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the case of unknown range data, we have presented an iterative algorithm for computing both the
motion and the depth coordinates of image points in the presence of measurement uncertainty.
The algorithm incorporates all of the constraints of general rigid body displacements. As far as
we are aware, this is the rst such algorithm to solve directly the relevant least-squares equation
simultaneously using analytic derivatives over the whole set of unknowns in the problem. We are
able to do this because of the powerful tools made available to us in the geometric algebra { for
instance, being able to minimize directly with respect to the rotors underpins the main part of the
algorithm, a facility which is not readily available in other frameworks. We have compared the
algorithm to one of the basic linear algorithms for estimating structure and motion showing that it
can improve the accuracy of the results by, in some cases, large factors. One of the main uses of the
algorithm may be as an additional stage in other quicker algorithms when more re ned estimates
are required.
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