How do I parametrise the surface $x^2 + y^2 + z^2 -2x =0$?
This is the beginning of a question where we have to work out a tangent plane at a point. I understand the rest of this method but somehow cannot work out how to parametrise tricky surfaces.
The other surfaces we have to parametrise are $x^2 + y^2 - z^2 = 2y + 2z$ where $−1 ≤ z ≤ 0$ and $(4 − x^2 + y^2)^2 + z^2 = 1$.
Obviously I don't expect someone to parametrise all these surfaces for me but any help would be appreciated!
In the first one: you can complete the square: if you add $1$ to both sides, you get $$ (x-1)^2+y^2+z^2=1. $$ So you have a sphere or radius $1$, centered at $(1,0,0)$. You can parametrize it with spherical coordinates: $$ x-1=\cos\theta\,\sin\varphi,\ \ \ y=\sin\theta\,\sin\varphi,\ \ z=\cos\varphi. $$ Thus $$ x=1+\cos\theta\,\sin\varphi,\ \ \ y=\sin\theta\,\sin\varphi,\ \ z=\cos\varphi,\ \ 0\leq\theta\leq2\pi,\ \ 0\leq\varphi\leq\pi. $$