Parametric equation of the intersection between surfaces

  1. [SOLVED] Parametric equation of the intersection between surfaces

    1. The problem statement, all variables and given/known data

    Given the following surfaces:
    S: z = x^2 + y^2
    T: z = 1 - y^2

    Find a parametric equation of the curve representing the intersection of S and T.

    2. Relevant equations


    3. The attempt at a solution

    The intersection will be:
    x^2 + y^2 = 1 - y^2
    x = (1 - 2y^2)^0.5

    At this point, I plug in the following parametrization:
    y = sin(t)

    Which yields:

    x = (1 - 2(sin(t))^2)^0.5
    y = sin(t)
    z = 1-(sin(t))^2 (from the equation for T)

    with t = 0..2*Pi.

    Judging from a Maple plot this seems to make sense; the curve is a projected ellipse, but due to the x term I have to split it into two separate segments. I'm pretty sure I should be able to use a more elegant solution with a single curve, but I haven't been able to figure it out - any help would be welcome.

  2. jcsd
  3. HallsofIvy

    HallsofIvy 41,260
    Staff Emeritus
    Science Advisor

    In a situation like that it is better not to solve for one of the variables.

    Instead, change x2+ y2= 1- y2 to x2+ 2y2= 1, the equation of an ellipse. Then use the "standard" parameterization of an ellipse: x= cos(t), y= sin(t)/[itex]\sqrt{2}[/itex]. Then, of course, you can have either [itex]z= cos^2(t)+ (1/2)sin^2(t)[/itex] or [itex]z= 1- (1/2)sin^2(t)[/itex].
  4. Great, thank you.
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