# Finding the spherical coordinates for the edge obtained by cutting a sphere with a plane

I am searching the spherical coordinates for the circular edge that are obtained when a sphere is cut at a certain position with a plane. The sphere has herby a radius $$r$$ and is focused at the center of a coordinate system. The plane cut is performed at a certain $$x, y,$$ or $$z$$ position (see an exemplary cut in the linked image).

What I am now interested in is finding the parametrization of the cutting edge, however not as parametrization of a circle, but instead in spherical coordinates of the sphere. This means I want to find the coordinates of every point on the cut, expressed in the spherical coordinate system. For a cut through the z-plane the solutions looks trivial with a azimuth angle changing between $$0$$ and $$2\pi$$ and a fixed elevation angle, as seen in the exemplary image. However the solution is not trivial for a cut through the $$x$$, or $$y$$ plane.

Does anyone know the solution for it?

• The cut is performed either perpendicular to the x, y or z axis. For a cut perpendicular to the z-axis the solution is trivial. So, the main challenge lies in finding the solution for the cut through the x or y axis or beyond that a generalized formula valid for a perpendicular cut through any of the 3 planes. Nov 9, 2019 at 19:36
• You mean the plane of small circle is not perpendicular to any of the three ${x,y,z}$ axes? Nov 10, 2019 at 9:31
• in my case the plane is perpendicular to one of the axes, however, a generalized solution might be of interested for some others. Nov 11, 2019 at 6:20
• You need to apply two Euler angle matrix rotations on rigid body (sphere). Nov 11, 2019 at 7:48
• @Narasimham Good idea for a second solution. As far as I understand the topic , Euler angle transformation should be connected to Rodrigues' rotation formula. Do you think such a solution will lead to a different result than the formula below? If yes, do you see a possiblity to derive it as second solution? Nov 13, 2019 at 14:10

The equations of the cut at some $$x_p$$ are given by $$x^2+y^2+z^2=r^2$$ and $$x=x_p$$ We can the write $$y^2+z^2=r^2-x_p^2$$ Now plug in the expression for $$y$$ and $$z$$ in polar coordinates:\begin{align}z&=r\cos\theta\\y&=r\sin\theta\sin\phi\end{align} You get $$\cos^2\theta+\sin^2\theta\sin^2\phi=1-\frac{x_p^2}{r^2}$$ To get the limits, note that $$\cos\theta$$ varies between $$\pm\sqrt{1-\frac{x_p^2}{r^2}}$$. Then you can write the expression for $$\sin\phi$$ in terms of $$\theta$$.
If you do a cut at $$y_p$$ instead, just replace $$x_p$$ with $$y_p$$ and use $$\cos\phi$$ instead of $$\sin\phi$$.
• Thanks i tried out the solution and it works ;) Do you see a possiblity to solve the last equation also for $\theta$ in terms of $\phi$? Nov 13, 2019 at 13:58
• To the solutions should be noted that the the squared cosine and sine terms and the squared cut $x_p$ lead to a paramterized quarter circle, however the other part of the circle can be easily concluded by symmetry. Nov 13, 2019 at 14:14
• To answer your first question, I don't have an easy way of saying what the limits for $\phi$ should be Nov 13, 2019 at 15:42