= *a A*^{2} + * b AB* +
* c B*^{2} + *d A* + *e B* + *f*

so the constant χ^{2} curves
are ellipses. Parameter error determination depends on understanding
the region where χ^{2} is
near its minimum value,
for example:

χ^{2}(*a*,*b*) ≤
χ^{2}_{min} + 1

In particular we need to find the maximum reach of
these ellipses, for example δ*A*
and δ*B* shown below:

If we shift the origin for the parameters *a* and *b*
to be at the center of the ellipse, using:

Δ*b* = *B* - *B*_{center}

the quadratic form can be simply expressed as a matrix equation:

We then generalize that a bit to include *N* re-origined parameters
α_{i} put together into a column
vector.

The symmetric matrix **M** is called the curvature matrix.
In those cases where χ^{2} is
a simple quadratic form, this result is exact. More generally we
can view this equation as the first non-zero term in a Taylor expansion of
Δχ^{2} at the minimum.
**M** is then
closely related to the matrix of second partial derivatives of
χ^{2} (which is known as the Hessian).
We will soon have use for the inverse of the matrix **M**, which
is known as the covariance matrix.

The points we seek (the extreme reach of these ellipses)
may be identified by gradient of χ^{2}
(wrt the parameters α_{i}):
at the extreme points the gradient points directly along the corresponding coordinate axis
as shown below:

Note that because all the ellipses are self-similar, the extreme points all lie along a ray (shown in dotted red above).

The gradient of χ^{2} is:

so the requirement that the gradient points directly along the *p*^{th}
coordinate axis amounts to the equation:

where *e*_{p} is the unit vector in the *p*^{th}
coordinate direction (e.g., *e*_{1}=(1,0,...,0),
*e*_{2}=(0,1,0,...,0), etc.) and λ
is the length of the vector. We can find the locations where the gradient is
along the *p*^{th} coordinate direction, by inverting the above
equation:

the result is a ray of locations parametrized by λ
with larger λ corresponding to larger values of the associated
Δχ^{2} ellipse. If we substitute the equation
for this ray into the equation for Δχ^{2}
we can associate the value of λ with the corresponding value
of Δχ^{2}:

so

and

As noted above, generally we are interested in Δχ^{2}=1,
hence the mantra: the error in a parameter is the square root of the corresponding
diagonal element of the covariance matrix. However I reproduce below a table
from *Numerical Recipes* which at least
shows that sometimes other values of Δχ^{2}
may be of interest.