- #1
BilalX
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[SOLVED] Parametric equation of the intersection between surfaces
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.
N/A
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-
Homework Statement
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.
Homework Equations
N/A
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-