(*^

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        Derivatives -- Introduction -- Curved Mirrors

The following cell draws a concave mirror described by the function

             mirror[x] = 8 - Sqrt[64 - x^2]

and n evenly spaced light rays coming down vertically and bouncing off the mirror.

WARNING:  This cell can take quite a while to evaluate.  Evaluate it now.   
;[s]
9:0,0;8,1;53,0;137,1;167,0;173,1;174,0;253,2;326,0;330,-1;
3:5,14,10,Courier,0,12,0,0,0;3,14,10,Courier,1,12,0,0,0;1,14,10,Courier,1,12,65535,0,0;
:[font = input; dontPreserveAspect; ]
n  = 30;                           (* Number of rays *)

mirror[x_] := 8 - Sqrt[64 - x^2]   (* Mirror *)

width := 5.0        (* x-axis runs from -width to width *)
height := 2 width   (* y-axis runs from 0 to height *)

Clear[slope]
slope[x_] := x/Sqrt[64 - x^2]  (* Slope of tangent *)

(*  The next block of code computes and draws the
    light rays coming in and bouncing off the mirror *)
    
rays = {}
For[x = -width; i = 0, i <= n, ++i; x = x + 2 width/n,
	y = mirror[x];
    rays = Join[{Line[{{x, height}, {x, y}}]}, rays];
    der = slope[x];
    If [Abs[der] > .001,  
        m = (der^2 - 1)/(2 der);  (* outgoing slope    *)
        dx = -Sign[der] * 0.05 * width/Sqrt[1 + m*m];
        rayx = x;
        rayy = y;
        oldrayx = rayx;
        oldrayy = rayy;
        sw = 0;
        While[sw == 0,
              rayx = rayx + dx;
              rayy = rayy + m * dx;
              Which[rayy < mirror[rayx], sw = 1,
                    Abs[rayx] > width, sw = 1,
                    rayy > height, sw = 1];
              If[sw == 0,
                 rays = Join[{Line[{{oldrayx, oldrayy},
                                 {rayx, rayy}}]}, rays];
                 oldrayx = rayx;
                 oldrayy = rayy]
             ]
        ]
]

Show[Plot[mirror[z],{z, -width, width},
                    DisplayFunction -> Identity],
     Graphics[{rays}],
     AspectRatio -> Automatic,
     PlotRange -> {0, height},
     Axes -> None,
     DisplayFunction -> $DisplayFunction]         
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You can modify this cell to try several experiments.

1.  If you hold a cup of coffee so that the sides of the
    cup are vertical and the light rays come over the edge
    of the cup at an angle then the picture above is not
    quite right.  In effect, the sides of the cup will look
    to the incoming light rays like an ellipse rather than
    a circle.  The function
    
         mirror[x] : = a (8 - Sqrt[64 - x^2])
       
    where a is a positive constant is an ellipse.  If a = 1
    then this is just a circle.  If a < 1 then
    the ellipse is short and fat.  If a > 1 then the
    ellipse is tall and thin.  Try different values of a
    to see what the resulting caustics look like.  Describe
    your results.
    
2.  If the mirror is a parabola rather than a circle or
    an ellipse then a different phenomenon appears. The
    function
    
         mirror[x] = x^2/25
       
    describes a parabola with dimensions similar to the
    mirrors we have been studying.  Try this mirror to see
    if your results from the exercises match the graphics.
;[s]
15:0,1;388,2;424,1;443,2;444,1;487,2;492,1;529,2;534,1;578,2;583,1;648,2;649,1;872,2;890,1;1073,-1;
3:0,14,10,Courier,0,12,0,0,0;8,14,10,Courier,0,12,0,0,65535;7,14,10,Courier,1,12,0,0,65535;
^*)