I originally solved this Euler problem on Feb 8, 2021. I recently revisited the solution to enhance the visualization and post it, because it's pretty neat!
In laser physics, a "white cell" is a mirror system that acts as a delay line for the laser beam. The beam enters the cell, bounces around on the mirrors, and eventually works its way back out.
The specific white cell we will be considering is an ellipse with the equation
4x^2 + y^2 = 100.
The section corresponding to -0.01 <= x <= +0.01 at the top is missing, allowing the
light to enter and exit through the hole.
The light beam in this problem starts at the point (0.0, 10.1) just outside the white cell, and
the beam first impacts the mirror at (1.4, -9.6).
Each time the laser beam hits the surface of the ellipse, it follows the usual law of reflection "angle of incidence equals angle of reflection." That is, both the incident and reflected beams make the same angle with the normal line at the point of incidence.
In this visualization, the red line shows the current two points of contact between the laser beam and the wall of the white cell; the blue line shows the line tangent to the ellipse at the point of incidence of the current bounce.
Click "Play" to see how many times the beam hits the internal surface of the white cell before exiting.
Try out "Party mode"!