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Spherical Coordinates and Mathematica

  1. Sep 20, 2008 #1
    Use spherical coordinates to draw the cone z=Sqrt[x^2+y^2].
    Hint: You will need to determine \[Phi].

    How would I go about finding phi?
    Below are the x y and z components, but I cannot figure out how to find the range of phi:

    z^2=x^2 + y^2
    (Rho)=sqrt(2x^2+2y^2)

    x = sqrt(2x^2+2y^2) sin [Phi] cos [Theta]
    y = sqrt(2x^2+2y^2) sin [Phi] sin [Theta]
    z = sqrt(2x^2+2y^2) cos [Phi]

    Mathematica Code:
    ParametricPlot3D[{sqrt(2x^2+2y^2) Sin[[Phi]]*Cos[[Theta]],
    sqrt(2x^2+2y^2)Sin[[Phi]] Sin[[Theta]], sqrt(2x^2+2y^2) Cos[[Phi]]}, {[Phi], 0,
    Pi}, {[Theta], 0, 2*Pi}, ViewPoint -> {6, 3, 2}]

    Any help would be appreciated.
    Thanks
    Stephen
     
    Last edited: Sep 20, 2008
  2. jcsd
  3. Sep 20, 2008 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Oh, please don't think of Mathematica as a Panacea. Like any computer program, GIGO! You are mixing Cartesian and Spherical coordinates there: you don't want "x" and "y" in your spherical coordinates formula.

    [tex]x= \rho cos(\theta)sin(\phi)[/tex]
    and
    [tex]y= \rho sin(\theta) sin(\phi)[/tex]
    so
    [tex]x^2+ y^2= \rho^2 sin^2(\phi)(cos^2(\theta)+ sin^2(\theta))= \rho^2 sin^2(\phi)[/tex]
    and so the equation becomes
    [tex]z= \rho cos(\phi)= \sqrt{\rho^2sin(\phi)}= \rho sin\phi[/tex]
    or simply
    [tex]cos(\phi)= sin(\phi)[/tex]
    [tex]tan(\phi)= 1[/itex]

    Of course, there is a much easier way to find [itex]\phi[/itex]! Since the equation is symmetric in x and y, just look in the xz-plane. If y= 0, the equation becomes z= x, a line which makes an angle of [itex]\pi/4[/itex] with the x-axis and so the entire cone makes an angle of [itex]\pi/2- \pi/4= \pi/4[/itex] with the z-axis. That is, [itex]\phi= \pi/4[/itex].
    (And, of course, [itex]tan(\pi/4)= 1[/itex].)
     
  4. Sep 20, 2008 #3
    so to graph it would I use
    x= (Rho)sin(phi)cos(theta)
    y=(Rho)sin(phi)sin(theta)
    z=(Rho)cos(phi)

    with theta going from 0 to 2pi
    phi = pi/4
    Rho= some range like (0,3)

    or is there a way to solve for Rho?
    Thanks
    Stephen
     
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