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Differential Geometry Proof (Need a Hand)

  1. Jan 13, 2007 #1
    1. The problem statement, all variables and given/known data

    Let alpha(t) be a regular curve. Suppose there is a point a in R^3 space s.t. alpha(t)-a is orthogonal to T(t) for all t. Prove that alpha(t) lies on a sphere.

    2. Relevant equations
    A regular curve in R^3 is a function alpha: (a,b)-->R^3 which is of class C^k for some k (greater than or equal to) 1 and for which the time derviative does not equal zero for all t in (a,b) Note: For this book all classes will be C^3, unless stated. (it is not stated otherwise in this problem so assume C^3).

    T(t)= Tangent vector field to a regular curve as defined by (d(alpha)/dt)/norm(d(alpha)/dt)).

    Hints: what should be the center of the sphere?
    3. The attempt at a solution

    And here is where I get stuck, I have not been given a crietria for a curve to lie on a sphere (and if I had oh so many years ago I have long since forgotten it). All of my attempts so far have fallen apart due to assumptions I have had to make, which without a guide towards showing something lies on a curve, I am not sure are justified.

    Just a hint, aside from the one given, is all I need to get some of my juices following. I know it isn't proper form to not show my solution; however, it won't be of use as I don't really know where I am heading with it.

    Any help would be great.
    Last edited: Jan 13, 2007
  2. jcsd
  3. Jan 13, 2007 #2


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    First find the answer to the hint given (what should be the center of the sphere?) and my hint is "if [itex]\vec{\alpha}(t)=(\alpha^1,\alpha^3,\alpha^3)[/itex] lies on the sphere then what is the relation btw the coordinates [itex]\alpha^i[/itex]?"
  4. Jan 14, 2007 #3


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    To prove [itex]\alpha[/itex] lies on a sphere, find a point C such that [itex]|\alpha (t) - C|[/itex] is a constant. 1) Do you see why this proves that [itex]\alpha[/itex] lies on a sphere? 2) Can you see, with the right guess for C, how to prove that [itex]|\alpha (t) - C|[/itex] is a constant?
  5. Jan 14, 2007 #4


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    It may be helpful to consider a simplfied example with a kinematical interpretation.
    Consider a particle travelling on a circle. What are the relations between the displacement vector [from the origin of the circle] and the velocity vector?
  6. Jan 14, 2007 #5
    Thanks for the help guys, I think I got it (I am going to run through it with someone else), then I will post it.

    Thanks for the help.
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