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I search for the maximum of a quadric for points on a sphere.

I have an affine transform A (4x4 matrix, in homogeneous coord.) and apply it to points on (and inside) a sphere [itex] x \in S_{m,r} \Leftrightarrow (x-m)^2<=r^2 [/itex]. (Although I think the extremum must be on the surface of the sphere?).

Now I want to find the maximum displacement of any point in/on S:

[itex] d^2 (x) = (A x - x)^2 = ( (A-E) x)^2 = x^T (A-E)^T (A-E) x [/itex]

The matrix [itex] B:= (A-E)^T (A-E) [/itex] is of course symmetric.

so what is:

[itex] max_{x \in S} \ d^2(x) = max_{x \in S} \ x^T B x [/itex]

Thanks for any help...

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# Maximization with constraint

Can you offer guidance or do you also need help?

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