Let
be an arbitrary point on the surface. The distance between
and the given point
is given by the function
Note that
and
attain their extrema, if they have any, at the same values of
. This allows us to consider the modified distance function,
So now you're minimizing
subject to the constraint
. This is a perfect candidate for applying the method of Lagrange multipliers.
The Lagrangian in this case would be
which has partial derivatives
Setting all four equation equal to 0, you find from the third equation that either
or
. In the first case, you arrive at a possible critical point of
. In the second, plugging
into the first two equations gives
and plugging these into the last equation gives
So you have three potential points to check:
,
, and
. Evaluating either distance function (I use
), you find that
So the two points on the surface
closest to the point
are
.
