Conformal Geometric Algebra in Stochastic Optimization Problems ...

140 CHAPTER 4. A PRIMER ON POSE ESTIMATION WITH CGA
where r denotes the radius of circle K. After some algebra it follows
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Rθ = cos(θ/2) + sin(θ/2) e1e2 + r (cos(θ) + 1)e1e + sin(θ)e2e
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= exp θ/2 Lθ . (4.2)
The element Lθ is a line representing the rotation axis of Rθ and plays the role of
the imaginary unit i of complex numbers, for L 2 θ = −13 . On writing
Rθ = P1(P3P2)P1
it can be recognized that Rθ is the reflection of R ′ θ := P3P2 in plane P1. Hence
Rθ rotates by an angle θ being twice the dihedral angle between P3 and P2. The
position of Rθ is depicted on the left of figure 4.5.
Fig. 4.5: Left: depending on the angle θ, the motor Rθ ‘orbits’ around circle K.
Right: each point x3 on the circle C guarantees a congruent triangle.
Recall that initially 3-point problems are to be tackled. Thus 3-point models (triangles)
have to be fitted to three corresponding projection rays. Clearly, a two-point
model can easily be fitted: simply follow the angle bisector of two projection rays
until the model forms an isosceles triangle with the projection rays, where O is
considered the optical center of the camera. Then one possible 2-point fit is accomplished.
Now let K be the unique circumcircle of that triangle, whence the
canonical coordinate system 4 and the motor Rθ can be defined. Thus a two-point
model can rigidly be moved such that the constituent model points {x1,x2} remain
on their respective projection rays, i.e. the 2-point fit may be varied w.r.t. θ. Note
that certain fits must necessarily be extendable to 3-point fits if these exist.
3 Not to be confused with one of the projection rays.
4 Notice that the canonical coordinate system is actually made use of.
Lθ