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August 8th, 2017, 02:43 PM  #1 
Newbie Joined: Aug 2017 From: Costa Rica Posts: 2 Thanks: 0  Optimization of lengths in space
i've been presented a problem of calculus in R3, i need help ASAP ! here's the problem 1. Let S be a cylindrical surface in space. Consider two different points, A and B in space, none of them in S. Find a point P over S so that the addition of the lengths of the segments AP and PB is minimal. 2. Enunciate possible extentions to part 1, propose solutions. 
August 15th, 2017, 05:41 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,576 Thanks: 667 
It looks like the setup, at least, is pretty straight forward. We can select a coordinate system in which the zaxis is the central axis of the cylinder and the intersection of the cylinder with the xyplane is the circle $\displaystyle x^2+ y^2= R^2$. We can use the angle, $\displaystyle \theta$, as parameter for the circle to write $\displaystyle x= R cos(\theta)$ and $\displaystyle y= R sin(\theta)$. We can take $\displaystyle z= t$ as the other parametric equation. Now, let one of the two points not on that cylinder be $\displaystyle (x_0, y_0, z_0)$ and the other point $\displaystyle (x_1, y_1, z_1)$. The distance from the first point to a point on the cylinder is $\displaystyle \sqrt{(R cos(\theta) x_0)^2+ (R sin(\theta) y_0)^2+ (t z_0)^2}$ and the distance from that point on the cylinder to the second point is $\displaystyle \sqrt{(R cos(\theta) x_1)^2+ (R sin(\theta) y1)^2+ (t z_1)^2}$. The total distance is the sum of those, $\displaystyle \sqrt{(R cos(\theta) x_0)^2+ (R sin(\theta) y_0)^2+ (t z_0)^2}+ \sqrt{(R cos(\theta) x_1)^2+ (R sin(\theta) y_1)^2+ (t z_1)^2}$. Find values of t and $\displaystyle \theta$ that minimize that. 
August 15th, 2017, 01:50 PM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,540 Thanks: 920 Math Focus: Elementary mathematics and beyond 
The shortest distance between two points is a straight line (triangle inequality). Choose P so that A, P and B are collinear and P is between A and B.

August 19th, 2017, 11:05 AM  #4 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,082 Thanks: 87 
This is a problem in Lagrange multipiers. Minimize $\displaystyle f(x,y,z)=P_{0}P+PP_{1}$ subject to $\displaystyle g(x,y,z)=x^{2}+y^{2}r^{2}=0$ let $\displaystyle h=f+\lambda g$ $\displaystyle \frac{\partial h}{\partial x}=0, \frac{\partial h }{\partial y}=0,\frac{\partial h}{\partial z}=0, g=0 $ 4 eqs in 4 unknowns 

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lengths, optimization, space 
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