Intersection of surfaces

**eddi** · April 15th 2009, 05:55 PM

I want to find the equation of the intersection between a sphere and cylinder (in the first octant) but it's kind of weird.

Sphere: x^2 + y^2 +z^2 = 4
Cylinder: x^2 + y^2 - 2y = 0

If I just sub one of them into the other I get:
2y + z^2 = 4

but that doesn't make sense to me since that is an equation of a surface since x can vary. There should be some sort of restriction on x... but how do I get this?

**Opalg** · April 16th 2009, 08:42 AM

Originally Posted by eddi

I want to find the equation of the intersection between a sphere and cylinder (in the first octant) but it's kind of weird.

Sphere: x^2 + y^2 +z^2 = 4
Cylinder: x^2 + y^2 - 2y = 0

If I just sub one of them into the other I get:
2y + z^2 = 4

but that doesn't make sense to me since that is an equation of a surface since x can vary. There should be some sort of restriction on x... but how do I get this?

The intersection will be a curve, not a surface, and that is best described parametrically. Write the equation of the cylinder as $x^2 + (y-1)^2=1$ , and you see that this can be parametrised as $x = \sin\theta,\:y = 1+\cos\theta$ . Then $x^2 + y^2 = 2+2\cos\theta$ , and if you substitute that into the equation of the sphere you see that $z^2 = 2(1-\cos\theta) = 4\sin^2(\theta/2)$ .

Therefore the part of the curve in the positive octant can be described by the parametric representation $(x,y,z) = (\sin\theta, 1+\cos\theta, \sin(\theta/2))\;(0\leqslant\theta\leqslant\pi)$ .

**eddi** · April 16th 2009, 09:57 PM

Originally Posted by Opalg

The intersection will be a curve, not a surface, and that is best described parametrically. Write the equation of the cylinder as $x^2 + (y-1)^2=1$ , and you see that this can be parametrised as $x = \sin\theta,\:y = 1+\cos\theta$ . Then $x^2 + y^2 = 2+2\cos\theta$ , and if you substitute that into the equation of the sphere you see that $z^2 = 2(1-\cos\theta) = 4\sin^2(\theta/2)$ .

Therefore the part of the curve in the positive octant can be described by the parametric representation $(x,y,z) = (\sin\theta, 1+\cos\theta, \sin(\theta/2))\;(0\leqslant\theta\leqslant\pi)$ .

Thanks. So to reiterate (for myself), I have to parametrize the cylinder aka:
x = sin(t)
y = 1 + cos(t)
z = p

Then transform my sphere to use t and p:
x^2 + y^2 + z^2 = 4 turns into sin(t)^2 + (1+cos(t))^2 + p^2 = 4.

Then I can solve for p in terms of t, substitute that back into the parametrization for the cylinder, and voila I have a set of parametric equations describing my curve. Haha cool.

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