# [SOLVED] Parametric equation of the intersection between surfaces

by BilalX
Tags: equation, intersection, parametric, solved, surfaces
 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-
 Math Emeritus Sci Advisor Thanks PF Gold P: 38,708 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)/$\sqrt{2}$. Then, of course, you can have either $z= cos^2(t)+ (1/2)sin^2(t)$ or $z= 1- (1/2)sin^2(t)$.
 P: 7 Great, thank you.

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