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I have a plane $P$ that intersects a sphere $S$ with a radius $R$. The result of this intersection is a circle $C$ with the distance between its center and the sphere's center is $s$. In my case is always $R \geqslant s$ so I get a circle or a point-circle as a result of this intersection.

Edit \begin{align*} P: && ux + vy + wz + d =0\\ S: && x^2 + y^2 + z^2 = R^2 \end{align*} The formula of $C$ is of the form: $$Ax^2 + Bxy + Cy^2 + Dx + Ey + F =0.$$

What is the formula to represent (equirectangular project) $C$ on a 2D plane ($xOy$ plane)? I would have the circle $C$ plotted so on a 2D plane (it would like look the day-night terminator as in map below:

Globally it looks like:

Thanks for any help :)

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  • $\begingroup$ Your formula for $C$ looks like a sphere to me. $\endgroup$
    – mvw
    Jun 17 '18 at 13:19
  • $\begingroup$ Sorry, you are right, I edited it $\endgroup$
    – Khaled
    Jun 17 '18 at 13:25
  • $\begingroup$ My goal is to represent a circle (not necesserily a great-circle) around a sphere (e.g. the globe) on a 2d plane (e.g. cylindrical equirectangular projection) $\endgroup$
    – Khaled
    Jun 17 '18 at 13:28
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    $\begingroup$ The you need to put all points from the circle through your projection. $\endgroup$
    – mvw
    Jun 17 '18 at 13:46

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