What is the intersective curve between sphere and a right cone?

Question

I am confused this picture : What the curve is? I think that the curve is not circle and not the ellipse too, What is the intersective curve?

Answer 1

Consider a plane $\alpha$ parallel to the plane of the red-green coordinate lines. Let this plane contain the center of the red sphere. The brownish right cone in question (if it is a circular cone) intersects $\alpha$ in an ellipse $\mathscr E$. The curve, $\mathscr C$, whose shape we are interested in is the inverse stereographic projection of $\mathscr E$ on the sphere.

In theory, the equation of this curve can be calculated by the inverse stereographic formula.

It is easy to see that $\mathscr C$ is not a planar curve. Consider the largest and the smallest circles in $\alpha$ centered at the center of $\mathscr E$ and are tangent to $\mathscr E$. The inverse stereographic images of these circles are circles on the sphere and $\mathscr C$ is tangent to these circles on the surface of the sphere. Obviously, the two circles are not in the same plane. This proves that our curve is not a planar one.

So, it is very probable that $\mathscr C$ is not a famous curve with a known name. Although, if we look at this curve (with one eye in the direction of the axis of the cone) from the North pole of the sphere we will see an ellipse because the generatrix lines (of the cone) coming out from our eye go through $\mathscr C$ and reach $\mathscr E$.

If the cone is not a circular cone, and it happens to intersect $\alpha$ in a circle then $\mathscr C$ is a circle.

Answer 2

The locus of points that are found at the intersection of a sphere and a right circular cone is in general a complicated nonplanar curve.

However, there are special cases when the locus is a circle. For example, if the cone axis is coincident with a diameter of the sphere, the locus is a circle. However, the example pictured is not one of the special cases, and the locus is not a circle.

So what is the curve? One can delve into the algebra and find relations between pairs ($x$ and $y$, or $y$ and $z$ in $xyz$ coordinates) of Cartesian coordinates. This amounts to finding projections of the locus into one of the $3$ coordinate planes, either the $xy, xz$, or $yz$ planes. These projections lead to relations that are generally $4$th degree polynomials in $2$ variables. Leo