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


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.
