Can Poncelet's invariant measure be generalized to pairs of quadrics in dimension 3?

Let $S\subset \mathbb R^3$ be a fixed sphere and let $E\subset \mathbb R^3$ be a fixed ellipsoid containing $S$. Consider tetrahedra

$$A_1A_2A_3A_4$$

such that

$$A_i\in E$$

and each face is tangent to $S$. Let $D_i\in S$ be the tangency point of the face opposite $A_i$.

Poncelet’s closure theorem in space states that if there exists a single polyhedron (in this case, a tetrahedron) inscribed in a quadric $E$ and circumscribed about a quadric $S$, then there is a continuous family of such polyhedra. In fact, any point on $E$ can be chosen as a vertex of one such tetrahedron.

If $E$ is a sphere concentric with $S$, then every such tetrahedron has the same three face angles at each contact point. For example, for the face $A_2A_3A_4$, the three angles

$$\angle A_2D_1A_3,\qquad \angle A_3D_1A_4,\qquad \angle A_4D_1A_2$$

are fixed, up to permutation, independently of the tetrahedron.

In the planar analogue, let the outer circle be

$$x^2+y^2=R^2$$

and let the inner circle be

$$(x-a)^2+y^2=r^2, \qquad 0<a,\quad a+r<R.$$

If

$$P(\theta)=R(\cos\theta,\sin\theta)$$

and $T(\theta)=P(\phi)$ is the next point on the outer circle such that the chord $P(\theta)P(\phi)$ is tangent to the inner circle, then there is a measure

$$d\mu=\rho(\theta)\,d\theta$$

on the outer circle such that

$$\int_\theta^\phi d\mu$$

is constant for every tangent chord. The density $\rho$ can be computed from the tangency equation. Namely, write the condition that the line through $P(\theta)$ and $P(\phi)$ is tangent to the inner circle, obtain the biquadratic relation

$$F(\theta,\phi)=0,$$

then

$$d\mu = \frac{d\theta}{\sqrt{\Delta(\theta)}},$$

where $\Delta(\theta)$ is the discriminant of $F(\theta,\phi)$ as an equation in $\phi$, after using a rational parameter such as

$$t=\tan\frac{\theta}{2}.$$

For concentric circles, this measure is just a constant multiple of ordinary angle measure.

In 2D, the relevant variety is

$$X=\{(P,L): P\in E,\ L\text{ tangent to }C,\ P\in L\},$$

where $E$ is the outer conic and $C$ is the inner conic.

Since tangent lines to $C$ form the dual conic $C^*$, we have

$$X\subset E\times C^* \cong \mathbb P^1\times \mathbb P^1.$$

The incidence condition $P\in L$ cuts out a curve of bidegree $(2,2)$. Hence, if smooth,

$$g(X)=(2-1)(2-1)=1.$$

So the 2D Poncelet correspondence lives on an elliptic curve.

What is the analogous measure in dimension $3$? More precisely, for a general ellipsoid $E$, is there a measure associated to the pair $(S,E)$ such that each tangent face of a tetrahedron inscribed in $E$ and circumscribed about $S$ has the same measured three-angle data, generalizing the fixed three face angles in the concentric sphere case?