set of centers of sphere inscribed in tetrahedron Having a sphere and three diffrent point $A,B,C$ on this sphere. Find set of all centers of spheres inscribed in a tetrahedron $ABCD$, where $D$ is some point on the given sphere. The problem reduced to 2-dimensions is trivial it's just sum of two arcs of some circle, but in 3-dimensions the set is not so simple. Checking in geogebra it's not sum of some parts of sphere. I don't know what this set looks and how it's described.

Here is a depiction of Robert Bryant's surface defined by the $56$-term polynomial
he details:

          


Note one component is the "triangular tea bag" discernable in 
my empirical investigation.
Here is the polynomial:
$$
2 x^5+x^4 y+x^4 z-7 x^4+2 x^3
   y^2+4 x^3 y z-10 x^3 y+2
   x^3 z^2-10 x^3 z+8 x^3+2
   x^2 y^3-12 x^2 y^2 z-15 x^2
   y^2-12 x^2 y z^2+6 x^2 y
   z+15 x^2 y+2 x^2 z^3-15 x^2
   z^2+15 x^2 z-2 x^2+x y^4+4
   x y^3 z-10 x y^3-12 x y^2
   z^2+6 x y^2 z+15 x y^2+4 x
   y z^3+6 x y z^2-6 x y z-4 x
   y+x z^4-10 x z^3+15 x z^2-4
   x z-2 x+2 y^5+y^4 z-7 y^4+2
   y^3 z^2-10 y^3 z+8 y^3+2
   y^2 z^3-15 y^2 z^2+15 y^2
   z-2 y^2+y z^4-10 y z^3+15 y
   z^2-4 y z-2 y+2 z^5-7 z^4+8
   z^3-2 z^2-2 z+1$$
The plot above restricts $(x,y,z)$ to lie on or in the sphere.
