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From your vague dismissal of David Lehavi's answer, it seems what you want is a distance between a pair of embeddings of ellipsoids. Or equivalently a metric on the 12-dimensional space "pairs of embedded ellipsoids mod rigid motions of $\mathbb{R}^3$".

Would the following idea be right for what you want? Given ellipsoids A and B in $\mathbb{R}^3$, first find the translation and rotation of $\mathbb{R}^3$ which moves ellipsoid A so that it's centered at the origin and has its principal axes aligned with the coordinate axes. That is, a rigid motion taking ellipsoid A to another embedding of an ellipsoid A' so that $C_{A'}=0$, $M_{A'}$ is diagonal with eigenvalues ordered by size. What we're doing here is putting ellipsoid A in a standard position.

Next take the transformation that took A to standard position and apply it to ellipsoid B which moves it to some new embedded ellipsoid B'. Ellipsoid B' is now characterized by $C_{B'}$ and $M_{B'}$ (something like $C_{B'} = C_B-C_A$, $M_{B'}= O^T M_B O$, where $O$ is the rotation diagonalizing $M_A$ as described previously)

One can now assign a "distance" between A and B which depends only on the eigenvalues of $M_A$ (3 parameters), the vector $C_{B'}$ (3 parameters) and the matrix $M_{B'}$ (6 parameters, which you could split into 3 eigenvalues+3 parameters of a rotation if you decomposed $M_{B'}$). By moving your ellipsoids together to this "standard position" we've quotiented out the space of "pairs of embedded ellipsoids" by the action of rigid transformations of $\mathbb{R}^3$, as presumably you want your distance function to be invariant under this group.

There are many distances one can assign to ellipsoids in standard position, depending on how you weight "shape" vs. "distance" vs. "orientation", etc. - what you choose is going to depend heavily on what you want. One family of examples is

$distance(A,B)=a_1|\lambda_A - \lambda_{B'}| + a_2|C_{B'}| + a_3 d_{SO(3)}(I,O)$,

where $\lambda_i$ is the (ordered) vector of eigenvalues of ellipsoid $i$, $|\cdot|$ denotes the norm on 3-vectors, $d_{SO(3)}(I,O)$ is the distance between the rotation $O$ and the identity matrix $I$ in the space of rotations $SO(3)$ and $a_1,a_2,a_3$ are weights.

So please do give your needs a little more thought.

From your vague dismissal of David Lehavi's answer, it seems what you want is a distance between a pair of embeddings of ellipsoids. Or something like a non-negative function on the 12-dimensional space "pairs of embedded ellipsoids mod rigid motions of $\mathbb{R}^3$".

Would the following idea be right for what you want? Given ellipsoids A and B in $\mathbb{R}^3$, first find the translation and rotation of $\mathbb{R}^3$ which moves ellipsoid A so that it's centered at the origin and has its principal axes aligned with the coordinate axes. That is, a rigid motion taking ellipsoid A to another embedding of an ellipsoid A' so that $C_{A'}=0$, $M_{A'}$ is diagonal with eigenvalues ordered by size. What we're doing here is putting ellipsoid A in a standard position.

Next take the transformation that took A to standard position and apply it to ellipsoid B which moves it to some new embedded ellipsoid B'. Ellipsoid B' is now characterized by $C_{B'}$ and $M_{B'}$ (something like $C_{B'} = C_B-C_A$, $M_{B'}= O^T M_B O$, where $O$ is the rotation diagonalizing $M_A$ as described previously)

One can now assign a "distance" between A and B which depends only on the eigenvalues of $M_A$ (3 parameters), the vector $C_{B'}$ (3 parameters) and the matrix $M_{B'}$ (6 parameters, which you could split into 3 eigenvalues+3 parameters of a rotation if you decomposed $M_{B'}$). By moving your ellipsoids together to this "standard position" we've quotiented out the space of "pairs of embedded ellipsoids" by the action of rigid transformations of $\mathbb{R}^3$, as presumably you want your distance function to be invariant under this group.

There are many distances one can assign to ellipsoids in standard position, depending on how you weight "shape" vs. "distance" vs. "orientation", etc. - what you choose is going to depend heavily on what you want. One family of examples is

$distance(A,B)=a_1|\lambda_A - \lambda_{B'}| + a_2|C_{B'}| + a_3 d_{SO(3)}(I,O)$,

where $\lambda_i$ is the (ordered) vector of eigenvalues of ellipsoid $i$, $|\cdot|$ denotes the norm on 3-vectors, $d_{SO(3)}(I,O)$ is the distance between the rotation $O$ and the identity matrix $I$ in the space of rotations $SO(3)$ and $a_1,a_2,a_3$ are weights.

