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Is there a name for linear transformations of the plane, that make $4$ points in general convex configuration co-circular, with the biggest circle through those points and, how can they be determined for a specific sample of such a quadruple of points?

Is there a name for linear transformations of the plane, that make $4$ points in general convex configuration co-circular, with the biggest circle through those points and, how can they be determined for a specific sample of such a quadruple of points?

edit: In reply to Michael Renardy's comment, I add the further restrictions, that the eigenvalues of the linear transformations shall be positive and one of them shall be $1$.

Is there a name for linear transformations of the plane, that make $4$ points in general convex configuration co-circular with the biggest circle and, how can they be determined for a specific sample of such a quadruple of points?

Is there a name for linear transformations of the plane, that make $4$ points in general convex configuration co-circular, with the biggest circle through those points and, how can they be determined for a specific sample of such a quadruple of points?

Is there a name for linear transformations of the plane, that make $4$ points in general convex configuration co-circular and, how can they be determined for a specific sample of such a quadruple of points?

Is there a name for linear transformations of the plane, that make $4$ points in general convex configuration co-circular with the biggest circle and, how can they be determined for a specific sample of such a quadruple of points?