This exercise explores fitting an ellipse to a data set, and an example of such data is given in Figure 8.30. The formula for the ellipse is                                 where  and  are positive...

This exercise explores fitting an ellipse to a data set, and an example of such data is given in Figure 8.30. The formula for the ellipse is

where
 and
 are positive and to be determined from the curve fitting. In what follows you are to find an error function and from this find and
.

(a) For a given and
, write down an expression that can be used as a measure of the error between a data point (

_i
,


_i
) and the curve. The requirements on this expression are: it is non-negative and only zero if (

_i
,


_i
) is on the ellipse.

(b) Using your expression from part (a), sum over the data points to produce an error function
().

(c) Minimize
() and find explicit formulas for
 and
. Note that if your error function does not result in you being able to find explicit formulas, then you need to redo parts (a) and (b) so you can.