Distance between two points on some surface

In summary, the conversation discusses finding the length of line M on surface Z, which is a function of variables x1, x2, x3, etc. The method suggested involves slicing M into small components and using directional derivatives at each point to integrate with respect to a variable such as dx1, dx2, dx3. The problem is compared to a "calculus of variations" problem, which involves finding extrema, but the speaker needs a more general expression for non-extrema problems. Some suggestions are to look into arc length or differential manifolds.
  • #1
exmachina
44
0
I have a surface Z, which is a function of the variables x1,x2,x3... etc. ie. Z(x1,x2,x3...) I have a point Z0 and a point Z1 which corresponds to some point on this surface. There is some line M that connects Z0 and Z1 on the Z surface, note that M does NOT have to be the shortest distance. However, M must be bound to the surface Z.

How do I find the length of the line M on this surface? Ideally the expression would be somehow linked to the directional derivative,

IE. I've been thinking of slicing the line M into tiny little components, and taking the directional directive at each point of M, the integrating with respect to.. something (maybe dx1, dx2, dx3...



Any tips? Some recommend arc length, others recommended differential manifolds (Which I have NO IDEA)
 
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  • #2
That looks to me like a "calculus a variations" problem. You might try to look that up. It's much to complicated to go into here.
 
  • #3
If I recall correctly, Variational calculus is mostly applied to extrema problems, I need a way to find a very general expression, even for the non-extrema problems.
 

What is the distance between two points on a surface?

The distance between two points on a surface is the length of the shortest path connecting the two points. This path is also known as the geodesic and is analogous to a straight line on a flat surface.

How is the distance calculated on a surface?

The distance on a surface is calculated using the surface's metric, which is a mathematical function that assigns a value to each point on the surface. This metric takes into account the curvature of the surface and allows for the calculation of the shortest path between two points.

What is the difference between distance on a surface and distance in Euclidean space?

The main difference between distance on a surface and distance in Euclidean space is that surfaces are curved while Euclidean space is flat. This means that on a surface, the shortest path between two points may not be a straight line, while in Euclidean space it always is.

Can the distance between two points on a surface be negative?

No, the distance between two points on a surface cannot be negative. Distance is always a positive value, representing the length of the path between two points.

Are there different methods for calculating distance on different types of surfaces?

Yes, there are different methods for calculating distance on different types of surfaces. For example, on a sphere, the distance is calculated using great circles, while on a plane, the distance is calculated using the Pythagorean theorem. The method used depends on the curvature and properties of the surface.

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