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### Alternate derivation: algebraic

To describe the relationship between R, , A1, and A2 more exactly, and to connect the above equations with those found in , we offer the following algebraic derivation.

Recall that a point produces an image through the equation . Without loss of generality, we can assume that is given with respect to the first camera's coordinate frame to yield the following two imaging equations: where and are scale factors, is the identity matrix and is the null vector. By letting ( is ), we achieve the following relation: (8) (9)

Geometrically, this equation says that the vector on the left is a linear combination of the two vectors on the right. Therefore, they are all coplanar, and the vector is perpendicular to that plane: which is identical to (6).

Similarly, the vector is perpendicular to the vectors in (8): This is a surprising result because it gives us a new and equivalent expression for F: (10)

which shows that F can be written as the product of an anti-symmetric matrix and an invertible matrix A2 R A1-1 .   Next: Alternate derivation: from the Up: Essential and fundamental matrices Previous: Essential and fundamental matrices
Stanley Birchfield
1998-04-23