  Ellipses, parabolas and hyperbolas all have beautiful reflective properties.
However, an intuitive explanation for why they have those properties has been
lacking. There exist many mathematical proofs, but they tend to involve several
analytical steps or geometrical constructions, making them unintuitive and hard
to understand. Here, a simpler explanation is presented which only requires
following the paths of light rays, and examining local paths that move from one
point on a conic to a nearby one. First, a light-ray is followed as it runs
along one of the legs of an isosceles triangle, and then reflects off a mirror
that is parallel to the triangle's base. It bounces back along the path of the
triangle's other leg. Next, a path is examined, moving from an arbitrary point
on a conic section curve to a nearby point on the same curve. This path
consists of two equal-length straight line steps, with each step following one
of the constraints that defines the curve. For example, on an ellipse, defined
by the sum of the distances to the two foci remaining constant, the path starts
at a point on the ellipse, moves a distance delta directly away from one focus,
then makes a second equal-length step directly towards the second focus. These
two steps form the legs of precisely the sort of isosceles triangle described
above, with its base running along the path of the curve. A light ray following
the legs of that triangle gets reflected directly from one focus to the other.
As the triangle shrinks towards zero, the reflection point converges onto the
actual curve. Exactly the same argument also explains the reflections of
parabolas and hyperbolas. Surprisingly, this explanation does not seem to have
appeared previously in the long history of writings about conics. It is hoped
that it will help to make the reflective properties of conic sections easier to
understand and to explain.
