Homework # 9b:  Elliptical Billiards

Below are a few simple exercises that will help you understand better some of the ideas and procedures used in Module 9b.  You may use the Documentation Center to assist you with these problems.

Please format your solutions appropriately.  Use text format, not input format, when you are typing text.  Write coherent sentences and paragraphs;  part of your grade will depend on how clearly you present your ideas.

You must work on these exercises by yourself, not in pairs.  If you get stuck and cannot find the solution in the Documentation Center, you may consult the labbie or the instructor during their office hours.

Elliptical Billiards

Here you are going to solve for the motion of a ball moving on an elliptical billiard table.

An ellipse centered at the origin of the coordinate system with the axes of the ellipse aligned along the x and y-axes is given by the equation

((x^2)/(a^2))+ ((y^2)/(b^2))= 1,

where a and b are the lengths of the semi-major and semi-minor axes.  If a>b, the foci of this ellipse are located at the points

F = {±(Sqrt[(a^2)-(b^2)]), 0}.

A normal to the ellipse at any point {x,y} on the ellipse is given by the expression

n[x,y] = {((2x)/(a^2)), ((2y)/(b^2))}.

Note that n[x,y] is not, in general, a unit normal.  To generate a unit normal, you must divide the vector n[x,y] by the length of n[x,y].

Problems

Write functions insideQ[{x,y}] and normal[{x,y}] for an elliptical billiards table.  The function insideQ[{x,y}] should determine if the point {x,y} lies inside or outside of the ellipse in Equation (1);  the  function normal[{x,y}] should compute the unit normal to the ellipse at the point {x,y} on the ellipse.

To represent an elliptical billiards table, you will need to define a graphics object ellipse[a,b] that represents an ellipse, where a and b are the lengths of the semi-major and semi-minor axes.  See the Documentation Center for how to represent an ellipse in Mathematica, and then write a function to display an elliptical billiards table.

Plot some trajectories for a ball moving on an elliptical billiards table.  Pick different starting points and different initial velocities.  
a.  What do you see if you start the ball at the center of the table?
b.  What do you see if you start the ball at one of the foci of the table?

Use Manipulate to animate the motion of a ball on an elliptical billiards table.