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I am looking for resources on the reflection properties of the circle, sphere, ellipse, and parabola. However, when looking for articles or entries, the same 3 examples keep coming up (Focus to focus in ellipse, parallel rays converge to focus in parabola etc...)

I am looking for more general studies on reflections within these shapes, in particular, orbits and bounds of the "Contact points" around the ellipse, and the path of the rays within it.

For example(conjecture): If a ray is cast between the two foci (the ray intersects the line between them), the ray (and subsequent reflections) are bounded by the hyperbola defined by the same two foci, that is also tangent to the first ray.

Likewise, if the ray is cast between either focus and the edge of the ellipse, the ray is bounded by a smaller ellipse with the same focus.

This holds in the case of a circle, where you end up with a smaller circle, or with the right set up, regular/star polygons.

Similar behaviors occur with two intersecting ellipse.

Example 2: Center a circle within an ellipse, where the circles radius is less than the minor axis of the ellipse. There are points were the reflections are bounded on both the circle and the ellipse, and others where the reflections become chaotic, and change rapidly with minor changes in the initial ray (sometimes settling into some attractor at the ends of the ellipse)

I feel like there is a body of work out there on this, but I am not a mathematics researcher (undergraduate math degree only), and have no idea where to even start looking for work on the subject. does anyone know of a name/paper/field/journal that has anything like this?

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    There is indeed a lot out there. Google "elliptical billiards". Here's one link: http://mathworld.wolfram.com/Billiards.html. Your hyperbola conjecture is there. – 2017-02-22
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    This is exactly what I was looking for. I would have never thought to look for billiards. Thank you so much! – 2017-02-22
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    You're welcome. I posted my comment as an answer so you can accept it and the question won't hang around unanswered. Perhaps someone else will have more suggestions. – 2017-02-22

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There is indeed a lot out there. Google "elliptical billiards". Here's one link: mathworld.wolfram.com/Billiards.html. Your hyperbola conjecture is there.