Reflections in Ellipse
Choose two points T_{1}, T_{2} on an ellipse. Imagine a light source at T_{1} sending a ray of light in the direction of T_{2}. Assuming the inner surface of the ellipse fully reflective, the ray will bounce off at T_{2} and reach the ellipse again at T_{3} where it will bounce again and then again at point T_{4} and so on.
What if applet does not run? 
There are three possibilities:

In the latter two cases, the straight lines T_{k}T_{k+1} are said to form an envelope of the ellipse (case 2) or of the hyperbola (case 3.)
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander BogomolnyReflections in Ellipse
The question is, What happens to a ray of light that hops off an elliptic mirror with a light source inside the ellipse?
One case, i.e. where the light source is located at a focus of the ellipse has been discussed from the first principles elsewhere. In this case, after every reflection the ray will proceed from one focus to the other indefinitely. This answer clearly applies to the case where the first the light source is not at a focus but emits a ray that passes through one.
Only other two possibilities are possible: either every piece of the light trajectory crosses between the foci of the ellipse or none does. In the latter case, the pieces envelope an ellipse, in the former a hyperbola. Bellow, we shall look into the case of ellipse.
What if applet does not run? 
Let the foci of the ellipse be F_{1} and F_{2}, and assume that the light source located at T_{1} on the ellipse emits a ray that reflects of the ellipse at point T_{2} and reaches it again at T_{3}. If T_{1}T_{2} does not cross F_{1}F_{2}, there is an ellipse confocal with the given one that touches T_{1}T_{2}. This is because the whole plane is split into the level curves of the function
We are going to prove that T_{2}T_{3} is tangent to the same ellipse.
Find F'_{1}, the reflection of F_{1} in T_{1}T_{2}, and F'_{2}, the reflection of F_{2} in T_{2}T_{3}. By the optical property of ellipse, the angles between the tangent at T_{2} and F_{1}T_{2} and F_{2}T_{2} are equal. By the construction, so are the angles between the tangent and T_{1}T_{2} and T_{3}T_{2}. It follows that the angles F_{1}T_{2}T_{1} and F_{2}T_{2}T_{3} are equal and so are their doubles:
F_{1}T_{2}F'_{1} = F_{2}T_{2}F'_{2}, 
implying that isosceles triangles F_{1}T_{2}F'_{1} and F_{2}T_{2}F'_{2} are similar. It then follows that triangles F_{1}T_{2}F'_{2} and F'_{1}T_{2}F_{2} are equal. In particular,
F_{2}F'_{1} = F_{1}F'_{2}. 
With respect to the constructed ellipse, the locus of points F'_{1} is a circle with center F_{2} and radius, say,
References
 V. Gutenmacher, N. Vasilyev, Lines and Curves: A Practical Geometry Handbook , Birkhauser, 1 edition (July 23, 2004)
Ellipse
 What Is Ellipse?
 Analog device simulation for drawing ellipses
 Angle Bisectors in Ellipse
 Angle Bisectors in Ellipse II
 Between Major and Minor Circles
 Brianchon in Ellipse
 Butterflies in Ellipse
 Concyclic Points of Two Ellipses with Orthogonal Axes
 Conic in Hexagon
 Conjugate Diameters in Ellipse
 Dynamic construction of ellipse and other curves
 Ellipse Between Two Circles
 Ellipse in Arbelos
 Ellipse Touching Sides of Triangle at Midpoints
 Euclidean Construction of Center of Ellipse
 Euclidean Construction of Tangent to Ellipse
 Focal Definition of Ellipse
 Focus and Directrix of Ellipse
 From Foci to a Tangent in Ellipse
 Gergonne in Ellipse
 Pascal in Ellipse
 La Hire's Theorem in Ellipse
 Maximum Perimeter Property of the Incircle
 Optical Property of Ellipse
 Parallel Chords in Ellipse
 Poncelet Porism in Ellipses
 Reflections in Ellipse
 Three Squares and Two Ellipses
 Three Tangents, Three Chords in Ellipse
 Van Schooten's Locus Problem
 Two Circles, Ellipse, and Parallel Lines
Conic Sections > Ellipse
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny71767371