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## Main Question or Discussion Point

Find a general equation of geodesics on cylinder's surface.

What's the name of these curves?

What's the name of these curves?

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Find a general equation of geodesics on cylinder's surface.

What's the name of these curves?

What's the name of these curves?

mathwonk

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just a guess, cycloids? or if they are unnamed we can call them "tehnos"!

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quasar987

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circular helixes?

A general param would be...

[tex]\gamma(t) = (a\cos(ct),a\sin(ct),bt)[/tex]

up to a rigid motion

A general param would be...

[tex]\gamma(t) = (a\cos(ct),a\sin(ct),bt)[/tex]

up to a rigid motion

Last edited:

George Jones

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Yes, because a cylinder is flat!circular helixes?

A general param would be...

[tex]\gamma(t) = (a\cos(ct),a\sin(ct),bt)[/tex]

up to a rigid motion

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For a given value of "flat"Yes, because a cylinder is flat!

[tex]\kappa_1 \kappa_2 = 0[/tex] to be exact.

Edit:

As a minor point of interest, if one considers that the gauusian curvature of the cylinder is zero, and thus that we can form a cylinder from a flat plane, then straight lines on the plane, become helixs on the cylinder.

But is it true that all geodesics are isomorphic under an isometry of this kind. If the gaussian curvature between two surface is equal, can we identify geodesics on one surface with those on the other?

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Well, the simply-connected cover of both surfaces is the plane and each geodesic of a given surface is the image of a line in the plane under the covering map. So, I think the answer is, yes (sort of): given a geodesic on one surface, we look at its inverse image in R^2. This will be a family of lines (maybe infinite, maybe not). Then take the geodesics in the second surface corresponding to those lines.But is it true that all geodesics are isomorphic under an isometry of this kind. If the gaussian curvature between two surface is equal, can we identify geodesics on one surface with those on the other?

So, we can associate a family of geodesics of one surface for every geodesic of the other.

mathwonk

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mathwonk

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it is a pretty easy problem since all you have to do is unroll the cylinder to a rectangle.

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But, what about surfaces with non zero curvature?So, I think the answer is, yes (sort of): given a geodesic on one surface, we look at its inverse image in R^2.

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Yeah, I guess you could do the same procedure for the non-zero curvature as long as its constant.But, what about surfaces with non zero curvature?

For constant negative Gaussian curv., the simply-connected cover would be the hyperbolic plane. For positive, it would be the 2-sphere.

of course, i'm assuming the two surfaces have the same curvature *and* that there isn't anything too aberrant about either's topology, i.e. both are connected and complete (when unioned with its boundary) etc. etc.

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That's why I classified the problem under "easy".it is a pretty easy problem since all you have to do is unroll the cylinder to a rectangle.

Helices ,as quasar987 said, is the correct answer.

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