What is the intersection of a plane and a sphere? Is it necessarily a circle? Can it be an ellipse?

Solution 1:

As already verified, all intersections of a circle and plane are circles either geodesic/great or small. I avoid symbols in the answer here.

If 2-parameter surface normal coincides with 1-parameter circle normal then the intersection is a great circle. The intersection is a geodesic.

From differential geometry standpoint $\psi$ is between arc and meridian. Cylindrical coordinates $(r,z)$ Clairaut constant derivative $\dfrac{d(r \sin \psi)}{dz}=0$ forms with respect to an arbitrary North-South polar axis.

If the 2-parameter surface normal does not coincide with 1-parameter circle normal, but makes an angle $\gamma$ of relative latitude then the intersection is a small circle. Intersection is like a parallel circle but not an equator.

Clairaut constant derivative $\dfrac{d(r \sin \psi)}{dz}$ is a constant $ =\tan \gamma$ forms a small circle with respect to an arbitrary North-South polar axis.

In stereographic projection all non-equatorial sections of a plane passing through North pole of the sphere at angle $\gamma$ to North-South polar plane are small circles.