The big advantage of use of algebraic distances is the gain in computational efficiency, because closed-form solutions can usually be obtained. In general, however, the results are not satisfactory. There are at least two major reasons.
Figure 1: Normalized conic
To understand the second point, consider a conic in the normalized system (see fig:normal_conic):
The algebraic distance of a point to the conic Q is given by :
where is the distance from the point to the center O of the conic, and is the distance from the conic to its center along the ray from the center to the point . It is thus clear that a point at the high curvature sections contributes less to the conic fitting than a point having the same amount of noise but at the low curvature sections. This is because a point at the high curvature sections has a large and its is small, while a point at the low curvature sections has a small and its is higher with respect to the same amount of noise in the data points. Concretely, methods based on algebraic distances tend to fit better a conic to the points at low curvature sections than to those at high curvature sections. This problem is usually termed as high curvature bias.