# Reflecting light ray between two curves in Mathematica

1. Jun 14, 2010

### interference

Hello everyone,

I am working on trying to model a bouncing light ray between two curves, and I would like to model it in Mathematica. However, I do not know what syntax or how to go about typing the code in. I tried to find examples on the Mathematica Documentation Centre but the examples there are not to relevant.

For example, I have two curves, lets say two parabolic curves one above the other. I want to simulate these as mirror and bounce a light ray between the two curves. Can anyone give me an idea on what Syntax to do it?

I understand people draw lines using the Line command, but I do not know how to proceed from there, such as how to program the line to reflect and calculate the slope and normal at the point where the light ray hits one curve and so on.

Thanks very much :)

2. Jun 14, 2010

### Redbelly98

Staff Emeritus
I don't know Mathematica, however:

If you have the equation of the line, you can find it's intersection with a parabolic surface. It's just solving a quadratic equation.

Once you have the intersection, you can find the slope of the parabola, and from that the slope of the normal at that point (taking the negative reciprocal).

Finally, since slope is the tangent of the angle of the line or normal, you can use the law of reflection to get the angle or slope of the reflected line. Since you have that slope, and a point on the reflected line, you can get its equation and find where i intersects the other surface.

Hope that helps. Feel free to ask more specific questions about the procedure I outlined here.

Last edited: Jun 14, 2010
3. Jun 15, 2010

### interference

Hi Redbelly98, yes I do know how to find the slope and all but I need help with the syntax with which to write a code so that Mathematica and do it.

Now for example, if I use NSolve to solve for a slope at a particular place, it gives me for example x-> 0.2. Does anyone know how I can use this value to be placed into another equation? Because if I put the NSolve function into a new function to solve, Mathematica cannot do that because it reads the value not as "0.2" but as "x->0.2".

Thanks!

4. Jun 15, 2010

### jackmell

You need to get familiar with drawing lines and curves and just graphics in general in Mathematica.

First just draw two parabolas. Say for example:

$$y1=x^2+2x-1$$

$$y2=-3x^2-x+4$$

Then to draw those, I'd code:

myplot=Plot[{x^2+2x-1,-3x^2-x+2},{x,-5,5},PlotRange->{{-5,5},{-5,5},PlotStyle->{Red,Blue}]

That's a good start. Get a print-out of that. Now just practice drawing dots on the parabolas. Say for example, draw a green dot at the point (1,2) on the first one:

mydot1=Graphics[{PointSize[0.008],Green,Point[{1,2}]}];

Show[{myplot,mydot1}]

Draw a dot on the second one:

mydot2=Graphics[{PointSize[0.008],Green,Point[{1,0}]}];

Now draw a straight line between the two points:

myline1=Graphics[{Yellow,Line[{{1,2},{1,0}}];

Now show everything:

Show[{myplot,mydot1,mydot2,myline1}]

Or if you wanted to draw a line with just the function, use:

Plot[m x+b,{x,b1,b2}]

and to dereference that x->2, try this example and then apply it to yours:

myroots = NSolve[(x - 2) (x + 2) == 0, x]
theroots = x /. myroots
firstone = theroots[[1]]

Last edited: Jun 15, 2010