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What is the implicit equation? resolved

Posted 9 years ago

enter image description here

POSTED BY: Simon Cadrin
3 Replies

If you want the solution in cartesian coordinates, which now appears what you want, you can use:

TransformedField["Polar" -> "Cartesian", r == Sqrt[Cos[\[CapitalTheta]]^2 + 1], {r, \[Theta]} -> {x, y}]

(*Sqrt[x^2 + y^2] == Sqrt[1 + Cos[\[CapitalTheta]]^2]*)

That still contains the $Theta$ so you can use

TransformedField["Polar" -> "Cartesian", r == Sqrt[Cos[\[CapitalTheta]]^2 + 1], {r, \[Theta]} -> {x, y}] /. \[CapitalTheta] -> 
  CoordinateTransform["Cartesian" -> "Polar", {x, y}][[2]]//FullSimplify

(*x^2 + y^2 == 1 + x^2/(x^2 + y^2)*)

To convert that to the equation you desire, you can multiply both sides by $(x^2+y^2)$:

Map[#*(x^2 + y^2) &, %] // FullSimplify

(* (x^2 + y^2)^2 == 2 x^2 + y^2*)

Cheers,

M.

POSTED BY: Marco Thiel

Here the implicit equation, it is a case of Booth oval. Thank you for your cooperation. enter image description here

POSTED BY: Simon Cadrin

(The question has changed since the answer was posted.)

Hi,

is this a Mathematica question or a general Mathematics question?

The equation you give is clearly not in Mathematica format and we need to make several assumptions here to guess what you are looking for. Let's suppose that you are looking for a curve given in polar coordinates, where $r$ is the radius and $\Theta$ is the angle. We could then plot the curve

PolarPlot[Sqrt[Cos[theta]^2 + 1], {theta, 0, 2 Pi}, AspectRatio -> 1]

enter image description here

It does look somewhat nicer if you square the $\Theta$ only and not the $Cos$:

PolarPlot[Sqrt[Cos[theta^2] + 1], {theta, 0, 2 Pi}, AspectRatio -> 1]

enter image description here

We can also rotate for more than $2 Pi$ and add a little color:

PolarPlot[Sqrt[Cos[theta^2] + 1], {theta, 0, 16 Pi}, AspectRatio -> 1,PlotPoints -> 500, ColorFunction -> Function[{x, y, t, r}, Hue[r]]]

enter image description here

Well, there are lot's of things you can do with Mathematica. It is worthwhile to have a look at Mathematica. You can, of course, also just type your equation into Wolfram Alpha like so:

http://www.wolframalpha.com/input/?i=r%3D%3DSqrt%5BCos%5Btheta%5D%5E2%2B1%5D

That gives you a whole lot of information about that implicit function. Here are the first couple of lines from the output:

enter image description here

You even get sliders etc, to better study the equation.

Cheers,

M.

POSTED BY: Marco Thiel
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