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Math Help - Straightness on Cylinders and Cones

  1. #1
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    Straightness on Cylinders and Cones

    1.) What lines are straight on a cylinder or cone, with respect to the surface? Why? Why not?

    2.) Can geodesics intersect themselves on cones and cylinders?

    3.) Can there be more than 1 geodesic joining 2 points on cones and cylinders?

    4.) What happens on cones with varying cone angles, including cone angles greater than 360 degrees?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by AfterShock View Post
    1.) What lines are straight on a cylinder or cone, with respect to the surface? Why? Why not?

    2.) Can geodesics intersect themselves on cones and cylinders?

    3.) Can there be more than 1 geodesic joining 2 points on cones and cylinders?

    4.) What happens on cones with varying cone angles, including cone angles greater than 360 degrees?
    Cones and cylinders have a Gaussian curvature of zero, so they can be cut
    and laid out flat. Then the Geodesics are straight lines on this flattened
    surface.

    That should be sufficient for questions 1 to 3. I don't understand what
    Q4 means.

    RonL
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    I need help with this question still:

    Can geodesics intersect themselves on cones and cylinders?

    Thanks!
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by AfterShock View Post
    I need help with this question still:

    Can geodesics intersect themselves on cones and cylinders?

    Thanks!
    Geodesics never intersect. (Though they may be finite in length because they "meet" themselves, eg a geodesic on a sphere is a circle.)

    -Dan
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    "Geodesics never intersect. (Though they may be finite in length because they "meet" themselves, eg a geodesic on a sphere is a circle.)"

    I disagree. I am stuck on the intersecting part, and how many geodesics there are connecting two points.

    For example, I know at 360 degrees there is only one.
    At 180 degrees, there are 2
    And at 90 degrees there are 3, but I have no way of proving this.

    As far as intersections go, according to my professor, there are 2 intersections from 180 degrees to 90 degrees, 3 from 90-60 degrees..how do I prove this. He tried drawing these R^2 planes that was a digraph of R^3, however, it made no sense to me.

    My professor agreed with me that there are any number of geodesics between any two points on a cylinder not on the same parallel. However, he disagreed when I said there is an infinite amount of geodesics between two points on a cone by simply choosing the degree of the cone (which I obtained from some online research). I got it from Geodesics on a Cone which states, "So there can be any number of geodesics between two points on a cone, provided you choose θ appropriately." It makes sense to me after reading it.

    I am really confused now.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by AfterShock View Post
    "Geodesics never intersect. (Though they may be finite in length because they "meet" themselves, eg a geodesic on a sphere is a circle.)"

    I disagree. I am stuck on the intersecting part, and how many geodesics there are connecting two points.

    For example, I know at 360 degrees there is only one.
    At 180 degrees, there are 2
    And at 90 degrees there are 3, but I have no way of proving this.

    As far as intersections go, according to my professor, there are 2 intersections from 180 degrees to 90 degrees, 3 from 90-60 degrees..how do I prove this. He tried drawing these R^2 planes that was a digraph of R^3, however, it made no sense to me.

    My professor agreed with me that there are any number of geodesics between any two points on a cylinder not on the same parallel. However, he disagreed when I said there is an infinite amount of geodesics between two points on a cone by simply choosing the degree of the cone (which I obtained from some online research). I got it from Geodesics on a Cone which states, "So there can be any number of geodesics between two points on a cone, provided you choose θ appropriately." It makes sense to me after reading it.

    I am really confused now.
    Hmmmm...my apologies. I had gotten my statement from my General Relativity course. I must have misremembered something. Wait! Of course I'm not remembering correctly because there are an infinite number of geodesics connecting two opposing points on a spherical surface. Obviously then your answer must depend on the geometry, which leads us back to your original question, which I am now obviously unable to answer. Sorry about that!

    -Dan
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    Not a problem! Does anyone know how many geodesics can connect two points (and why my professor said ANY number of geodesics is wrong, which the site clearly states). Also, how many intersecting lines intersect on a cone, and cylinder (depending on the cone angle), and how to prove such.
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  8. #8
    Grand Panjandrum
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    Quote Originally Posted by AfterShock View Post
    Not a problem! Does anyone know how many geodesics can connect two points (and why my professor said ANY number of geodesics is wrong, which the site clearly states). Also, how many intersecting lines intersect on a cone, and cylinder (depending on the cone angle), and how to prove such.
    Your page says: For any n you can find a cone which has points which
    can be connected by n, and no more geodesics, and no pair of points
    can be connected by more than n geodesics.

    It does not claim that there is a cone with pairs of points that can be
    connected by n and no more geodesics for any n. It may be what your
    prof thinks you are claiming is this rather than the above.

    I think the difference between you and your professor is purely a semantic
    difference. I think your wording may be confusing him as to what exactly
    you are claiming, if you word your statement more carefully I think he will
    agree with you.

    RonL

    RonL
    Last edited by CaptainBlack; September 18th 2006 at 10:16 PM.
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