From Affine to Euclidean Space

Here, by Euclidean space we actually mean space under the group of
similarity transforms, *i.e.* we allow uniform changes of scale in
addition to rigid displacements. This permits a very elegant algebraic
formulation, and in any case scale can never be recovered from images,
it can only be derived from prior knowledge or calibration. (You can
never tell from images whether you are looking at the real world or a
reduced model). In practice, Euclidean information is highly
desirable as it allows us to measure angles and length ratios.

Last century, Laguerre showed that Euclidean structure is given by the
location in the plane at infinity
of a distinguished
conic, whose equation in a Euclidean coordinate system is

This is known as the

As in the projective to affine case, prior Euclidean information is
needed to recover Euclidean structure from affine space. Perhaps the
easiest way to do this is to reconstruct known circles in 3D space.
Algebraically, each such circle intersects
in exactly
two complex points, and these always belong to [23].
itself can be reconstructed from three
such circles. Let the resulting equation be

A change of coordinates is needed to bring into the form of equation (4.1). As the matrix Q is symmetric, there is an orthogonal matrix P such that:

Setting

With a further rescaling along each axis, we get equation (4.1).

Another way to proceed is to use the basic extended Euclidean
invariant: the angle
between two coplanar lines *L* and *L*'.
Such angles also put a constraint on
and can be used to
compute it. Let *A* and *A*' be the intersections of the two lines
with
.
Let *I* and *J* be the intersections with
of the (line at infinity in the) plane defined by these two
lines. Laguerre's formula states that:

We can write

A polynomial constraint on Q is easily derived. Eliminate

extract

and substitute into (4.2) to provide a quadratic polynomial constraint on

Theoretically, the absolute conic can be computed from the above constraint given 5 known angles. However, in practice there does not seem to be a closed form solution and in our experiments we have used different Euclidean constraints. But the above discussion does clearly show the relationships between the different layers of projective, affine and Euclidean reconstruction, and specifies exactly what structure needs to be obtained in each case.