Convert space curve to cartesian

  1. if R = sinti+sqrt(2)costj+sintk, 0<=t<=Pi/2
    please eliminate t to determine the cartesian equation of R(t). Put limits on the variables and verbally describe the curve
     
  2. jcsd
  3. x= sint, y=sqrt(2)cost, z=sint

    u can clearly see that
    x2 + y2 +z2=2{sin2t +cos2t}

    =2

    x2 + y2 +z2=2
     
  4. matt grime

    matt grime 9,396
    Science Advisor
    Homework Helper

    And eqaully clearly, surely you can see there is more to it than that? You've just replaced a locally 1-d structure (a curve) with a locally 2-d structure, a sphere.

    yes, the x, y, and z coordinates necessarily satisfy that, but that isn't sufficient. You need to intersect with the plane x=z (or similar) at the very least.

    generally the equation is [tex]x=z=(1-y^2)^{1/2}/\sqrt 2[/tex]
     
  5. Thanks matt grime, i've checked yours is correct.
    but can you show me how the above equation is reached.
     
  6. HallsofIvy

    HallsofIvy 40,218
    Staff Emeritus
    Science Advisor

    If matt grime will forgive me for sticking in my oar:

    x= sint, y=sqrt(2)cost, z=sint so obviously x= z.

    x2= sin2t= (1-cos2t). But
    y2= 2 cos2t so cos2t= y2/2. That is x2= 1- y2/2 and
    x= z= &radic;(1- y2/2).
     
    Last edited: Feb 17, 2004
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