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Parametric equation of the intersection between surfaces 
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#1
Feb1508, 03:26 AM

P: 7

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 N/A 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. Thanks 


#2
Feb1508, 05:34 AM

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P: 39,346

In a situation like that it is better not to solve for one of the variables.
Instead, change x^{2}+ y^{2}= 1 y^{2} to x^{2}+ 2y^{2}= 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]. 


#3
Feb1508, 09:30 AM

P: 7

Great, thank you.



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