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? 


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  
