Geometric Reasoning for Reflections on Conics

From MathWiki

It is common in the study of conics that students will be introduced to a conic as a locus:

 an ellipse is the locus of points such that the sum of the distances to two fixed point - the foci - is a constant. 

Then they are asked to explore applications - which involve reflective properties of conics:

  rays of light from one focus will reflect off the curve back to the other focus

However they are offered no reasoned mathematical connection between the properties introduced, and the physical/ geometric properties of reflection.

We offer a simple strategy, based on the simplest geometric reasoning about rate of change (no formulae and no derivatives) which demonstrates this connection.

Consider the GSP sketch for the ellipses Ellipse Reflection

Here is an extension related to paper-folding and sand pouring Ellipse Paper Folding

Here is the GSP sketch for the parabola Parabola Reflection

Here is the GSP sketch for the Hyperbola Hyperbola Reflection

Here is an extension related to paper-folding and sand pouring for the hyperbola Hyperbola Paper Folding


As a bonus, here is a GSP sketch for observing the impact of projective transformations on a plane conic. It was produced by one of my undergrad students (Jonathan Slater) some time ago: Projections of Conics

Questions

A few extra observations:

If you have vertical parallel lines meeting the outside of a parabola - the light will reflect to look like it came from the focus. This appears to be something people doing ray-tracing in animation know.

There are similar properties for the 'exterior' of a hyperbola and and ellipse.

Question: The locus of points equadistant from two skew lines in space is an hyperbolic paraboloid. If we have a light source along one line, which emits a plane of light rays perpendicular to the first line, I think it reflects to look like the light came from the second line. My initial investigation is for the special case where the lines are skew but perpendicular. Is there a nice geometric story for this class of quadrics and their reflective properties.

Contact

For further information, contact Walter Whiteley: mailto://whiteley@mathstat.yorku.ca, 416-736-2100 ext 22598.

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