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Math Help - Parallel transport frame

  1. #1
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    Parallel transport frame

    Hi there,

    A curve is parameterized by: r(t)={x(t),y(t),z(t)}
    x, y and z are differentiable infinity times.
    {T0,N10,N20} is an initial frame at r(0).

    How to determine the frame through the curve by parallel transporting?

    I've been surfing the internet, trying to figure out how. Wasn't exactly a success...

    Thanking you in anticipation

    Edit: Why didn't I post this under differential geometry..

    Moved to better location ... CB
    Last edited by CaptainBlack; November 5th 2011 at 07:01 AM.
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  2. #2
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    Re: Parallel transport frame

    fx. if r(t) is set {sin[t], cos[t],t}, the frame will rotate around the curve with unchanged velocity.

    How to show that velocity depends on the steepness?
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  3. #3
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    Re: Parallel transport frame

    How do you define "parallel"? In Euclidian space E^3 "parallel" just means constant.
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  4. #4
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    Re: Parallel transport frame

    The frame is said to be parallel to the curve r(t) if its derivative is tangential along the curve.

    I found this picture showing the parallel transport frame on a spiral.

    Parallel transport frame-frame.jpg
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  5. #5
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    Re: Parallel transport frame

    So the natural frame (T, N, B) described by the Frenet satisfies you requirement.
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  6. #6
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    Re: Parallel transport frame

    Quote Originally Posted by xxp9 View Post
    So the natural frame (T, N, B) described by the Frenet satisfies you requirement.
    Frenet frame is just one of solutions...

    Some info about parallel transportation frames can be found in book: "Game Programming Gems 2". Preview of it is available online at google books, see page 215... there is a nice explanation why the Frenet Frame is NOT the best solution in some cases.

    Great source about this topis is the article by Wang et al. "Computation of Rotation Minimizing Frames" - there is even algorithm given. Article can be found at:
    http://www.ag.jku.at/pubs/2007wjzl.pdf
    or
    http://research.microsoft.com/en-us/...g%20frames.pdf

    Just today, i found paper by Hanson and Ma: "Parallel Transport Approach
    to Curve Framing" - I haven't read whole paper yet, but looks promising. See:
    http://citeseerx.ist.psu.edu/viewdoc...=rep1&type=pdf
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