I was asked to find 3 different atlas for the surface sphere of rad 1 ($S^2$). (As an Atlas is a collection of local parametrizations of surface, whose Union gives the whole surface, or they combine covers the whole surface.)
I have managed to find 2 atlas for sphere of radius 1... That are given below...
First Atlas is $\{ f_i : i = 1,2, ... 6\}$ and $U_1= \{ (x,y) \in \mathbb R^2 ; x^2+y^2<1 \}$ With $ f_1,f_2,f_3,f_4,f_5,f_6 : U_1 \to S^2 $ S.t $f_{1,2} (x,y) = (x,y, \pm \sqrt {1-x^2-y^2} )$ $f_{3,4} (x,y) = (x, \pm \sqrt {1-x^2-y^2} , y)$ $f_{5,6} (x,y) = (\pm \sqrt {1-x^2-y^2} , x, y)$
And the second atlas is $\{f_7,f_8\}$ Where $f_7,f_8 : U_2 \to S^2$ with $U_2 = \{ (x,y) ; 0<x<π , 0<y<2π\}$ and $f_7(x,y) = (\sin x \cos y , \sin x \sin y , \cos x )$ $f_8(x,y) = (-\sin x \cos y , \cos x , - \sin x \sin y)$
Now I m trying to find the 3rd one... If any one have any mapping in mind, which maps an open set in $\Bbb R^2$ to the sphere or some part of that sphere of rad 1, please share...
For a third atlas, you can just take the union of the two atlases you have considered already, namely: $$\{f_1,f_2,f_3,f_4,f_5,f_6\} \cup \{f_7,f_8\} = \{f_1,f_2,f_3,f_4,f_5,f_6,f_7,f_8\} $$