### Euclidean calibration of a plane

• An Euclidean transformation leaves invariant as a set.
• Consequently, under any Euclidean transformation, there are two points on which are fixed. These are called the circular points , with coordinates

and

• To see that they are fixed under Euclidean transformations

(Similarly for )
• The conic is dual to the the circular points. In Euclidean terms it is given by

• The conic is fixed under actions of any Euclidean transformation.
• is the null vector of . This is because .
• For lines and with normals parallel to and respectively, in Euclidean geometry the angle between them is given as

• Projectively, this is given as

• Thus the projective ( ) and the affine ( ) components are completely determined by the image of , but the similarity component is undetermined.
• Once is known metric calibration of a plane is possible. Writing the SVD of as

is the rectifying projectivity up to a similarity.
• can be computed from five independent angle consrtaints on a plane.
Subhashis Banerjee 2008-01-20