Ellipse Exhibit

(Plane Curves)

Brief Description:

If a and b are two positive real numbers, with a greater than or equal to b, then the standard ellipse in the Cartesian plane, having semi-major axis length a and semi-minor axis length b, is given parametrically by the equations x = a * cos(t) and y = b * sin (t), with t between 0 and 2 pi, or implicitly by the equation (x/a)^2 + (y/b)^2 = 1.

The eccentricity of the ellipse is the non-negative real number e = sqrt( 1 - (b/a)^2). Note that if a = b = r, then the ellipse is the circle of radius r centered at the origin. Thus a circle is the special case of an ellipse for which the eccentricity is 0.

In polar coordinates (r,theta), if l = a * (1 - e^2) is the so-called semi-latus rectum, then the equation of the ellipse is r * (1 + e * cos(theta)) = l. (In classical mechanics, this is the form in which one finds the ellipse when one derives Kepler's famous "First Law"---that the planets travel in ellipses with the Sun at one focus---from Newton's Laws of Motion and Newtons Law of Gravitation---that the force the Sun exerts on a planet is directed toward the Sun and its magnitude is proportional to the reciprocal of the square of the distance from the Sun to the planet.

What You Will See When You Select the Exhibit:

You will see an ellipse with the its foci marked by crosses, and one of them labelled "F". Call the second (unmarked) vertex F'. There is a blue circle with center F' and with radius the major-axis length L. A point marked "S" moves around this circle; call the point where the radius through S meets the ellipse "Q". (It is not marked in the Exhibit.) The distances QS and QF stay visibly equal, indicating that the ellipse is the locus of points equidistant from F and the circle. Another segment is drawn from Q to F, and as Q moves around the ellipse, it should be reasonably evident the sum of their lengths |QF| + |QF'| is L, i.e., the ellipse is also the locus of points the sum of whose distances from F and F' is L. Moreover these two segments, QF and QF', make equal angles with the (green) tangent line to the ellipse at Q, illustrating that a light ray emitted at one focus and reflecting off the ellipse will pass though the other vertex.

More Details:

A classic (and ancient) definition of an ellipse is as follows: Choose two points F and F' in a plane P; these will be the two so-called foci of the ellipse (each is called a focus). Also choose a positive real number L, greater than the distance between the two foci. Then the ellipse E with foci F and F' and major axis length L is the locus of points in the plane P for which the sum of its distances from the two foci equals L. Intuitively speaking, place pins at F and F', and tie a piece of string of length L joining them. Now use a sharp pencil point to stretch the string taut, and trace out all the points that the pencil can reach. The eccentricity of the ellipse E is the ratio, e, of the distance |FF'| to the major axis length L. Define a Cartesian coordinate system with its origin at the midpoint of F and F' and with x-axis the line joining them. Let a = L/2 be the semi-major axis length and let b = a * sqrt(1 - e^2). Then the two foci are at (e * a, 0) and (- e * a, 0) and the ellipse is given as above by (x/a)^2 + (y/b)^2 = 1.

A less familiar but equivalent geometric definition of an ellipse is as follows. Draw a circle of radius L centered at one focus, say F'. Then the ellipse E is the locus of points Q such that the distance of Q to the other focus, F, equals the distance of Q from the circle C.

Still another geometric definition of the ellipse is 3-dimensional in nature. It says that an ellipse in the plane P can always be realized as the intersection of P with a right circular cone K with vertex v outside the plane---in other words, an ellipse is a conic section. Another way to think of this is that an ellipse is the shadow on P of a circle (a section of the cone) thrown by a light at the vertex v.

There is also an interesting "rolling construction" of the ellipse. Let a circle C1 of radius r roll along the inside of a circle C2 of radius R. Choose a radius of C1 that moves along with it as C1 rolls inside C2 and let Q be the point of distance d from the center along this radius. Then the path traced out by Q is called a hypertrochoid. A remarkable fact is that if r = R/2, then the resulting hypertrochoid is an ellipse. (The semi-major axis length is d + r and the semi-minor axis length is d - r.)

A useful properties of an ellipse, called the reflection property, is that a ray of light emitted from one vertex and reflecting off the ellipse will pass through the other vertex. This just means that if you draw line segments from each focus to the same point Q on the ellipse, then the normal to the ellipse at Q bisects the angle FQF', or equivalently that QF and QF' make equal angles with the tangent to the ellipse at Q. This accounts for the "whispering gallery" effect of an elliptically shaped room; a word whispered softly at one focus can be heard clearly at the other focus, but not elsewhere in the room.

Things to Try:

Try the various items in the Action menu, and see About This Gallery for the Plane Curve Gallery for a discussion of these items. Also, try Morphing the ellipse, using the Animate menu.

Links to Further information:



http://www-groups.dcs.st-and.ac.uk/~history/Curves/Ellipse.html http://www.du.edu/~jcalvert/math/ellipse.htm