minimising the square of a distance

**Mppl** · July 2nd 2010, 03:42 AM

Well because of an exercise involving a curve and a point not in that curve I found out that the minimum of the distance between the curve and the point is found at a point such that the derivative of the curve vector is perpendicular to the vector (curve-external point vector). I could find it cause being the curve X(t) and the point P the distance between the curve and the point is abs(X(t)-Q) and setting the derivative to zero to get the critical point we get that to have a critical point X'(t) is perpendicular to X(t)-Q but i cant prove thats a minimum and not a maximum... how can i do it?

help would be apreciated...thanks

**Failure** · July 2nd 2010, 07:44 AM

Originally Posted by Mppl

Well because of an exercise involving a curve and a point not in that curve I found out that the minimum of the distance between the curve and the point is found at a point such that the derivative of the curve vector is perpendicular to the vector (curve-external point vector). I could find it cause being the curve X(t) and the point P the distance between the curve and the point is abs(X(t)-Q) and setting the derivative to zero to get the critical point we get that to have a critical point X'(t) is perpendicular to X(t)-Q but i cant prove thats a minimum and not a maximum... how can i do it?

help would be apreciated...thanks

I'm not exactly sure about your notation. I assume that you have a curve $\vec{x}(t)$ parametrized by $t$ , and a point $P$ with position vector $\vec{p}:= \vec{OP}$ . If so, why don't you just consider the square of the euclidian distance of $\vec{x}(t)$ and $\vec{p}$ , that is
$d^2(t) := (\vec{x}(t)-\vec{p})^2$

Its first derivative is $\frac{d}{dt}d^2(t)=2\cdot (\vec{x}(t)-\vec{p})\cdot \dot{\vec{x}}(t)$

Obviously, the condition that this derivative be equal to 0 is equivalent to your condition that the vector $\vec{x}(t)-\vec{p}$ be perpendicular to the tangent vector $\dot{\vec{x}}(t)$ .

To distinguish between points $\vec{x}(t)$ of shortest (local) distance to $P$ from those of greatest (local) distance, you can now check the sign of the second derivative of $d^2(t)$

**Mppl** · July 2nd 2010, 09:21 AM

I already knew that is just that I'm wondering there must be a way of proving that in the general case that this very crit point we found is a minimum and not a maxmimum, and as I compute my second derivative I can't find a reason for it to be always positive,is it possible to prove it on the general case?
thank you

**Failure** · July 2nd 2010, 10:00 AM

Originally Posted by Mppl

I already knew that is just that I'm wondering there must be a way of proving that in the general case that this very crit point we found is a minimum and not a maxmimum, and as I compute my second derivative I can't find a reason for it to be always positive,is it possible to prove it on the general case?
thank you

No, because those critical points $\vec{x}(t)$ can be points of locally minimal or locally maximal distance from $P$ , as the case may be.

For example, let $\vec{x}(t)$ be the parametrization of a circle and let $P$ be a point outside the circle. There are two points where the derivative of $d^2(t)$ is zero: one has minimal distance and the other has maximal distance from $P$ .

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