25 Dec 2008

Parabola Obtained from Ellipse through a Limiting Process, in Hilbert and Cohn-Vossen's Geometry and the Imagination

[The following is largely quotation. It offers further explanation for the geometrical example in Leibniz' letter on his law of continuity.]

Parabola Obtained from Ellipse
through a Limiting Process


Hilbert and Cohn-Vossen

Geometry and the Imagination

We first define a parabola:

A parabola is the set of all points in a plane equidistant from a fixed point F and a fixed line L in the plane. The fixed point F is called the focus, and the fixed line L is called the directrix. A line through the focus perpendicular to the directrix is called the axis, and the point on the axis halfway between the directrix and focus is called the vertex.


An ellipse is the set of all points P in a plane such that the sum of the distances of P from two fixed points in the plane is constant. Each of the fixed points, F ' and F, is called a focus, and together they are called foci. Referring to the figure, the line segment V 'V through the foci is the major axis. The perpendicular bisector B'B of the major axis is the minor axis. Each end of the major axis, V ' and V, is called a vertex. The midpoint of the line segment F ' F is called the center of the ellipse. (789)

We can imagine, then, that if we move F away from F ', the ellipse will gradually become more and more like a parabola.

We turn to Hilbert and Cohn-Vosson's Geometry and the Imagination for specifically how this is done.

[The following is quotation.]

We may obtain a parabola from an ellipse by means of a limiting process.

To this end, we fix one focus, say F1, and the vertex S nearest to this focus (where the vertices of the ellipse are defined to be the two points of intersection of the ellipse with the line joining the foci). Let us consider the ellipses that result when the second focus F2 keeps moving further and further away from F1 on the extension of the line SF1. These ellipses approach a limiting curve, and this is the parabola.

Parabola image from:

Parabola and Ellipse text, and Ellipse image from:

Barnett, Ziegler, and Byleen. Precalculus: Functions and Graphs. New York: McGraw Hill, 2001.

Hilbert, David, and S. Cohn-Vosson. Geometry and the Imagination. AMS Bookstore, 1999.
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