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 

.