# The Geometry of the Conic Sections

## Louis A. Talman, Ph.D.

### Introduction

Let L and M be distinct lines in space, and suppose that they meet at a point P. If we rotate the line L, using the line M as the axis of rotation and keeping the angle that L makes with M constant, the surface C that we generate in this way is called a right-circular cone with vertex P and axis M. The smaller of the angles from M to L at P is called the vertex angle of the cone. We will generally orient our pictures so that the axis of our cone is vertical. Note that a cone, as we have defined it, consists of two of the surfaces that are commonly called cones—placed vertex to vertex and sharing the same axis. Any line that lies entirely in C necessarily passes through P and is called a generator of the cone.

A conic section is a curve formed when a plane intersects a cone. If the intersecting plane passes through the vertex of the cone, the intersection may be a single point, a single line, or a pair of crossing lines. Such "curves" are called degenerate conic sections; we will not concern ourselves with degenerate conic sections. We assume, instead, that our intersecting plane does not pass through the vertex of the cone.

The resulting intersection is then one of three distinct kinds of curve:

1. A hyperbola, which results when the angle made by the intersecting plane with the axis of the cone is less than the vertex angle,
2. A parabola, which results when the angle made by the intersecting plane with the axis of the cone equals the vertex angle,
3. An ellipse, which results when the angle made by the intersecting plane with the axis of the cone exceeds the vertex angle.

Our purpose on this page is to reveal some of the elementary geometry of these curves through animated depictions. All animations are in QuickTime format. (A QuickTime viewer is included in Macintosh operating systems, and one for Windoze and its successors can be obtained, free, from Apple's website.) Some of the animations on this site are a dozen or more megabytes in size, so you will need a broadband connection (or plenty of patience) to view them. Most Web browsers will optionally download these files to your hard-drive, where you can store them for use off-line. Permission is granted for non-commercial educational use; all other rights are reserved.

### Other Definitions

Some textbooks define the conic sections in other ways than we have done. In particular, many students encounter the focus-directrix definitions for all three non-degenerate conic sections. For the ellipse and the hyperbola, they often encounter definitions based on the distances from a pair of points called the foci of the curve to a point on the curve. In this section, we examine these alternate definitions visually and we show how a construction due to Pierre Germinal Dandelin can be used to show that the curves we have defined to be conic sections also satisfy the requirements of these alternate definitions. In particular, the Dandelin construction shows us how to determine foci and directrices of non-degenerate conic sections from the geeometry of cone and plane.

#### The Ellipse

Focus-directrix definition of ellipse:

Let F denote a point and let L denote a line that does not pass through F. The ellipse with focus F and directrix L is the set of all points P lying in the plane determined by F and L for which the ratio of the distances PF and PL is a certain constant e < 1. (The constant e is called the eccentricity of the ellipse; ellipses are conic sections of eccentricity less than one.)

Sum-of-the-distances definition of ellipse:

Let F and F' denote points in a plane. An ellipse with foci F and F' is the set of all points P lying in the plane for which the sum of the distances PF and PF' is a constant.

#### The Parabola

Focus-directrix definition of parabola:

Let F denote a point and let L denote a line that does not pass through F. The parabola with focus F and directrix L is the set of all points P lying in the plane determined by F and L for which the distance PF is equal to the distancePL. (Parabolae are conic sections of eccentricity 1.)

#### The Hyperbola

Focus-directrix definition of hyperbola:

Let F denote a point and let L denote a line that does not pass through F. The hyperbola with focus F and directrix L is the set of all points P lying in the plane determined by F and L for which the ratio of the distances PF and PL is a certain constant e > 1. (The constant e is called the eccentricity of the hyperbola; hyperbolae are conic sections of eccentricity less than one.)

Difference-of-the-distances definition of hyperbola:

Let F and F' denote points in a plane. A hyperbola with foci F and F' is the set of all points P lying in the plane for which the magnitude of the difference of the distances PF and PF' is a constant.

