Algorithm to determine if a 3D ellipsoid is contained within another?Can anyone point me to an algorithm for how to efficiently check if a 3D ellipsoid is contained within another one? We can assume their origins are collocated.I am dealing with covariance ellipsoids constructed from matrices.

Question

Veronica Riddle · Accepted Answer

Since the two ellipsoid share the same center, then we can take this center to be the origin of the coordinate system, and then the equations of the two ellipsoids will berTQ1r=1&amp;nbsp;and&amp;nbsp;rTQ2r=1Diagonalizing&amp;nbsp;Q1&amp;nbsp;so that&amp;nbsp;Q1=R1D1R1T, thenrTR1D1R1Tr=1Define&amp;nbsp;u=D1(12)R1Tr&amp;nbsp;as a change of variable, then&amp;nbsp;uTu=1&amp;nbsp;so that the first ellipsoid is transformed into the unit sphere. Applying the same transformation to the second ellipsoid, we getuTD1(−12)R1TQ2R1D1(−12)u=1which is of the form&amp;nbsp;uTQ2′u=1&amp;nbsp;withQ2′=D1(−12)R1TQ2R1D1(−12)Now diagonalize&amp;nbsp;Q2′&amp;nbsp;into&amp;nbsp;Q2′=R′D′R′TFinally compute the diagonal matrix&amp;nbsp;D&amp;nbsp;″=D′(−12)The diagonal entries of D&#039;&#039; are the lengths of the semi-axes of the second ellipsoid after transformation.If the maximum of the diagonal entries of D&#039;&#039; is less than 1 then the second ellipsoid is totally inside the first ellipsoid. If the minimum of the diagonal entries of D&#039;&#039; is greater than 1 then the second ellipsoid totally contains the first ellipsoid. Otherwise, the two ellipsoids intersect.

Algorithm to determine if a 3D ellipsoid is

Answered question

Answer & Explanation

New Questions in Integral Calculus