will change slowly from one type to the next (Fig. 27).  For instance, as the foci move closer together an ellipse gradually becomes a circle.  Then as the ellipse gets longer and its foci moves further apart, it gradually becomes a parabola.

 

Fig. 27 – A plane system of conic sections by Johannes Kepler.
 

Giovanni Battista Benedetti (1530 – 1590) further proposed that if a cone is cut by two planes parallel to one another, then the two conic sections will be similar (Fig. 28).

 

Fig. 28 – Two parallel sections yield similar ellipses.

Specifically, Fig. 28 shows ellipses with the same eccentricity.  Benedetti also proved that a plane intersecting two cones with different vertical angles will yield two different conics.