### The Dandelin Construction

In 1822, the Belgian geometer Pierre Germinal Dandelin published a construction that can be used to show that the curves which we have called conic sections meet the criteria of the other definitions we have just seen for ellipses, parabolae, and hyperbolae. Dandelin observed that when a plane passes through a cone, but not through its vertex, it is always possible to inscribe one or two spheres in the cone so that they are tangent to the cone along a circle and also tangent to the intersecting plane at a point. As we see in each of the videos in this section, this fact gives a simple and elegant way of showing that every conic section must be an ellipse, a parabola, or a hyperbola according to either a focus-directrix definition or a distances-from-the-foci definition.

#### Ellipses and the Dandelin Construction

The Dandelin construction and the focus-directrix definition of ellipse:

How to use the Dandelin construction to see that a plane whose angle from the vertical is greater than the vertex angle of a cone meets that cone in a curve the satisfies the conditions of the focus-directrix definition of an ellipse.

The Dandelin construction and the sum-of-the-distances definition of ellipse:

How to use the Dandelin construction to see that a plane whose angle from the vertical is greater than the vertex angle of a cone meets that cone in a curve the satisfies the conditions of the sum-of-the-distances definition of an ellipse.

#### Parabolae and the Dandelin Construction

The Dandelin construction and the focus-directrix definition of parabola:

How to use the Dandelin construction to see that a plane whose angle from the vertical is the same as the vertex angle of a cone meets that cone in a curve the satisfies the conditions of the focus-directrix definition of a parabola.

#### Hyperbolae and the Dandelin Construction

The Dandelin construction and the focus-directrix definition of hyperbola:

How to use the Dandelin construction to see that a plane whose angle from the vertical is less than the vertex angle of a cone meets that cone in a curve the satisfies the conditions of the focus-directrix definition of a hyperbola.

The Dandelin construction and the difference-of-the-distances definition of hyperbola:

How to use the Dandelin construction to see that a plane whose angle from the vertical is less than the vertex angle of a cone meets that cone in a curve the satisfies the conditions of the difference-of-the-distances definition of a hyperbola.

### The Reflective Properties of the Conic Sections

The conic sections, when used to generated surfaces of revolution, have specific and important reflection properties involving their foci. In this section, we illustrate those properties, both in three dimensions and in cross-section.

#### Ellipses and reflection

An ellipsoidal reflector:

An ellipse, rotated about its long (or major) axis, forms an ellipsoid of revolution. When a light pulse is emitted from one focus of this figure, it is reflected to the other focus.

An ellipsoidal reflector in cross-section:

A plane cross-section of an ellipsoidal reflector, the plane of the cross-section being one that contains the major axis of the ellipse.

#### Parabolae and reflection

A paraboloidal reflector:

A parabola, rotated about its axis, forms a paraboloid of revolution. When plane wave enters such a figure, it is reflected to the focus. On the other hand, if a pulse of light is emitted from the focus, it is reflected by the paraboloid as a plane wave.

A paraboloidal reflector in cross-section:

A plane cross-section of a paraboloidal reflector, the plane of the cross-section being one that contains the axis of the parabola.

#### Hyperbolae and reflection

A hyperboloidal reflector:

A hyperbola, rotated about its axis, forms a hyperboloid of revolution. When a pulse of light is emitted from one focus of such a figure, it is reflected as though it had come from the other focus. Here, the left-hand focus emits a pulse of red light. Somewhat later, the right-hand focus emits a green pulse of light. The two pulses are timed so that they both reach the vertex of the right branch of the hyperboloid at the same time. Can you predict what happens next?

A hyperboloidal reflector in cross-section:

A plane cross-section of a hyperboloidal reflector, the plane of the cross-section being one that contains the axis of the hyperbola.

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Comments are welcome. Write to talmanl@mscd.edu.