Conic Sections

Reflection Properties of Conic Sections

by James W. Swift

Ellipses

An ellipse is defined as the set of points in the plane for which the sum of the disance to two fixed points (the foci) is a constant. As a result, a pulse of sound from one focus will reflect from the ellipse and converge to the other focus. The distance between the foci, divided by the constant sum of the distances to the foci, is the eccentricity of the ellipse.
ellipse eccentricity 0.7
An ellipse with eccintricity 0.7: Click the image to get an animation in a new tab. Or, you can click here to get an animation in the current tab, and use back button to return.
ellipse eccentricity 0.7
An ellipse with eccintricity 0.2: Click the image to get an animation in a new tab. Or, you can click here to get an animation in the current tab, and use back button to return.

The Parabola

A parabola is defined as the set of points in the plane for which the distance to a fixed point (the focus) is equal to the distance to a fixed line (the directrix). Every parabola is similar to every other parabola; their size may differ but not their shapes. As a result, a pule of sound from the focus will be reflected from the parabola in a flat pulse.

Parabolic Telescope
Parabolic mirror in an telescope. The light from a star is focused at the vertex of the parabola. The transmitted light reaches the directrix of the parabola at the time it would hit the focus.

Parabolic Flashlight
Parabolic mirror in a flashlight. The intensity of the light is indicated by the thickness of the line.

Hyperbolas

A hyperbola is the set of points in the plane for which the distance to one fixed point (a focus) minus the distance to another fixed point (another focus) is a constant.
Hyperbolic mirror
Hyperbolic mirror.
2 hyperbolic mirrors
Two hyperbolic mirrors.

Hyperbolas
Interference.

Hyperbolas in wave interference
Hyperbolas in the interference of waves.


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e-mail: Jim.Swift@nau.edu