I have a sphere $S^2$, how can I define it?

Question

I'm not sure how to say, using proper mathematical notation, a sphere $S^2$ is equal to the basic unit sphere $1=x^2 + y^2 + z^2$. Just looking for a quick answer, thank you.

Answer 1

Using set builder notation:

$$S=\{(x,y,z)\in \Bbb R^3\mid x^2+y^2+z^2=1\}$$

If it is clear that the ambient space is $\Bbb R^3$ and you don't feel that it's necessary to specify it, a possible shortcut is simply

$$S=\{x^2+y^2+z^2=1\}$$

Answer 2

Yes for $(x,y,z) \in \mathbb{R^3}$ the cartesian equation for a sphere centered at the origin and with radius $R$ is

$$x^2+y^2+z^2=R^2$$

and more in general with center in $C=(a,b,c)$

$$(x-a)^2+(y-b)^2+(z-c)^2=R^2$$

Note that the equation is derived from Pytagorean Theorem, that is

$$R= \sqrt{(x-a)^2+(y-b)^2+(z-c)^2}$$

Answer 3

There seems to be a difference in notation between geometers (the number denoting the dimension of the coordinate space) and topologists (the number denoting the dimension of the surface).

See introduction here.