- #1
Gianmarco
- 42
- 3
I was working on a problem on this domain:
<br /> E=[x,y,z)\:s.t. \: \sqrt{x^2+y^2}\leq z\leq \sqrt{3x^2+3y^2},\: x^2+y^2+z^2\leq 2]<br />
and at some point I wanted to find the intersection of the internal cone(##\sqrt{3x^2+3y^2}=z##) with the sphere of radius ##\sqrt{2}## to find the height z of the circle. I did so by equating the two in cylindrical coordinates as follows:
<br /> 3x^2+3y^2-z^2=x^2+y^2+z^2-2\\x=rcos\theta,\: y=rsin\theta,\: z=z\\2r^2-2z^2+2=0 \rightarrow z^2=r^2+1<br />
This result confuses me. The intersection of a cone along the z-axis and a sphere centered in (0,0,0) should be a circle around the z-axis(and so with a fixed z). The radius(##\sqrt{x^2+y^2}## is of course also fixed. Then why do I get a function of z and r? Isn't the intersection of a cone and a sphere a circle?
<br /> E=[x,y,z)\:s.t. \: \sqrt{x^2+y^2}\leq z\leq \sqrt{3x^2+3y^2},\: x^2+y^2+z^2\leq 2]<br />
and at some point I wanted to find the intersection of the internal cone(##\sqrt{3x^2+3y^2}=z##) with the sphere of radius ##\sqrt{2}## to find the height z of the circle. I did so by equating the two in cylindrical coordinates as follows:
<br /> 3x^2+3y^2-z^2=x^2+y^2+z^2-2\\x=rcos\theta,\: y=rsin\theta,\: z=z\\2r^2-2z^2+2=0 \rightarrow z^2=r^2+1<br />
This result confuses me. The intersection of a cone along the z-axis and a sphere centered in (0,0,0) should be a circle around the z-axis(and so with a fixed z). The radius(##\sqrt{x^2+y^2}## is of course also fixed. Then why do I get a function of z and r? Isn't the intersection of a cone and a sphere a circle?
