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!

 

[Phrak]

ContourPlot[(3/2) Sqrt[(30 - x)^2 + y^2] + Sqrt[x^2 + y^2], {x, -20, 20}, {y, -20, 20}]

This will return a contour plot in mathematica. However, attempting

ContourPlot[(3/2) Sqrt[(30 - x)^2 + y^2] + Sqrt[x^2 + y^2]==7/20, {x, 10^100, -10^100}, {y, 10^100, -10^100}],

returns a blank plot. Apparently there is no solution of 7/20 within this region of the plane.

 

[oswaler]

I can understand your frustration with trying to obtain a specific Cartesian oval using mathematical software. However, it is important to keep in mind that these software programs are not infallible and may not always give the desired output. In this case, it seems like there may be a technical issue with the syntax of your input, as both Mathematica and Maple13 are giving errors.

One potential solution could be to try using a different command or function in the software to plot implicit functions. You could also try simplifying the equation and inputting it in a different format to see if that gives a better result.

Additionally, it may be helpful to consult with experts or forums dedicated to using these specific software programs for mathematical purposes. They may be able to provide more specific advice and troubleshooting tips.

In terms of the scientific implications of the Cartesian oval, it is a useful concept in the field of physical optics for perfectly stigmatic imaging. Understanding and being able to accurately plot its equation can help in designing optical systems for various applications. Keep persevering and seeking assistance, and hopefully you will be able to achieve the desired output.