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:[font = title; inactive; preserveAspect; startGroup]
Analyzing projected light on a flat screen from a point source of light inside an elliptical cylinder filled with water
:[font = section; inactive; preserveAspect; startGroup]
BRIEF ABSTRACT
:[font = subsection; inactive; preserveAspect; endGroup]
We attempt to ascertain how fast the light cast by a centrally located
ball of light sitting in the middle of an elliptical tank of water is
passing along a flat panel which is tangent to the cylinder.  We use
some physics and the fact that light travels at different speeds in
different media.
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
GENERAL INFORMATION
:[font = subsection; inactive; preserveAspect; endGroup]
FileName:    TURNOUT

Full title:  Analyzing projected light on a flat screen from a point source of light inside an elliptical cylinder filled with water. 
Developer:  
Brian J. Winkel, Department of Mathematics,  Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA.

Contact:  
Brian J. Winkel, Department of Mathematics,  Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA.  Phone:  812-877-8412.  Email:  winkel@rose-hulman.edu.   FAX:  812-877-3198.

Support:  
The production of this material is supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849:  Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering and the Arvin Foundation of Columbus IN.
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
STATEMENT OF THE PROBLEM
:[font = subsection; inactive; preserveAspect]
Consider a rotating light source sitting in the center (at the origin in an axis system) of an elliptical cylinder of water whose boundary is defined by the relation (x/2)^2 + y^2 = 9.   2 meters beyond the radius of the cylinder there is a screen 8 meters wide.  The screen is perpendicular to y-axis at a distance of 5 m from the origin with center at (0, 5, 0), i.e. along the line y = 5.  Light is known to travel at v1 m/sec in the water in the cylinder and v2 m/sec between the cylinder and the screen off to the right rear..
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
The following code generated the picture above.
:[font = input; preserveAspect]
ellipse = ParametricPlot3D[{6 Cos[t], 3 Sin[t],u},
				{t,Pi/4, 2 Pi - Pi/2},{u,-5,5},
		          Axes->False,Boxed->False,AspectRatio->Automatic]
:[font = input; preserveAspect]
plane = ParametricPlot3D[{t,5, u}, {t,-4,4},{u,-4,4},
			Axes->False,Boxed->False,AspectRatio->Automatic]
:[font = input; preserveAspect]
sphere = ParametricPlot3D[{.5 Cos[t] Cos[u],.5 Sin[t] Cos[u], .5 Sin[u]},
			{t,0, 2 Pi},{u,0,2 Pi},
			Axes->False,Boxed->False,AspectRatio->Automatic]
:[font = input; preserveAspect; endGroup]
Show[sphere,ellipse,plane,ViewPoint->{1.688, -1.288, 0.909}]
:[font = subsection; inactive; preserveAspect]
The speed of light in a vacuum is 2.99776 10^(10) cm/sec; in air (standard temperature 0 degrees C and pressure 1 atm) is va = 2.99689 10^(10) cm/sec; in water at 20 degrees C is  vw = 2.25395 10^(10) cm/sec.   We offer these speeds in m/sec to go with our units in the problem at hand.
v1 = 2.99689 10^(8); v2 = 2.25395 10^(8).
:[font = subsection; inactive; preserveAspect]
The light can be pinpoint directed from the center of the ellipse to any point along the perimeter of the ellipse and the light is rotating at a speed of 60 rpm.   The light starts facing diametrically away from the screen and rotates clockwise.
:[font = subsection; inactive; preserveAspect]
Describe the motion of the light which reaches the screen as a function of time t during the first revolution of the light.  Offer a plot to convince yourself that your mathematical description is right.
:[font = subsection; inactive; preserveAspect; endGroup]
Finally determine when (time and position) the light is moving across the screen horizontally at a rate of 64 m/sec. 
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
KEYWORDS  
:[font = subsection; inactive; preserveAspect; endGroup]
Parametric equations, ellipse, composition of functions, Snell's Law, speed of light, refraction, angle, vector projections, dot product, angle between vectors, rates.
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
TEACHER NOTES
:[font = subsection; inactive; preserveAspect]
ISSUES RELATED TO THE PROBLEM
:[font = subsection; inactive; preserveAspect]
Prerequisites
:[font = subsection; inactive; preserveAspect]
Students need a working knowledge of parametric equations, lines and slopes, and vectors to determine the x coordinate of where the light strikes the screen in terms of where it struck the ellipse, which is in terms of the time during the rotation of the light.
:[font = subsection; inactive; preserveAspect]
Time allotment - time management
:[font = subsection; inactive; preserveAspect]
Expectations 
:[font = subsection; inactive; preserveAspect]
Future payoffs
:[font = subsection; inactive; preserveAspect]
Extensions
:[font = subsection; inactive; preserveAspect]
Have the tank of water have a sinusoidal front to it between the screen and the light.
:[font = subsection; inactive; preserveAspect; endGroup]
References and Sources
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
POSSIBLE SOLUTION
:[font = subsection; inactive; preserveAspect]
We first must determine the deflection of the light according to Snell's Law which we could derive (but do not here!) and apply in this case.

