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How to define a parabola in 3d coordinate system.

  1. Jul 18, 2010 #1
    Currently im using a graphying application called "Autograph" and modeling a building with a dome shaped roof on top. I need to define parabolic shapes in 3d system.
    But i cant do it ( my math knowledge is pretty elementary)
    What would be the basic parabolic function in 3d that i can base my model on?
    I have three sets of coordinates that have to be on the parabola.
     
  2. jcsd
  3. Jul 18, 2010 #2
    in the x y z system, i want this parabolic functionto go through (-75, 36, 0) (0,36, 36) (75,-36,0)
     
  4. Jul 18, 2010 #3

    disregardthat

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    If we start out with [tex]z=ax^2[/tex] in the xz-plane, we can translate and rotate it to the parabola in 3d. We can obviously get all 3-dimensional parabolas by varying a and doing rotations and translations.

    The parameterization is the following: [tex]P(t) = [t,0,at^2][/tex].

    http://en.wikipedia.org/wiki/Rotation_matrix#Three_dimensions

    Now rotate it around the x,y and z axis by angles v,u and w respectively. We do this by matrix multiplication, so we must multiply [tex]P(t)[/tex] by [tex]R_z(w)R_y(u)R_x(v)[/tex]. After that we perform a translation by the arbitrary vector r.

    So the general parabola is [tex]G(t) = R_z(w)R_y(u)R_x(v)P(t)+r[/tex]. You can find this on vector-form by multiplying out the matrices.
     
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