Optical Property of Ellipse: What is it?
A Mathematical Droodle


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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.


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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 F1 and F2. The segments F1T and F2T 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 F1 and F2, in the plane define a function

f(M) = |F1M| + |F2M|

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: f(M) = const. The level curves of f(M) split the plane into a family of confocal (i.e. sharing the same foci) ellipses. As the ellipses expand the function grows.

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: f(T) < f(M), for all M outside the selected ellipse. In particular that is true for all points of the tangent at T. But then, by Heron's theorem, the focal radii at T form equal angles with the tangent.

Other properties follow from this one. For example, denote P2 the pedal point (the foot of the perpendicular) of F2 on the tangent and R2 the reflection of F2 in the tangent. P2 is the midpoint of F2R2. Also,

|F1R2| = |F1T| + |TR2| = |F1T| + |F2T| = C = const.

Therefore all such reflections of F2 in the tangents to the ellipse lie at the same distance from F1, i.e., on a circle with center at F1.

It follows that the midpoints P2 of the segments F2R2 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 f(M) = C, serves as the 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

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

Conic Sections > Ellipse

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