Parametrization/Ellipse

DeFacto · #1 $Old$ May 20th, 2009, 06:20 AM

"Let Σ denote that part of the cone x^2 + y^2 = z^2, z > 0 which lies beneath the plane x + 2z = 1.
Let F(x, y, z) = (0,x,0).
Show that the projection of ∂Σ vertically to the xy-plane is an ellipse. Parametrise ∂Σ."

I have no idea how they get an ellipse from this. I've substituted the plane equation into the cone equation and i get a set which isn't an ellipse and hence don't know how to parametrize this. I must be doing something wrong!

Any help would be greatly appreciated.

Thanks.

artvandalay11 · #2 $Old$ May 20th, 2009, 06:31 AM

$z=\frac{1-x}{2}$ so plug that into the cone equation to get $x^2+y^2=\frac{(1-x)^2}{2}=\frac{1-2x+x^2}{2}=\frac{1}{2}-x+\frac{x^2}{2}$ so

$\frac{1}{2}x^2+x+y^2=\frac{1}{2}$ so $x^2+2x+2y^2=1$ and now complete the square and add $1$ to both sides so

$x^2+2x+1+2y^2=\frac{1}{2}+1$

$(x+1)^2+2y^2=\frac{3}{2}$ and this is an ellipse which can be parameterized with sin and cos similar to how you would parameterize a circle

DeFacto · #3 $Old$ May 20th, 2009, 06:41 AM

yeah, cool. This is just a horizontal translation of an ellipse in the x-y plane right. thanks.