The alignment of a pair of planar patches,
and
, provides 3 constraints on the 6 parameter rigid transformation,
, where:
In homogeneous coordinates this can be expressed as the product of a
rotation,
, and a translation,
.
The rotation constraint is the set of rotations that align the surface
patch normals,
with
. This is determined by finding an arbitrary rotation that aligns
with
and then rotating around
. We can align
with
by rotating by
radians about the bisector,
, to
and
. This gives the value of the constraint at
.
where
describes the rotation around an axis defined by
by
radians. A good reference for this can be found in [
11
]. The constraint is then found by rotating around
.
The translation that aligns a pair of planes is only constrained in the direction of the plane normals. Note that the first plane has now been rotated so that it is parallel with the second.
We now define a pair of orthogonal vectors
and
, which are mutually orthogonal to the plane normals
, such that:
The translation constraint is then given by:
Anthony Ashbrook