So please do give your needs a little more thought.

From your vague dismissal of David Lehavi's answer, it seems what you want is a distance between a pair of embeddings of ellipsoids. Or equivalently a norm on the 12-dimensional space "pairs of embedded ellipsoids mod rigid motions of $\mathbb{R}^3$".

Would the following idea be right for what you want? Given ellipsoids A and B in $\mathbb{R}^3$, first find the translation and rotation of $\mathbb{R}^3$ which moves ellipsoid A so that it's centered at the origin and has its principal axes aligned with the coordinate axes. That is, a rigid motion taking ellipsoid A to another embedding of an ellipsoid A' so that $C_{A'}=0$, $M_{A'}$ is diagonal with eigenvalues ordered by size. What we're doing here is putting ellipsoid A in a standard position.

Next take the transformation that took A to standard position and apply it to ellipsoid B which moves it to some new embedded ellipsoid B'. Ellipsoid B' is now characterized by $C_{B'}$ and $M_{B'}$ (something like $C_{B'} = C_B-C_A$, $M_{B'}= O^T M_B O$, where $O$ is the rotation diagonalizing $M_A$ as described previously)

One can now assign a "distance" between A and B which depends only on the eigenvalues of $M_A$ (3 parameters), the vector $C_{B'}$ (3 parameters) and the matrix $M_{B'}$ (6 parameters, which you could split into 3 eigenvalues+3 parameters of a rotation if you decomposed $M_{B'}$). By moving your ellipsoids together to this "standard position" we've quotiented out the space of "pairs of embedded ellipsoids" by the action of rigid transformations of $\mathbb{R}^3$, as presumably you want your distance function to be invariant under this group.

There are many distances one can assign to ellipsoids in standard position, depending on how you weight "shape" vs. "distance" vs. "orientation", etc. - what you choose is going to depend heavily on what you want. One family of examples is

$distance(A,B)=a_1|\lambda_A - \lambda_{B'}| + a_2|C_{B'}| + a_3 d_{SO(3)}(I,O)$,

where $\lambda_i$ is the (ordered) vector of eigenvalues of ellipsoid $i$, $|\cdot|$ denotes the norm on 3-vectors, $d_{SO(3)}(I,O)$ is the distance between the rotation $O$ and the identity matrix $I$ in the space of rotations $SO(3)$ and $a_1,a_2,a_3$ are weights.

So please do give your needs a little more thought.

From your vague dismissal of David Lehavi's answer, it seems what you want is a distance between a pair of embeddings of ellipsoids. Or equivalently a metric on the 12-dimensional space "pairs of embedded ellipsoids mod rigid motions of $\mathbb{R}^3$".

Would the following idea be right for what you want? Given ellipsoids A and B in $\mathbb{R}^3$, first find the translation and rotation of $\mathbb{R}^3$ which moves ellipsoid A so that it's centered at the origin and has its principal axes aligned with the coordinate axes. That is, a rigid motion taking ellipsoid A to another embedding of an ellipsoid A' so that $C_{A'}=0$, $M_{A'}$ is diagonal with eigenvalues ordered by size. What we're doing here is putting ellipsoid A in a standard position.

Next take the transformation that took A to standard position and apply it to ellipsoid B which moves it to some new embedded ellipsoid B'. Ellipsoid B' is now characterized by $C_{B'}$ and $M_{B'}$ (something like $C_{B'} = C_B-C_A$, $M_{B'}= O^T M_B O$, where $O$ is the rotation diagonalizing $M_A$ as described previously)

One can now assign a "distance" between A and B which depends only on the eigenvalues of $M_A$ (3 parameters), the vector $C_{B'}$ (3 parameters) and the matrix $M_{B'}$ (6 parameters, which you could split into 3 eigenvalues+3 parameters of a rotation if you decomposed $M_{B'}$). By moving your ellipsoids together to this "standard position" we've quotiented out the space of "pairs of embedded ellipsoids" by the action of rigid transformations of $\mathbb{R}^3$, as presumably you want your distance function to be invariant under this group.

