Eccentricity - Introduction
A general definition of eccentricity for conic sections is where is the distance from any point on a conic section to a focus and is the distance from that point to the corresponding directrix. A connection that is seldom mentioned in the literature between eccentricity and the namesake of conic sections (i.e., the intersection of a plane and a cone) is explored in these two applets. Namely,
where is the angle of inclination of the plane intersecting the cone and is the base angle of the cone.
The above relationship is illustrated in the first applet which initially shows just a conic section in two forms: as the intersection of a plane with a double-napped cone and as a 2-D graph. Moving the Display slider to the right:
(1) Adds the foci and directrix lines to the 2-D figure.
(2) Adds the foci and directrix lines to the 3-D figure with the help of Dandelin spheres.*
(3) Displays the eccentricity, calculated as the ratio of distances, in the 2-D figure.
(4) Displays the eccentricity, calculated as the ratio of sines of the angles, in the 3-D figure.
The user can vary the angles of the plane and cone as well as other parameters to see that the two ratios are equal for non-degenerate conic sections.
*Note: Dandelin spheres were discovered by Germinal Pierre Dandelin in 1822. The points of tangency between the Dandelin spheres and the green plane (blue dots) are the foci of the conic section. The points of tangency between the Dandelin spheres and the cone, indicated with dashed circles, lie in the horizontal yellow planes. The intersections of the yellow and green planes (blue lines) are the directrix lines of the conic section.
The major steps in proving the above relationship are given in the second applet.