Shortest Path On Cylinder

The middle geodesic path is not a minimal path 1.865, while the other two paths are the shortest path due to symmetry. Figure 10.6 shows how the initial approximation path the right-most thick solid line converges gradually to the final solution wavy thick solid line. The intermediate paths are illustrated by the thin solid lines.

An ant starts at a point P on the bottom edge of a right circular cylinder of radius R and height H.If the ant makes n complete circuits around the cylinder and finishes at a point at the top edge directly above its starting point, find, with justification, the length of its shortest possible path.

The shortest path problem These notes will locally solve the shortest path problem on an ndimensional manifold MRm. More precisely, given a point pin M, we will show that if qis close enough to pthen there is a Let Rgt0 be given, and let pand qbe two points on a circular cylinder of radius R. Use the parametrization ,z Rcos

92begingroup tomi The OP doesn't say so explicitly, but context strongly suggests the problem concerns a path along the curved surface of a cylinder, in which case this answer is perfectly correct, because a line segment gives the shortest distance between two points in a rectangle. 92endgroup

OK, after thinking about it for a while, I came up with an answer. Step 1 Find the caps of the cylinder, in other words two closed disjoint paths along the graph's borders. Step 2 Find a path along the face graph from one cap to the other. Step 3 Create a new sub-graph by removing all edges which lie along the path found in step 2. Keep track of the edges removed as they will be used later.

You're right. The shortest path is not planar. It's part of a helix. Think about even a very short trip along the thread of a bolt. This is from my 1959 edition of Hugo Steinhaus' Mathematical Snapshots A planar path on a cylinder is an ellipse. Steinhaus shows that's a sine curve when you unroll it.

You may assume that the given points must be on the surface of the given cylinder. All input numbers are non-negative and less than or equal to 1000. Output. For each test case, output only one line contains the length of the shortest path on the surface of cylinder.

We have two points on a cylinder's curved surface, one inside and the other outside. How can we find the shortest path between the two points?To do that, wat

The shortest path between any two points on a curved surface is called a geodesic, to clarify the title of this article and the video above. You could imagine expressing the generalized coordinates 92q_j92 in Cartesian coordinates as 92x,y,z92 which will represent any curve on the cylinder between the coordinate points 92x_1,y_1,z_192 and

3 Geodesic Shortest path on a Circular Cylinder, Three Ways 3.1 Using One Dependent Variable z Show that the shortest path between two points on a circular cylinder is along a helix. Do this with the assumption that, z may be written as a function of in cylindrical coordinates rz with r being a constant. The equation of a helix