• analytic solution?

    From Michael Koch@21:1/5 to All on Tue Dec 18 08:42:16 2018
    Hi all,

    I'd like to know if there exists an anlytic solution for this problem.
    A beam starts at height = 0 with angle alpha, and passes through air of thickness A and then through a glass window of thickness D. When exiting the window, the beam's height is C.
    A, D, C and the index of refraction are known. Is there an analytic solution for angle alpha? I think it doesn't exist, but maybe I'm overlooking something.

    http://www.astro-electronic.de/Brechung.jpg

    Thanks,
    Michael

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  • From Lu Wei@21:1/5 to Michael Koch on Wed Dec 19 09:29:22 2018
    On 2018-12-19 0:42, Michael Koch wrote:
    Hi all,

    I'd like to know if there exists an anlytic solution for this problem.
    A beam starts at height = 0 with angle alpha, and passes through air of thickness A and then through a glass window of thickness D. When exiting the window, the beam's height is C.
    A, D, C and the index of refraction are known. Is there an analytic solution for angle alpha? I think it doesn't exist, but maybe I'm overlooking something.

    http://www.astro-electronic.de/Brechung.jpg

    Yes there is.
    To reduce the use of brackets, post notation is used afterwards.
    As in your draft, we have
    α tan= B/A, β tan= (C-B)/D,
    and
    α sin= n * β sin (eq.1)
    Since
    α sin ^2 = 1-α cos ^2= 1- 1/(1+ α tan ^2)
    Square eq.1 and substitute tangent terms, an equation about B, which is
    up to the degree of 4, could be obtained. The general analytic root of
    such an equation is given by: https://en.wikipedia.org/wiki/Quartic_function#General_formula_for_roots
    The general radical form is formidable, yet maybe this equation belongs
    to those which could be transformed to a simpler case.
    After B is solved, Then α = (B/A) atan.

    --
    Regards,
    Lu Wei
    IM: xmpp:luweitest@riotcat.org
    PGP: 0xA12FEF7592CCE1EA

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  • From Lu Wei@21:1/5 to Michael Koch on Thu Dec 20 17:11:19 2018
    On 2018-12-20 16:44, Michael Koch wrote:
    The general radical form is formidable, yet maybe this equation belongs
    to those which could be transformed to a simpler case.

    unfortunately it doesn't look like one of the simpler cases...


    Let computer do the calculation then. Maybe not what you expected though.

    --
    Regards,
    Lu Wei
    IM: xmpp:luweitest@riotcat.org
    PGP: 0xA12FEF7592CCE1EA

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  • From Michael Koch@21:1/5 to All on Thu Dec 20 00:44:30 2018
    The general radical form is formidable, yet maybe this equation belongs
    to those which could be transformed to a simpler case.

    unfortunately it doesn't look like one of the simpler cases...

    Michael

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