The following is a construction of an ellipse using this method (of course in this case the string is made out of pixels on your computer screen). Based on this definition one can construct an ellipse using a piece of string, something to hold the string in place with and a pencil. These two fixed points are called the foci. The effect that such a room creates is that if one person is standing at one of the foci, a person standing at the other focus can hear even the slightest whisper spoken by the other.įinding the tangent to a point on an ellipseĪn ellipse can be defined as the locus of points such that the sum of the distances from two fixed points to a point on the ellipse is constant. Paul's Cathedral in London, England was designed by architect and mathematician Sir Christopher Wren and contains one such whisper gallery. Such buildings are referred to as whisper galleries or whisper chambers. In fact there are many famous buildings designed to exploit this property.

Both light and sound are affected in this way. Most notable of which is its significance to physics, primarily optics and acoustics. Now, apart from being mathematically interesting, what makes this property so fascinating? Well, there are several reasons. The angle of incidence of ray is measured from the normal to the plane it intersects and congruent angles have congruent angles have congruent complements, so it follows that the angles of incidence are congruent.The Reflective property of an ellipse is simply this: when a ray leaves one of the foci and meets a point on that ellipse, it will reflect off of the ellipse and pass through the other focus. Angle AMO and NMB' are vertical angles and vertical angles are congruent. It can be proven triangles MNB and MNB' are congruent by the SAS rule of congruent triangles. Let P be a point also on the mirror on the opposite of M from N but the same distance. Now consider a line dropped from B to B' and call where it intersects the mirror N. This line also intersects the mirror at M.

The argument is reversible starting from B and traveling to a reflected point of A, call it A'. So the line from A to B' intersects the mirror at a point M that minimizes A to M to B travel distance/time.

The shortest distance between two points is a straight line and by the Pythagorean Theorem it can be shown that the distance form M to B and the distance from M to B' are the same for any point M. One could also consider the problem if a point B' is added as a reflection of the point B. One can use the Pythagorean theorem and calculus to solve this. Find a point M on the mirror between these two such that the sum of the distance from A to M and from M to B is minimized. Suppose you have two points A and B above a mirror at different heights above the mirror. These will serve to guide a geometric argument for equal angles. If the speed of light remains constant in the material, the principle of Least Time becomes a principle of least distance. In the case of physical principles, some of these will be just reliable observations.įermat's Least Time Principle tells us that light take the least time path between possible trajectories. A proof requires axioms, basic statements that can't be proven.

Angle of reflection ellipse