Vectors between 0 and PI

  1. Mar 12, 2014 #1
    Hello all,

    Sorry about the crappy title. I'm actually not sure what the call the thing I'm here to ask about, which is why I'm here.

    In the process of reading about gravitational lensing, I've run across an odd mathematical thing that I just don't know how to handle. When a spherical galaxy get's lensed by another galaxy it gets distorted from its original spherical shape into an elliptical shape. People define a parameter known as the shear, which in some sense is a vector whose magnitude tells you the semi-major axis of the elliptical shape, and the direction is the direction of the ellipse from some arbitrary reference point.

    Now the issue is, the shear points in two directions. Think of a vector along the major axis of an ellipse that points in both directions. This means this isn't really a vector at all, and as such it has to be handled differently. This usually results in lots of random 2's thrown into equations to account for this property.

    I've never really come across this mathematical object before and don't even know what to call it. I heard someone mention the name "polar" but googling that doesn't get me anywhere. Does anyone have any idea what object I'm talking about and can explain them further or reference me to a good site/book about them?

    Thanks for your help.
  2. jcsd
  3. Mar 20, 2014 #2
    Sounds a lot like elements of a real projective space (take the n-dimensional vectors, 0 excluded, and call two of them "equal" if they're scalar multiples of each other.) For instance, a vector v is "equal" to -v since v=-1*-v.

    Not quite, though, since two vectors are still considered "distinct" in this context if they have different magnitudes. You'd probably want the product space of RP##^n## with the positive real numbers to allow for said distinct magnitudes.
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