To overcome the problems with the algebraic distances, it is natural to replace them by the orthogonal distances which are invariant to transformations in Euclidean space and which do not exhibit the high curvature bias.
The orthogonal distance 
between a point 
and a
conic Q(x,y) is the Euclidean distance between 
and the point
in the conic whose tangent is orthogonal to the
line joining and 
(see fig:orth-dis).
Figure 2: Orthogonal distance of a point to a conic
Given n points ( 
), the orthogonal distance fitting is
to estimate the conic Q by minimizing the following function
However, as the expression of is very complicated (see below), an
iterative optimization procedure must be carried out. Many techniques are
readily available, including Gauss-Newton algorithm, Steepest Gradient
Descent, Levenberg-Marquardt procedure, and simplex method. A software
ODRPACK (written in Fortran) for weighted orthogonal distance regression is
domain public and is available from NETLIB (netlib@ornl.gov).
Initial guess of the conic parameters must be supplied, which can be obtained
using the techniques described in the last section.
Let us now proceed to compute the orthogonal distance . The subscript i
will be omitted for clarity. Refer again to fig:orth-dis. The conic
is assumed to be described by
Point 
must satisfy the following two equations:
Equation (9) merely says the point is on the conic, while
(10) says that the tangent at is orthogonal to the vector
.
Let 
and 
. From (9),
where 
. From (10),
Substituting the value of 
(11) in the above equation leads to the
following equation:
where
Squaring the above equation, we have
Rearranging the terms, we obtain an equation of degree four in 
:
where
The two or four real roots of (13) can be found in closed form. For one
solution , we can obtain 
from (12), i.e.:
Thus, is computed from (11). Eventually comes the orthogonal
distance d, which is given by
Note that we possibly have four solutions. Only the one which gives the smallest distance is the one we are seeking for.
