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Math Help - Reflections and Reflection Groups - Basic Geometry

  1. #1
    Super Member Bernhard's Avatar
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    Reflections and Reflection Groups - Basic Geometry

    I am reading Kane's book on Reflections and Reflection Groups and am having difficulties with some basic geometric notions - see attachment Kane paes 6-7.

    On page 6 in section 1-1 Reflections and Reflection Groups (see attachment) we read:

    " We can define reflections either with respect to hyperplanes or vectors. First of all, given a hyperplane  H \subset E through the origin, let L = the line through the origin that is orthogonal to H. So  E = H \oplus L "

    My question is why/how is  E = H \oplus L ???

    Can anyone help?

    (see my intuitive diagrams - my notion of  H \oplus L is a line going through a plane)

    Maybe I need to read some basic differential geometry?

    Peter
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    Re: Reflections and Reflection Groups - Basic Geometry

    this has nothing to do with differential geometry (we're not considering differentiability of anything). where you're going wrong is in thinking about which direction L is going. L is perpendicular to the hyperplane H, in physics, one often hears the term "normal vector (at 0)" to describe L. in other words, L is coming "straight up" (perpendicular) out of the plane. for example, if H was the xy-plane in euclidean 3-space, then L would be the z-axis.

    Plane + perpendicular line = 3-space.

    actually, we could use ANY line, not lying in the plane H but then we lose orthogonality. and orthogonality makes the arithmetic easier (it makes our basis for V look like the standard basis "rotated" to line up with H and L).

    in other words, one way to describe a point in 3-space, is to pick a point in a plane (any plane will do, imagine it being the xy-plane "tilted" by some amount), and to pick a distance "up off the plane" (how far along the line L we've come).

    Reflections and Reflection Groups - Basic Geometry-hyperplane.jpg
    Thanks from Bernhard and mash
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    Super Member Bernhard's Avatar
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    Re: Reflections and Reflection Groups - Basic Geometry

    Yes, take your point re differential geometry - my comment was rather careless ... was thinking that a differential geometry text may cover some of this in the preamble to winding into the subject ... but I suppose an algebra text covering vector spaces would be more likely to help!

    Thanks for help around this roadblock ... not quite sure why I did not see it ...


    Can now take Reflections and Refection Groups a bit further thanks to your help

    Peter
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