Your projection function looks a bit different to the one I derived from a cone (https://www.desmos.com/3d/oh3w8e08o9 (last 5 lines)), probably equivalent though.
Ashandelle
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I and some friends of mine have some level suggestions
Duospindle (no idea if that's its common name)
The the convex hull of two circles in orthogonal planes, it is also the dual of duocylinder
I think the equation is sqrt(x^2+y^2)+sqrt(z^2+w^2)=r
Sphone (sphere cone)
A cone with a sphere base, I think it might be fairly boring aside from the end point
The equation should be sqrt(x^2+y^2+z^2)+sqrt(w^2)=r (actually a bicone, w>=0 for cone)
Tiger (you might have this one already but I didn't see it)
Its another generalization of the torus
The equation is pretty complex, the parametric is nicer
Cone torus #1
A cone with a torus base, very similar to the spheritorus
sqrt((c-sqrt(x^2+y^2))^2+z^2)+sqrt(w^2)=a (actually a bicone, w>=0 for cone)
Cone torus #2
I only have an equation for this one, it might be similar to revolving a triangle around the origin but I'm not sure
sqrt(z^2+w^2)<=sqrt(x^2+y^2)<=1
Pinched duocylinder (needs a better name)
When rolled properly it should trace out a torus on the ground like a cone frustum traces an annulus
1-sqrt(x^2+y^2)>=z, sqrt(z^2+w^2)<=1
The one me and my friends are most interested in is the duospindle