BINARY OPERATIONS WITH ELLIPSOIDS

The sum of two ellipsoids

• Given the ellipsoidal sets E1 and E2, their sum is defined by E = { x+y | x in E1 and y in E2 }.
• The sum of ellipsoids is normally not an ellipsoid.
• In order to represent the sum of two ellipsoids there is a family Y of ellipsoids contained in the sum so that the union of Y is the sum exactly. Similarly, there is a family X of ellipsoids covering the sum so that the intersection of X is the sum exactly.
• The largest volume ellipsoid inside the sum is in Y and the smallest volume ellipsoid covering the sum is in X.

Figures

• Sum represented as intersection of the family X of external ellipsoids
• Sum represented as union of the family Y of internal ellipsoids
• Both families X and Y displayed
• The largest volume ellipsoid inside the sum (the largest area in 2D),
• The smallest volume ellipsoid containing the sum

The difference of two ellipsoids

• Given the ellipsoidal sets E1 and E2, their difference is defined by E = { x | x + y in E1 for all y in E2 }.
• The difference of ellipsoids is normally not an ellipsoid and it may be empty.
• In order to represent the difference of two ellipsoids there is a family Y of ellipsoids internal to the difference so that the union of Y is the difference exactly. Similarly, there is a family X of ellipsoids covering the difference so that the intersection of X is the difference exactly.
• The largest volume ellipsoid inside the difference is in Y and the smallest volume ellipsoid covering the difference is in X.

Figures

• Difference represented as intersection of the family X of external ellipsoids
• Difference represented as union of the family Y of internal ellipsoids
• Both families X and Y displayed
• The largest volume ellipsoid inside the difference (the largest area in 2D),
• The smallest volume ellipsoid containing the difference

The ELLDIF.M routine of GBT can be used to compute the approximating ellipsoids of all types above.