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Math Help - Projection

  1. #1
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    Projection

    Proposition: Let  S be a circle on  \displaystyle \Sigma and let  T be its projection on  \mathbb{C} . Then (a) if  S contains  (0,0,1) ,  T is a line; (b) if  S does not contain  (0,0,1) ,  T is a circle.


    If  S contains  (0,0,1) then the circle in not entirely contained on the sphere. Is this an intuitive reason why the projection is a line? Whereas a circle that does not contain  (0,0,1) is a circle completely contained on the sphere. Thus its projection is a circle.
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  2. #2
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    Quote Originally Posted by manjohn12 View Post
    Proposition: Let  S be a circle on  \displaystyle \Sigma and let  T be its projection on  \mathbb{C} . Then (a) if  S contains  (0,0,1) ,  T is a line; (b) if  S does not contain  (0,0,1) ,  T is a circle.


    If  S contains  (0,0,1) then the circle in not entirely contained on the sphere. Is this an intuitive reason why the projection is a line? Whereas a circle that does not contain  (0,0,1) is a circle completely contained on the sphere. Thus its projection is a circle.
    I cannot think of a great intuitive explanation why lines and circles go to lines and circles, I think you need to do some algebraic calculations. However, this is how I remember it. If a circle passes through (0,0,1) on \Sigma then it passes through the point at infinity. Thus, when you map it onto \mathbb{C} (i.e. project) it means the image becomes unbounded. Since circle are bounded it cannot be a circle. It has to be a line then. And if a circle is on \Sigma and not through (0,0,1) it means it stays away from the point at infinity, therefore it stays bounded. Thus, such a circle gets projected back onto a circle.

    For intuition you might want to think in 2d. Imagine a circle sitting on top of the real number line. Can you visualize this?
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  3. #3
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    Yes. E.g. the circle is homeomorphic to  [0,1] . I was thinking that since  (0,0,1) is the point at infinity, a circle that contains it also contains points not on the sphere. The part that is contained on the sphere can be "stretched" out into a line on  \mathbb{C} .

    Or maybe think of lines as infinitely long circles.
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