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Intersection of a sphere and plane

  1. May 15, 2013 #1
    1. The problem statement, all variables and given/known data
    Show that the circle that is the intersection of the plane x + y + z = 0 and the
    sphere x2 + y2 + z2 = 1 can be expressed as:

    x(t) = [cos(t)-sqrt(3)sin(t)]/sqrt(6)
    y(t) = [cos(t)+sqrt(3)sin(t)]/sqrt(6)
    z(t) = -[2cos(t)]/sqrt(6)


    2. Relevant equations



    3. The attempt at a solution
    At first, I tried to let z=-x-y then sub it into the equation of the sphere:
    x2+y2+(-x-y)2=1
    x2+y2+x2+2xy+y2=1
    2x2+2xy+2y2 = 1
    And i stuck at here, I can't make it into a circle equation.

    So, I changed another way.
    I equated two equations, so it becomes:
    x2 + y2 + z2 -1 = x+y+z
    x2+ x + y2 + y + z2 +Z -1 =0
    (x-1/2)2 + (y-1/2)2 + (z-1/2)2 = 7/4 *By completing the square

    But isn't this is an equation of a sphere??
    Shouldn't it just a circle?

    Any help is appreciated!
     
  2. jcsd
  3. May 15, 2013 #2

    tiny-tim

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    hi yy205001! :smile:
    why don't you just bung the three parameter formulas into the two original equations, and show that they work? :confused:
    but that also gives you the intersections of x + y + z = k and the
    sphere x2 + y2 + z2 = 1 + k, for all values of k ! :wink:
     
  4. May 15, 2013 #3

    LCKurtz

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    That would show the parametric curve lies on the circle. But is it the whole circle? Looks like there is more to do.
     
  5. May 15, 2013 #4
    And I am thinking is there anything do with the normal vector of the plane.
     
  6. May 16, 2013 #5

    tiny-tim

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    what does that have to do with the parametric equations? :confused:
     
  7. May 16, 2013 #6
    So, I just sub in those 3 parametrize equations into the equation for the plane and sphere, then show they satisfy x2+y2+z2=1 and x+y+z=0?
     
  8. May 16, 2013 #7

    tiny-tim

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    yes

    the question starts "show", so all you need to do is show that the answer is correct, you don't have to find the answer from scratch :smile:

    (and, as LCKurtz says, you'll also have to prove that that parametrisation gives the whole intersection)
     
  9. May 16, 2013 #8
    I got it!
    tiny-tim, LCKurtz, thank you so much for the help!
     
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