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Math Help - Mirror surface reflection question

  1. #1
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    Question Mirror surface reflection question

    The surface given by parametric equations r(phi,z) = (2*sqrt(z)*cos(phi), 2*sqrt(z)*sin(phi), 2z) is a mirror where 0<z<2 and 0<phi<2pi. A light beam has direction (0,0,-1), and hits the mirror at position (sqrt(z0), 0, z0). Find the direction of the reflected beam, and show that all beams with an incident direction parallel to the z-axis get reflected such that they all intersect the z-axis at the same point irrespective of z0. Find this intersection point.
    I don't really know where to start with this. Any help would be appreciated!

    DottyK
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  2. #2
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    Re: Mirror surface reflection question

    Hey DottyK.

    The first thing you need to do is find the normal vector at a given point on your object (and also on the right side of the object - the one that is being hit). Can you do this? (Hint: Find partial derivatives with respect to each parameter and take the cross product).
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  3. #3
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    Re: Mirror surface reflection question

    Hi,

    Thanks for the help.

    The partial derivative with respect to phi is (-2*sin(phi)*sqrt(z) , 2*cos(phi)*sqrt(z), 0) and with respect to z is
    (cos(phi)/sqrt(z) , sin(phi)/sqrt(z), 2) The cross product dr/d(phi) cross dr/dz is then (4*sqrt(z)*cos(phi), -4*sqrt(z)*sin(phi) , -2) and the unit normal is this divided by its modulus. How do I know if this is for the right side of the object? For the direction part would I then say the component of the beam parallel to the mirror is conserved, and the perpendicular component is reversed, as I can work these these out from the normal? What should I do after this?
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  4. #4
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    Re: Mirror surface reflection question

    To check whether its on the right side of the object you translate each point by the point you are testing and do a dot product with the normal vector and the point.

    If your normal vector is defined correctly, then the result should be positive (meaning that the point is approaching the surface). If its negative then it means it will not approach the surface (or that you got the normal wrong).

    Also once you find the normal, then you are simply reflecting that point about that normal. This involves creating a vector (that represents incoming rays) and rotating it to get the outgoing vector which you can use for your answer.
    Last edited by chiro; December 3rd 2013 at 02:29 PM.
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