My Math Forum  

Go Back   My Math Forum > College Math Forum > Real Analysis

Real Analysis Real Analysis Math Forum

LinkBack Thread Tools Display Modes
August 8th, 2017, 02:43 PM   #1
Joined: Aug 2017
From: Costa Rica

Posts: 2
Thanks: 0

Arrow 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.
edugonzaa is offline  
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 z-axis is the central axis of the cylinder and the intersection of the cylinder with the xy-plane 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 $\displaystyle \sqrt{(R cos(\theta)- x_1)^2+ (R sin(\theta)- y-1)^2+ (t- z_1)^2}$.

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.
Country Boy is offline  
August 15th, 2017, 01:50 PM   #3
Global Moderator
greg1313's Avatar
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.
greg1313 is online now  
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.
$\displaystyle f(x,y,z)=P_{0}P+PP_{1}$
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
zylo is offline  

  My Math Forum > College Math Forum > Real Analysis

lengths, optimization, space

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Need Help Finding the lengths. Inkaho Trigonometry 2 May 13th, 2015 08:03 PM
Lengths and angles Zman15 Geometry 2 February 22nd, 2015 03:21 PM
Find lengths of the triangle ? zgonda Algebra 9 August 27th, 2010 02:18 PM
Container Space Optimization c8h8r8i8s8 Algebra 1 March 1st, 2009 10:51 AM
How can I calculate the lengths of all 3 sides? SecretSamurai Algebra 6 February 16th, 2009 04:21 AM

Copyright © 2017 My Math Forum. All rights reserved.