We shall place the origin at the center of the light and put the screen as a line segment from (-6, 5) to (6, 5) m on the same axis system.
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
Speed of light in vacuum is 2.99776 10^(10) cm/sec; in air (standard temperature 0 degrees C and pressure 1 atm) is va = 2.99689 10^(10) cm/sec; in water at 20 degrees C is  vw = 2.25395 10^(10) cm/sec.   We offer it in m/sec to go with our units in the problem at hand.
:[font = input; preserveAspect; endGroup]
v1 = 2.99689 10^(8); v2 = 2.25395 10^(8);
:[font = subsection; inactive; preserveAspect]
We need to find out how the light bends at the interface between the cylinder of water and the air outside.
:[font = subsection; inactive; preserveAspect]
From Snell's Law we know that the relationship between the incident (incoming) light and refracted (outgoing) light at a planar interface is:

                            Sin[theta1]/v1 = Sin[theta2]/v2.
:[font = subsection; inactive; preserveAspect]
We consider the typical  place of interface along the circular perimeter, say, (x0,y0) where y0 = Sqrt[3^2 - x0^2].  We need to find the angle of incidennce of the light coming from the center with the tangent line at this point (x0, y0) and from this predict the angle of refraction based on Snell's Law.
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
We enter the equation of the top half of the ellipse.
:[font = input; preserveAspect; endGroup]
f[x_] = Sqrt[3^2 - (x/2)^2];
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
Direction from center of circle to point p0 = (x0, y0):
:[font = input; preserveAspect; endGroup]
dc[x_] = {x, f[x]};
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
Direction of tangent line at (x, y):
:[font = input; preserveAspect; endGroup]
dt[x_] =  {1, f'[x]}; 
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
We compute the angle of incidence between the ray of light coming to the perimeter of the ellipse and the tangent line to the ellipse at the point of incidence.  A plot is reassuring.
:[font = input; preserveAspect]
inangle[x_] = Pi/2 - ArcCos[Simplify[dc[x].dt[x]/
			(Sqrt[dc[x].dc[x]] Sqrt[dt[x].dt[x]])]];
:[font = input; preserveAspect; endGroup]
Plot[inangle[x],{x,0,6},PlotRange->{0, Pi}]
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
We use Snell's Law to compute the angle of refraction at this same point on the perimeter of the  ellipse.  Again a plot is reassuring.
:[font = input; preserveAspect]
outangle[x_] = ArcSin[v2/v1 Sin[inangle[x]]];
:[font = input; Cclosed; preserveAspect; startGroup]
Plot[outangle[x],{x,0,6},PlotRange->{0, Pi}]
:[font = message; inactive; preserveAspect]
                                 1
Power::infy: Infinite expression -- encountered.
                                 0.
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Graphics["<<>>"]
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-Graphics-
:[font = subsection; inactive; preserveAspect]
We now need to compute the slope, equation, and intercept with the line of the screen (y = 5) of the line of the refracted light.
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
We compute the perpendicular to the tangent line to the perimeter of the ellipse at (x, f(x)).
:[font = input; preserveAspect; endGroup]
perpSlope[x_]  =  {-dt[x][[2]],dt[x][[1]]};
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
And we compute the angle with the horizontal this perpendicular line makes.
:[font = input; preserveAspect; endGroup]
perpangle[x_] = ArcTan[perpSlope[x][[2]]/
			perpSlope[x][[1]]];
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
We compute the angle which the line of refraction makes with the horizontal. 
:[font = input; preserveAspect; endGroup]
lineangle[x_] = perpangle[x] - outangle[x];
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
We finally have the slope of the refraction light at a point on the perimeter of the ellipse.
:[font = input; preserveAspect; endGroup]
slopeline[x_] = Tan[lineangle[x]];
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
Equation of line through the point (x0, f(x0)) in the direction of the refracted light:
:[font = input; preserveAspect; endGroup]
rl[x_,x0_] = slopeline[x0] (x - x0) + f[x0]; 
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
We solve for the x value when the y value of the refracted light line hits the screen.
