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I have a continuous closed parametric curve $$ \begin{align} x &= \arccos\left(-\frac{Q × \sqrt{\frac{1}{3}} \left(\tan\left(\frac{π}{4} u\right)^2 - 1\right) - \sqrt{\frac{2}{3}} \left(\tan\left(\frac{π}{4} u\right) + 1\right)}{Q × \left(\tan\left(\frac{π}{4} u\right)^2 - 1\right) + 2}\right), \\ y &= \arctan\left( Q × \sqrt{\frac{2}{3}} \left(\tan\left(\frac{π}{4} u\right)^2 - 1\right) + \sqrt{\frac{1}{3}} \left(\tan\left(\frac{π}{4} u\right) - 2\right)\right) + \frac{π}{6} k, \\ -2& ≤ u ≤ 2, \end{align} $$ where $Q$ is a positive constant and $k$ is an integer.

I want to convert it into an implicit curve, but I'm struggling to eliminate the parameter $u$. Asking WolframAlpha to solve either equation for $u$ produces multiple results, but substituting each of those results into the remaining equation only produces partial curves instead of a continuous closed curve, and I haven't been able to figure out a way to algebraically manipulate any of them to get the closed curve I need.

The curve is the projection of a circle from 3D space and spherical coordinates $(r, θ, φ)$ into 2D space, with the inclination $θ$ as the $x$-value and the azimuth $φ$ as the $y$-value. The 3D circle lies on the surface of a sphere of radius 1 centered on the origin, and is the intersection between that sphere and the plane given by the equation $$ -\sqrt{\frac{29 - 2 \sqrt{2}}{51}} × x - \sqrt{\frac{5 + 2 \sqrt{2}}{17}} × y - \sqrt{\frac{7 - 4 \sqrt{2}}{51}} × z = \sqrt{\frac{7 - 4 \sqrt{2}}{17}} $$

I am hopeful that a solution exists in this case because I was able to successfully convert a different circle (sitting on the same sphere as the one in this question) from parametric to implicit form.

How should I go about this? Is there a way to tell if a solution does or does not exist?

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    $\begingroup$ Do you have any reason to believe a solution exists? What's the context? $\endgroup$ Commented Jul 22 at 17:03
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    $\begingroup$ Please delete all the $\times$ signs. And yes, there's no reason to believe that you can do what you want. Theory just tells you it's possible locally, and not necessarily explicitly. $\endgroup$ Commented Jul 22 at 17:05
  • $\begingroup$ @CyclotomicField I added more context and a reason I think a solution exists. $\endgroup$ Commented Jul 22 at 17:49
  • $\begingroup$ What was the original circle in spherical coordinates? I am suspicious. $\endgroup$ Commented Jul 22 at 17:51
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    $\begingroup$ So what is the cartesian equation of the plane with which you are slicing the unit sphere? $\endgroup$ Commented Jul 22 at 18:30

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This is one of those questions where the answer to the original problem is quite simple but the question is posed in terms of an intermediate form that makes everything immensely more difficult. The solution is to ignore the intermediate form and solve the original problem.

To summarize the question, you have the equation of a plane in 3D Cartesian space, $$ -x \sqrt{\frac{29 - 2 \sqrt{2}}{51}} - y \sqrt{\frac{5 + 2 \sqrt{2}}{17}} - z \sqrt{\frac{7 - 4 \sqrt{2}}{51}} = \sqrt{\frac{7 - 4 \sqrt{2}}{17}}, $$ which is equivalent to $$ x \sqrt{\frac{29 - 2 \sqrt{2}}{3}} + y \sqrt{5 + 2 \sqrt{2}} + z \sqrt{\frac{7 - 4 \sqrt{2}}{3}} = - \sqrt{7 - 4 \sqrt{2}}.\tag1 $$

This plane intersects the unit sphere in a circle. You wish to take the polar angle $\theta$ and azimuth $\varphi$ of each point on that plane and plot these values as $x$ and $y$ values of a 2D Cartesian graph.

Assuming that's an accurate retelling of the question, we proceed as follows.

Recall the conversion of spherical coordinates with radius $r,$ polar angle $\theta,$ and azimuth $\varphi$ to 3D Cartesian coordinates, \begin{align} x &= r\sin\theta \cos\varphi,\\ y &= r\sin\theta \sin\varphi,\\ z &= r\cos\theta. \end{align}

For points on the surface of the unit sphere, $r = 1,$ so these equations simplify to $$\begin{align} x &= \sin\theta \cos\varphi,\\ y &= \sin\theta \sin\varphi,\\ z &= \cos\theta. \end{align}\tag2$$

We can then simply make the substitutions for $x,$ $y,$ and $z$ indicated by Equation $(2)$ into Equation $(1)$ to obtain

$$ \begin{multline} \sqrt{\frac{29 - 2 \sqrt{2}}{3}} \sin\theta \cos\varphi\\ + \sqrt{5 + 2 \sqrt{2}} \sin\theta \sin\varphi + \sqrt{\frac{7 - 4 \sqrt{2}}{3}} \cos\theta \\ = -\sqrt{7 - 4 \sqrt{2}}. \end{multline}\tag3 $$

This is an implicit equation in $\theta$ and $\varphi$ for the selected circle on the sphere. Now if you want to plot this on an $x,y$ plane with $\theta$ plotted along the $x$ axis and $\varphi$ plotted along the $y$ axis, simply substitute $x$ for $\theta$ and $y$ for $\varphi$ in Equation $(3).$

You can view the resulting graph here. I have plotted the spherical coordinates of the three points mentioned in one of your comments, $(-\sqrt{1/3}, 0, \sqrt{2/3}),$ $(-\sqrt3(52 - 3\sqrt2)/102, (4 + 5\sqrt2)/34, \sqrt3(3 + 8\sqrt(2))/51),$ and $(-\sqrt{3/4}, 1/2, 0),$ as a check on the procedure.

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    $\begingroup$ This is also called an XY problem $\endgroup$ Commented Jul 22 at 20:35

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