The relation between a, b and c is illustrated on the next picture. The length of the major axis AA' is 2a and the length of the minorĪxis BB' is 2b. Given the value 2a and calling 2c the distance between the foci F1 and F2 we are easily led to The given points F1 and F2 are called the foci or focal points of the ellipse, PF1 and PF2 are called the focal radii of the point P.įor all points outside the ellipse the sum of the distances to the focal points is larger than 2a, for all points inside the ellipse it is smaller than 2a. The distances to F1 and F2 equals a given value 2a. Given two fixed point F1 and F2 in a plane, an ellipse is the locus of all points P in that plane for which the sum of The animations on this page have been made using Maple.įirst we remind some facts about ellipses.ĭefinition and a useful property of an ellipse The book also contains an extensive bibliography.
![angle of reflection ellipse angle of reflection ellipse](https://www.geogebra.org/resource/W2sFu6Fj/rHSFgtjvjHjld3fx/material-W2sFu6Fj.png)
A nice geometrical proof of this property is mentioned further. As to elliptical billiards this leads in a short and elegant way to the property that a shot passing through one focus is reflected through the other focus. The bouncing law is linked to the reflection law in optics. In this book (186 pages) the author treats different billiard table shapes and the (often not so elementary) mathematics behind them.
![angle of reflection ellipse angle of reflection ellipse](http://i.ytimg.com/vi/c3Dnt6z2ifc/0.jpg)
I started this page in 2002, but only in 2007 I discovered the book Geometry and Billiards by Serge Tabachnikov. We'll consider a billiard table that takes the shape of an ellipse. Dodgson, better known as Lewis Carroll, the author of "Alice in Wonderland" thought and wrote about circular billiard Both angles can be considered as the angles formed by the side of the table involved Reflection equals the angle of incidence. Moreover, this ball is reduced to a point that moves in a straight line until it hits an edge and bounces back in such a way that the angle of On a "mathematical billiard table" we consider only one ball. The two locations are the foci of an ellipse.Normally a billiards game is played on a rectangular table.
![angle of reflection ellipse angle of reflection ellipse](https://i.stack.imgur.com/uCqJx.png)
This is the principle behind rooms designed so that a small sound made at one location can be easily heard at another, but not elsewhere in the room. The reflective property of an ellipse: If an ellipse is thought of as a mirror, any ray which passes through one focus and strikes the ellipse will be reflected through the other focus. The formula is an elliptic integral which can be evaluated only by approximation. The perimeter: There is no simple formula for the perimeter of an ellipse. The area: The area of an ellipse is given by the simple formula PIab, where a and b are the semimajor and semiminor axes. This angle is the arc cosine of the eccentricity. It is the acute angle formed by the major axis and a line passing through one focus and an end point of the minor axis. The angle measure of eccentricity: Another measure of eccentricity. All ellipses having the same eccentricity are geometrically similar figures. When the eccentricity is close to zero, the ellipse is almost circular when it is close to 1, the ellipse is almost a parabola.
![angle of reflection ellipse angle of reflection ellipse](https://i.stack.imgur.com/7hyH7.png)
These two definitions are mathematically equivalent. It is the ratio e in Definition 2, or the ratio FC/VC (center-to-focus divided by center-to-vertex). The eccentricity: A measure of the relative elongation of an ellipse.