# Optical Property of Ellipse: What is it?

A Mathematical Droodle

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Copyright © 1996-2018 Alexander Bogomolny### Optical Property (Reflection Law)

One of the definitions of *ellipse* is that it is a locus of points the sum of whose distances from the given two points is constant. The latter are called the *foci* of the ellipse. Under other definitions, the distance preservation becomes a property, among many others. As such, it is known as the *Focal property* of the ellipse.

One remarkable property is common to all conic sections and has been established for parabola independently. Let T be a point on an ellipse with foci F_{1} and F_{2}. The segments F_{1}T and F_{2}T are known as *focal radii* of point T. The property we are going to prove is alternatively called the *Optical property*, *Mirror property*, *Reflective property*, and *Reflection law*. It says that

The focal radii of a point on an ellipse form equal angles with the tangent to the ellipse at that point.

Light reflects off a curved surface by forming equal (incidence and reflection) angles with the tangent at the point it falls on the surface. So locally a surface (curve) behaves like a flat plane (straight line). In the latter case of a straight line, the optical property has an expression in Heron's theorem, which finds a point on a line the sum of whose distances to two given points (on the same side from the line) is minimum. Heron's theorem helps explain the reflective property of ellipse.

Two points, say F_{1} and F_{2}, in the plane define a function

f(M) = |F_{1}M| + |F_{2}M|

which is the sum of distances from the two points to a point M. By definition, ellipse is a *level curve* of that function, i.e., a curve where the function is constant:

Choose one ellipse from the family, a point T on it and the tangent to the ellipse at T. The tangent has a single point in common with the selected ellipse. All its other points are outside that ellipse, i.e. on the ellipses with the values of f(M) greater than at T:

Other properties follow from this one. For example, denote P_{2} the pedal point (the foot of the perpendicular) of F_{2} on the tangent and R_{2} the reflection of F_{2} in the tangent. P_{2} is the midpoint of F_{2}R_{2}. Also,

|F_{1}R_{2}| = |F_{1}T| + |TR_{2}| = |F_{1}T| + |F_{2}T| = C = const.

Therefore all such reflections of F_{2} in the tangents to the ellipse lie at the same distance from F_{1}, i.e., on a circle with center at F_{1}.

_{2}of the segments F

_{2}R

_{2}lie on the circle with center at the center O of the ellipse and the radius equal to half the constant C. This constant C/2 is equal to the

*major semiaxis*of the ellipse. Thus the circle of radius C/2 and center at the center of the ellipse

*pedal curve*of the ellipse. (The pedal curve of a point and a curve is the locus of the feet of the perpendiculars from the point onto the tangents to the curve.) This circle is the smallest circle encompassing the ellipse. It's known as the ellipse's

*major circle*.

### References

- V. Gutenmacher, N. Vasilyev,
*Lines and Curves: A Practical Geometry Handbook*, Birkhauser; 1 edition (July 23, 2004) - R. C. Yates,
*Curves and Their Properties*, NCTM, 1974 (J. W. Edwards, 1959) - C. Zwikker,
*The Advanced Geometry of Plane Curves and Their Applications*, Dover, 2005

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Copyright © 1996-2018 Alexander Bogomolny