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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?

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 $1$.

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    $\begingroup$ Which circle is the biggest circle? $\endgroup$
    – Ben McKay
    Commented Oct 7, 2016 at 10:17
  • $\begingroup$ @BenMcKay I wrote "biggest circle" because I am not sure about the uniqueness; it may depend on the direction, in which one stretches the plane. My concerns about uniqueness are motivated by the fact, that one always can find a stretching along one of the major axes, that makes the points co-cyclic or, equivalently formulated, finds an ellipse in standard orientation through the points. $\endgroup$
    – Manfred Weis
    Commented Oct 7, 2016 at 11:09
  • $\begingroup$ There is something here I must be misunderstanding. If the have a linear transformation that puts the four points on a circle, cannot you then combine that linear transformation with a dilation that makes the circle as big as you want? $\endgroup$
    – Michael Renardy
    Commented Oct 7, 2016 at 11:44
  • $\begingroup$ @MichaelRenardy good point; I have edit accordingly. $\endgroup$
    – Manfred Weis
    Commented Oct 7, 2016 at 13:37

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