Shortest Point on Cylinder Vertices

  • Context: Undergrad 
  • Thread starter Thread starter injun_joe
  • Start date Start date
  • Tags Tags
    Shortest distance
Click For Summary
SUMMARY

The shortest path between opposite vertices on a cylinder can be determined by parametrizing points using angle and height coordinates. To find this path, one must plot the start and end points on a plane, draw a straight line between them, and translate this line to the cylinder's surface. It is essential to account for the non-uniqueness of angle-height coordinates by plotting each point twice, once at the original angle and once at an angle 360º less. The shortest path can be visualized by unrolling the cylinder into a rectangle and measuring the straight line distance between the points.

PREREQUISITES
  • Understanding of cylindrical coordinates
  • Familiarity with geometric visualization techniques
  • Basic knowledge of calculus for path optimization
  • Experience with parametric equations
NEXT STEPS
  • Research cylindrical coordinate systems and their applications
  • Learn about geometric transformations and unrolling surfaces
  • Study optimization techniques in calculus for shortest paths
  • Explore parametric equations and their graphical representations
USEFUL FOR

Mathematicians, physics students, and engineers interested in geometric optimization and surface pathfinding on cylindrical shapes.

injun_joe
Messages
4
Reaction score
0
What will be the shortest point on the opposite "vertices" of a cylinder?
 

Attachments

  • untitled.JPG
    untitled.JPG
    3.1 KB · Views: 406
Physics news on Phys.org
Do you mean the shortest PATH along the surface of the cylinder?

I imagine if you parametrize points by angle and height, plot your start and end points on a the plane, draw a straight line between them, and then translate those to the points on the surface to take the length.

At least, that'd be my starting point. You have to consider that angle-height coordinates aren't unique. When you plot the points, you need to plot one twice. One at one angle and one 360º less. You draw two paths and pick the shortest.

Then you'd have to prove it. I'm not sure how you'd do that, but it seems obvious enough.
 
Cut the cylinder vertically and unroll it to a rectangle. Draw the straight line between the points (having cut it so that can be done). Take its length.
 

Similar threads

MHB APC.3.1.2 shortest distance between curve and origin
  • · Replies 4 ·
Replies
4
Views
2K
MHB Shortest distance between (18,0) and y=x^2
  • · Replies 2 ·
Replies
2
Views
2K
High School Shortest distance between two points
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad Does Light Travel the Shortest Path in Curved Space-Time Around a Neutron Star?
  • · Replies 82 ·
3
Replies
82
Views
5K
MHB Shortest distance between line and point
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Why do we need to imagine a varied path in the calculus of variations?
  • · Replies 2 ·
Replies
2
Views
1K
Which Path is Quickest? Investigating Ball Bearings on an Inclined Tube
  • · Replies 14 ·
Replies
14
Views
3K
Undergrad Distance from origin to a surface
  • · Replies 4 ·
Replies
4
Views
10K
Distance from a point on a circle to an arbitrary axis
  • · Replies 6 ·
Replies
6
Views
2K
Graduate Finding Shortest Distance between two 3d Parametrized Curves
  • · Replies 6 ·
Replies
6
Views
6K