Question

Parametrize the intersection between the sphere \[
x^2+y^2+z^2=1
\]And the plane \[
x+y+z=0
\]
Is there any good trick for this?

My intuition is to start with \[
(\cos(t),\sin(t),0)\quad t\in [0,2\pi]
\]With a normal vector \(<0,0,1>\) and then rotate the parametrization to align with the normal vector of the plane \(<1,1,1>\)

Answer 1

Another idea I have it to project the intersection onto the \(xy\) plane and then after parametrizing \(x\) and \(y\), using the equations to parametrize \(z\).

Answer 2

It's been a while since I've parametrized intersections. I solved for z in the second equation and substituted, getting:

\[2(x^{2} + xy + y ^{2}) = 1\]
This equation seems awfully familiar to me but I can't remember how to parametrize.

Answer 3

I have no idea how to parametrize that.

Answer 4

Anyone got any ideas?

Answer 5

The rotation matrix for \(45^\circ\) is \[
\begin{bmatrix}
\cos(45^\circ) & -\sin(45^\circ) \\
\sin(45^\circ) & \cos(45^\circ)
\end{bmatrix} = \begin{bmatrix}
\sqrt{1/2} & -\sqrt{1/2} \\
\sqrt{1/2} & \sqrt{1/2}
\end{bmatrix}
\]

Answer 6

I think your rotation strategy is a good idea, I'm just not familiar with the implementation.

Answer 7

Something like this. Transformations are not tough to do on parametrizations.

Answer 8

We have this sphere intersected by a plane so I know it's some sort of rotated circle.