:[font = input; preserveAspect; endGroup]
sol = Solve[rl[x,x0]==5,x];
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
We identify the point on the screen (screenpoint(x0)) which the light hits when the light beam hits (x0, f(x0)) on the perimeter of the ellipse.
:[font = input; preserveAspect; endGroup]
screenpoint[x0_] = {x,5}/.sol[[1]];
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
We make some plots leading to a time animation of the light beams as well as  the points on the perimeter of the ellipse and the screen.
:[font = input; preserveAspect; endGroup]
ellipse = Plot[f[x],{x,-6,6},AspectRatio->Automatic]
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
We identify the motion of the light's beam on the perimeter of the ellipse as a function of time (sec) - going clockwise from the point (0, -3) at a rate of one revolution per second.
:[font = input; preserveAspect]
xp[t_] = -6 Sin[2 Pi t]; yp[t_] = -3 Cos[ 2 Pi t];
:[font = input; preserveAspect; endGroup]
ParametricPlot[{xp[t],yp[t]},{t,0, .6},
				AspectRatio->Automatic]
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
We make some plots leading to a time animation of the light beams as well as  the points on the perimeter of the ellipse and the screen.
:[font = input; preserveAspect; endGroup]
ellipse = Plot[f[x],{x,-6,6},AspectRatio->Automatic]
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
We identify the motion of the light's beam on the perimeter of the ellipse as a function of time (sec) - going clockwise from the point (0, -3) at a rate of one revolution per second.
:[font = input; preserveAspect]
xp[t_] = -6 Sin[2 Pi t]; yp[t_] = -3 Cos[ 2 Pi t];
:[font = input; preserveAspect; endGroup]
ParametricPlot[{xp[t],yp[t]},{t,0, .6},
				AspectRatio->Automatic]
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
We find the times when the light first hits the screen on the left and when it last hits the screen on the right.
:[font = input; preserveAspect]
xlo  = x/.FindRoot[screenpoint[x][[1]] == -6,{x,-5}][[1]];
:[font = input; Cclosed; preserveAspect; startGroup]
leftpt = t/.FindRoot[xp[t] == xlo,{t,.4}][[1]] 
:[font = output; output; inactive; preserveAspect; endGroup]
0.4025129884495336
;[o]
0.402513
:[font = input; preserveAspect]
xro = x/.FindRoot[screenpoint[x][[1]] == 6,{x,5}][[1]];
:[font = input; Cclosed; preserveAspect; startGroup]
rightpt = t/.FindRoot[xp[t] == xro,{t,.5}][[1]] 
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
0.5974870115504661
;[o]
0.597487
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
Finally, we offer a full time animation of the light beam from when it first strikes the screen to when it leaves the screen.
:[font = input; preserveAspect; endGroup]
Do[Show[ellipse,Graphics[{PointSize[.02],Point[
		screenpoint[xp[t]]],Point[{xp[t],f[xp[t]]}],
				Point[{0,0}],Line[{{0,0},{xp[t],f[xp[t]]}}],
					Line[{{xp[t],f[xp[t]]},screenpoint[xp[t]]}],
						Line[{{-6,5},{6,5}}]}
		        ],PlotRange->{{-7,7},{-1,6}}],
		        	{t,leftpt,rightpt,.01}]
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
We find when the image of the light on the screen is moving along at 64 m/sec.  This occurs at two times - once on the left and once on the right.
:[font = input; preserveAspect]
derscreenpoint[t_] = D[screenpoint[xp[t]][[1]],t];
:[font = input; preserveAspect]
Plot[derscreenpoint[t],{t,leftpt,rightpt}]
:[font = input; Cclosed; preserveAspect; startGroup]
FindRoot[derscreenpoint[t]==64,{t,.4}]
:[font = output; output; inactive; preserveAspect; endGroup]
{t -> 0.4256558543556602}
;[o]
{t -> 0.425656}
:[font = input; Cclosed; preserveAspect; startGroup]
FindRoot[derscreenpoint[t]==64,{t,.57}]
:[font = output; output; inactive; preserveAspect; endGroup]
{t -> 0.5743441456443398}
;[o]
{t -> 0.574344}
:[font = subsubsection; inactive; preserveAspect; endGroup; endGroup]
Thus at t = 0.425656 sec and again at t = 0.574344 sec the image of the light on the screen moves at 64 m/sec.
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
ISSUES IN SOLUTION
:[font = subsection; inactive; preserveAspect; endGroup; endGroup]
The goal is to get the x coordinate of where the light strikes the screen in terms of where it struck the ellipse, which is in terms of the time during the rotation of the light.  Getting the equation of the refracted light is thus paramount and not direct - at least we did not find it to be direct!
^*)