![]() ![]() Case 2: a > b.įor this case we can show that the curve satisfies the Two Focus Property with foci at the points (± c, 0), where c = √( a 2 − b 2). From any point on the circle, the distance to the center is a constant, so the sum of distances to the two foci is also a constant. Here the curve is a circle and we take both foci to be at the center of the circle. There are three possible cases, depending on whether a is equal to, greater than, or less than b, and we will consider each case separately. To get started, let us see that a curve given by the standard ellipse equation (1)Īlways satisfies the Two Focus Property. That will complete the overall demonstration. That means showing that the set of points defined by the Two Focus Property for a particular choice of the foci and d is necessarily an ellipse. Then we still have to consider the converse. That in itself requires two parts: showing that points of the ellipse satisfy the Two Focus Property, and also showing that no other points do. As a first step, we show that every ellipse is made up of exactly the points defined by the Two Focus Property for a suitable choice of the foci and the constant d. The details are a bit involved, so here is a preview. That means we have to show that every ellipse has the two focus property, and that any curve with the two focus property is an ellipse. The goal for this page is to show that the Two Focus Property is a valid alternate definition of an ellipse. It has the following official statement: Two Focus Property of an Ellipseįor every ellipse E there are two distinguished points, called the foci, and a fixed positive constant d greater than the distance between the foci, so that from any point of the ellipse, the sum of the distances to the two foci equals d. For reference, we will call this the Two Focus Property of an ellipse. The fact that every ellipse can be defined using two foci and a specified number d will be used frequently in the discussions to follow. To avoid this situation we insist that d is greater than the distance between the given points A and B. This corresponds to a degenerate ellipse. Then the pencil slides along the taut string, tracing the line segment. In the extreme case, with a string exactly long enough to reach from A to B, the string becomes a straight line segment. In general, we need a string longer than the distance between A and B. The pencil and string construction in the graphic would not be possible with a string less than 5 units long. As the point moves around the ellipse, the two red line segments always combine to the same total length. This idea can be visualized in the animated graphic below, which is located at Wolfram MathWorld. Now move the pencil, keeping the string taut, and you will trace the ellipse. This shows that your pencil point is on the ellipse. From the pencil point to A and B the distances must total 7 units, since together these lengths use up the entire string. Stretch the string taut with the point of your pencil, and put the pencil on the paper. One way to imagine drawing the ellipse in this example is to cut a piece of string, 7 units long, and tack the ends to the points A and B. ![]() The two fixed points that were chosen at the start are called the foci (pronounced foe-sigh) of the ellipse individually, each of these points is called a focus (pronounced in the usual way).įor example, if you specify points A and B in the plane, 5 units apart, then look for every possible point C whose distances from A and B add up to 7, these points form an ellipse. The set of all such points is an ellipse. Now consider any point whose distances from these two points add up to a fixed constant d. The Most Marvelous Theorem in Mathematics, Dan Kalman Two focus definition of ellipseĪs an alternate definition of an ellipse, we begin with two fixed points in the plane. The Journal of Online Mathematics and Its Applications, Volume 8 (2008) ![]()
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