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Math Help - Spherical coordinate system- sketching curves for constant parameters

  1. #1
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    Exclamation Spherical coordinate system- sketching curves for constant parameters

    x=coshEsinTcosQ y=coshEsinTsinQ z=sinhEcosT

    where (E,T,Q) refers to the spherical coordinate system.

    a) Describe in words and sketch the curves on which E and T are constant.

    b) Describe and sketch the curves on which E and Q are constant.

    c) Describe and sketch the SURFACES on which E is constant.
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  2. #2
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    anyone have any ideas? i'm really struggling
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  3. #3
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    Quote Originally Posted by choccookies View Post
    x=coshEsinTcosQ y=coshEsinTsinQ z=sinhEcosT

    where (E,T,Q) refers to the spherical coordinate system.

    a) Describe in words and sketch the curves on which E and T are constant.

    b) Describe and sketch the curves on which E and Q are constant.

    c) Describe and sketch the SURFACES on which E is constant.
    Hi. That's not spherical coordinates right. That's just a user-defined coordinate system:

    x=\cosh(e)\sin(t)\cos(q)
    y=\cosh(e)\sin(t)\sin(q)
    z=\sinh(e)\cos(t)

    and with e and t constant, you have:

    x=k\cos(q)
    y=k\sin(q)
    z=c

    and as q ranges in some interval say (-10,10), the x and y coordinates just go round a circle of radius k. and the z coordinate is constant say z=5. Then the first set is the circle a distance c above the x-y plane with a radius of k.
    Looks to me anyway.
    Last edited by shawsend; November 25th 2009 at 03:48 AM. Reason: corrected formulas for x and y
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  4. #4
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    hi, thanks for your help.
    yeah i didnt know how to type the real symbols for the sphericals. Oh also I forgot to put the conditions;

    E>= 0, Pi>=T>=0, 2Pi>=Q>=0

    How would I draw the surface? If E was constant I;d get
    x=sinTcosQ
    y=sintTsinQ
    z=cosT

    This seems too tricky to plot?
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  5. #5
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    Quote Originally Posted by choccookies View Post
    hi, thanks for your help.
    yeah i didnt know how to type the real symbols for the sphericals. Oh also I forgot to put the conditions;

    E>= 0, Pi>=T>=0, 2Pi>=Q>=0

    How would I draw the surface? If E was constant I;d get
    x=sinTcosQ
    y=sintTsinQ
    z=cosT

    This seems too tricky to plot?
    Build it up piece by piece. First let t=\pi/2. Then we have:

    x=k\cos(q)
    y=k\sin(q)
    z=0

    so that's obviously some curve in the x-y plane right? And the sine and cosine thing means it's a circle with radius k. So draw that one. Now, how about letting t=0? Then do a few more and see what it starts looking like. Also, compare your user-defined coordinate system with the standard spherical coordinate transformations and note the similarities:

    x=\rho\sin(\phi)\cos(\theta)
    y=\rho\sin(\phi)\sin(\theta)
    z=\rho\cos(\phi)

    That's kinda' tough for me. Here's what it looks like in Mathematica as e is varied between 0 and 1. Try and get a machine to run the code on it, see what's happening, then try and explain it analytically:

    Code:
    Manipulate[
     ParametricPlot3D[{Cosh[e] Sin[t] Cos[q], Cosh[e] Sin[t] Sin[q], 
       Sinh[e] Cos[t]}, {t, 0, \[Pi]}, {q, 0, 2 \[Pi]}, 
      PlotRange -> {{-3, 3}, {-3, 3}, {-3, 3}}], {e, 0, 1}]
    Last edited by shawsend; November 25th 2009 at 04:28 AM. Reason: learning as I go
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