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Parametric equation of the intersection between surfaces

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BilalX
#1
Feb15-08, 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-
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HallsofIvy
#2
Feb15-08, 05:34 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,490
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].
BilalX
#3
Feb15-08, 09:30 AM
P: 7
Great, thank you.


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