- 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 coordinatesand - To see that they are fixed under Euclidean transformations
- 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 -
can be computed from five independent angle
consrtaints on a plane.