Cartesian Oval in Maple13 OR Mathematica7

[Questioneer]

I'm attempting to get an output of a specific Cartesian oval (or oval of descartes, the perfect imaging system in physical optics- viz. perfectly stigmatic imaging). Algebraically I've deduced the equation to be

3/2 Sqrt[(30 - x)^2 + y^2] + Sqrt[x^2 + y^2] == 7/20 (Mathematica input).

I'm kind of new to math programs, so for Mathematica I used the ContourPlot command- the website said it would work fine for implicit functions. I tried it on a simple circle and it worked fine. For the parameters I set up (-20<x<20, -20<y<20) it showed nothing.

For Maple13, I input

> implicitplot(7/(20)=sqrt( x^(2)+y^(2)) + 3/(2)sqrt((30-x)^(2)+y^(2)), x =-20..20, y=-20..20);

and it reads

Error, invalid >

.

For the oval there are four things that need to be known. index of refraction of both medium, and the object distance and image distance. From this point I'm solving for a triangle that has a base of the object and image distance summed, and the arms are equal to the distance traveled through air (medium 1) and the unknown medium 2 with refraction index 1.5. The object distance is 20 cm and the image distance is 10 cm. It first travels through air which has an index of 1.00

Using pen and paper I see no reason why it would be an invalid function, even though implicit. For example, by definition of my cartesian system, the smallest value x (the distance of the object to the interface, which I define to be s0=20 cm) should yield a y value of 0. It does.

The general function for any cartesian oval or oval of descartes is:

n1*Sqrt[x^2+y^2] + n2*Sqrt[(s0+s1-x)^2 + y^2] == n1s0 + n2s1. The general function is attached as a .png file.

Please help- this is the most frustrating feeling!
Attached Thumbnails

 

[Jhero]

Thank you for sharing your experience and the equation you are trying to solve. I understand how frustrating it can be when a mathematical program does not produce the desired output.

After reviewing your equation and the steps you have taken, I have a few suggestions that may help you find a solution.

1. Check the syntax of your equation: Make sure that all the brackets, parentheses, and operators are correctly placed. In your Mathematica input, there seems to be a missing closing bracket after "x^2". Also, in your Maple input, the "=" sign should be replaced with "==". These small errors can cause the program to produce an error.

2. Try different plotting functions: Instead of using the ContourPlot function in Mathematica, you could try using the RegionPlot or Plot3D functions. In Maple, you could try using the plot3d or implicitplot3d functions. These functions may give you a better visualization of your oval.

3. Experiment with different parameters: In your equation, the parameters n1 and n2 represent the refractive indices of the two media. You could try changing these values to see how it affects the shape of your oval.

4. Consider using a numerical solver: Since your equation is implicit, it may be easier to solve it numerically using a solver such as FindRoot in Mathematica or fsolve in Maple. These solvers may give you a more accurate solution compared to plotting functions.

I hope these suggestions help you in your quest to find the solution to your equation. If you continue to face difficulties, please do not hesitate to reach out for further assistance. Good luck!