There are many distances one can assign to ellipsoids in standard position, depending on how you weight "shape" vs. "distance" vs. "orientation", etc. - what you choose is going to depend heavily on what you want. One family of examples is

$distance(A,B)=a_1|\lambda_A - \lambda_{B'}| + a_2|C_{B'}| + a_3 d_{SO(3)}(I,O)$,

where $\lambda_i$ is the (ordered) vector of eigenvalues of ellipsoid $i$, $|\cdot|$ denotes the norm on 3-vectors, $d_{SO(3)}(I,O)$ is the distance between the rotation $O$ and the identity matrix $I$ in the space of rotations $SO(3)$ and $a_1,a_2,a_3$ are weights.

So please do give your needs a little more thought.

From your vague dismissal of David Lehavi's answer, it seems what you want is a distance between embeddings of ellipsoids. So would the following idea be right for what you want? Given two embedded ellipsoids in $\mathbb{R}^3$, translate and rotate $\mathbb{R}^3$ so that $C_A$ is $\vec{0}$ and $M_A$ is diagonal with eigenvalues ordered by size. What we're doing here is putting ellispoid A in a standard position.

We can take the transformation that took A to standard position and apply it to ellipsoid B which now moves to some new position. Ellipsoid B is now characterized by $C_B'$ and $M_B'$ ($C_B' = C_B-C_A$, $M_B'= O^T M_B O$, where $O$ is the rotation diagonalizing $M_A$ as described previously)

One can now assign a "distance" between A and B which depends only on the eigenvalues of $M_A$, the vector $C_B'$ and the matrix $M_B'$. By moving your ellipsoids together to this "standard position" we've quotiented out the space of "pairs of embedded ellipsoids" by the action of rigid transformations of $\mathbb{R}^3$, as presumably you want your distance function to be invariant under this group.

There are many distances one can assign to ellipsoids in standard position, depending on how you weight "shape" vs. "distance" vs. "orientation", etc. - what you choose is going to depend heavily on what you want. So please do give your needs a little more thought.

From your vague dismissal of David Lehavi's answer, it seems what you want is a distance between a pair of embeddings of ellipsoids. Or equivalently a norm on the 12-dimensional space "pairs of embedded ellipsoids mod rigid motions of $\mathbb{R}^3$".

Would the following idea be right for what you want? Given ellipsoids A and B in $\mathbb{R}^3$, first find the translation and rotation of $\mathbb{R}^3$ which moves ellipsoid A so that it's centered at the origin and has its principal axes aligned with the coordinate axes. That is, a rigid motion taking ellipsoid A to another embedding of an ellipsoid A' so that $C_{A'}=0$, $M_{A'}$ is diagonal with eigenvalues ordered by size. What we're doing here is putting ellipsoid A in a standard position.

Next take the transformation that took A to standard position and apply it to ellipsoid B which moves it to some new embedded ellipsoid B'. Ellipsoid B' is now characterized by $C_{B'}$ and $M_{B'}$ (something like $C_{B'} = C_B-C_A$, $M_{B'}= O^T M_B O$, where $O$ is the rotation diagonalizing $M_A$ as described previously)

One can now assign a "distance" between A and B which depends only on the eigenvalues of $M_A$ (3 parameters), the vector $C_{B'}$ (3 parameters) and the matrix $M_{B'}$ (6 parameters, which you could split into 3 eigenvalues+3 parameters of a rotation if you decomposed $M_{B'}$). By moving your ellipsoids together to this "standard position" we've quotiented out the space of "pairs of embedded ellipsoids" by the action of rigid transformations of $\mathbb{R}^3$, as presumably you want your distance function to be invariant under this group.

There are many distances one can assign to ellipsoids in standard position, depending on how you weight "shape" vs. "distance" vs. "orientation", etc. - what you choose is going to depend heavily on what you want. One family of examples is

$distance(A,B)=a_1|\lambda_A - \lambda_{B'}| + a_2|C_{B'}| + a_3 d_{SO(3)}(I,O)$,

where $\lambda_i$ is the (ordered) vector of eigenvalues of ellipsoid $i$, $|\cdot|$ denotes the norm on 3-vectors, $d_{SO(3)}(I,O)$ is the distance between the rotation $O$ and the identity matrix $I$ in the space of rotations $SO(3)$ and $a_1,a_2,a_3$ are weights.

So please do give your needs a little more thought.