# Shortest distance

• Sep 25th 2012, 08:14 AM
LChenier
Shortest distance
Hi every one

There is a question that is bugging me since a while. I remember this to be very easy to do when I was studying Calculus, but since this was many years ago, I cant recall how to do.

I tried to figure it myself, but no luck

I looked in old books and google it, but with no luck.

any way here is the question:

Given two function f(x,y) and g(x,y), where f(x,y) don't intersect g(x,y) at any point.

What is the shortest distance between the two functions and what is the equation of this shortest vector?

If it is simpler to show with f(x) and g(x), do so ... If I get the idea I can probably find how to do with f(x,y) and g(x,y)

I would really appreciate if you can take a minute and refresh my memory. Otherwise, I guess that this question will bug me for the rest of my life

Thank you
• Sep 25th 2012, 11:38 AM
richard1234
Re: Shortest distance
Given two points $\displaystyle (x_1,y_1,f(x_1,y_1))$ and $\displaystyle (x_2, y_2, g(x_2, y_2))$, the distance D is equal to

$\displaystyle D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (g(x,y) - f(x,y))^2}$

Also,

$\displaystyle D^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 + (g(x,y) - f(x,y))^2$

Basically, minimizing $\displaystyle D^2$ is the same as minimizing $\displaystyle D$.
• Sep 25th 2012, 02:19 PM
LChenier
Re: Shortest distance
Got it to work

I knew that it was simple.

Thank you