Parametrization of a curve(the intersection of two surfaces)

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Homework Statement


I am looking to find the parametrization of the curve found by the intersection of two surfaces. The surfaces are defined by the following equations: z=x^2-y^2 and z=x^2+xy-1

Homework Equations




The Attempt at a Solution


I can't seem to separate the variables well enough to find parametric equations of this curve. I really don't like asking for answers on homework, but help defining one of the variables would be appreciated.
 

Answers and Replies

  • #2
Delta2
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Solve easily for x=f(y). Paremetrization will be y=t, x=f(t).
 
  • #3
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Solve easily for x=f(y). Paremetrization will be y=t, x=f(t).
So I already have x=(1/y)-y^2 and y=t and therefore x=(1-t^2)/t. Then z=(1-2t)/t^2. Is this the parametrization? If so, I have been sitting here for hours trying to find a trig relationship. It just looks too simple.
 
  • #4
Delta2
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The parametrization of a curve is done with 1 variable t, but the parametrization of a surface needs 2 variables.
 
  • #5
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The parametrization of a curve is done with 1 variable t, but the parametrization of a surface needs 2 variables.
Wait, so am I thinking about my parametrization wrong? Can I not define x, y, and z in terms of parameter t? In what ways is my answer wrong for my parametrization? I'm sorry if I'm asking so many quick questions but I really would like to understand this concept. So I had solved to x(t), y(t), and z(t) by manipulating the equations of two surfaces, z=x^2-y^2 and z=x^2+xy-1, and from this is gained a parametrization of x(t)=(1-t^2)/t, y(t)=t, and z(t)=(1-2t)/t^2 and I write this as r(t)=<x(t),y(t),z(t)> which is defined when t does not equal 0. Thank you for all your help so far.
 
  • #6
Delta2
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Ok i just thought you were trying to parametrize the surface z=... but i see now what you were after.
 
  • #7
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Ok i just thought you were trying to parametrize the surface z=... but i see now what you were after.
So this form I found is a good final form? I feel like I'm missing something? Thank you for everything! If this is true I can finally sleep before work!
 
  • #8
Delta2
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slight mistake for z(t) it should be z(t)=(1-2t^2)/t^2. i guess probably a typo.
 

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