http://calc3.askbot.com/questions/Open source question and answer forum written in Python and DjangoenCopyright Askbot, 2010-2011.Tue, 01 Jul 2014 00:12:08 -0500http://calc3.askbot.com/question/29/how-do-i-find-the-intersection-of-a-sphere-and-an-arbitrary-plane/It is very easy to find the intersection of a sphere and a plane like the $xy$ plane, but harder to find the intersection of a plane in general. The example given in class today was: > Describe the intersection between the unit sphere and the plane $x + 2y + z = 2$. Here is the work I have done on this problem so far. The general parametrization of a circle in space is: $$ \vec p(t) = \vec c + r\cos(t) \vec u + r\sin(t) \vec v$$ Where $\vec c$ is the center, $r$ is the radius, $\vec u \perp \vec v$, and $||\vec u|| = ||\vec v||$. The normal vector of the plane is: $$ \vec n = \langle 1, 2, 1 \rangle $$ I now let $\vec u$ and $\vec v$ equal a random vector on the plane. $$ \vec u = \langle 1, 0, -1 \rangle $$ $$ \vec v = \vec u \times \vec n = \langle 2, -2, 2 \rangle $$ Normalizing these vectors, I obtain: $$ \vec u = \langle 1, 0, -1 \rangle / \sqrt{2} $$ $$ \vec v = \langle 2, -2, 2 \rangle / \sqrt{12} $$ Then to find the center of the circle, I create a line that has the normal vector as its direction. $$ \vec l(t) = \langle 1, 2, 1 \rangle t $$ Next, I find the $t$-value intersection of this line and the plane, by plugging in the components into the equation of the plane. $$ t + 2(2t) + t = 2 $$ $$ t = \frac{1}{3} $$ The center is therefore equal to $\frac{1}{3} \langle 1, 2, 1 \rangle = \langle \frac{1}{3}, \frac{2}{3}, \frac{1}{3} \rangle $. Are these steps correct? From here, where do I go? *Comment*: Good question and looks great so far. As far as where to go next, take a look at your general parametrization of a circle and address what you're missing. Update ====== See [this question](http://calc3.askbot.com/question/36/sphereswill-this-workin-class-3/) for the answer.Mon, 30 Jun 2014 15:46:39 -0500http://calc3.askbot.com/question/29/how-do-i-find-the-intersection-of-a-sphere-and-an-arbitrary-plane/http://calc3.askbot.com/question/29/how-do-i-find-the-intersection-of-a-sphere-and-an-arbitrary-plane/?answer=33#post-id-33Very good question! And something like this is likely to be on the exam - so I hope ya'll figure it out!!Mon, 30 Jun 2014 20:39:29 -0500http://calc3.askbot.com/question/29/how-do-i-find-the-intersection-of-a-sphere-and-an-arbitrary-plane/?answer=33#post-id-33http://calc3.askbot.com/question/29/how-do-i-find-the-intersection-of-a-sphere-and-an-arbitrary-plane/?answer=34#post-id-34So I was going to upvote your question as I am curious of the same thing, but since my karma isn't high enough I'll just have to comment. What you have so far looks great, the center you found is the same answer we got in class working it out on the board. So take this vote of confidence as an answer to the first part of your question, unfortunately I am stumped as to how to continue. Hopefully someone can add a more complete answer to the question but I'll be sure to post again if I happen to figure it out. Tue, 01 Jul 2014 00:12:08 -0500http://calc3.askbot.com/question/29/how-do-i-find-the-intersection-of-a-sphere-and-an-arbitrary-plane/?answer=34#post-id-34