This is a question from Barrett O'Neill's Elementary Differential Geometry. Let $S^2$ be a sphere of radius $r$ centered at the origin. Following are three maps defined on $S^2$. The question is to describe the effect of these maps on the meridians and parallels of $S^2$. Let $p=(p_1,p_2,p_3)$ be a point on $S^2$.
$F(p)=-p$
$G(p_1,p_2,p_3)=(p_3,p_1,p_2)$
$H(p)=(\frac{p_1+p_2}{\sqrt 2},\frac{p_1-p_2}{\sqrt 2},-p_3)$
My thoughts : I am checking what these functions do to a single point of the sphere. I suspect that all these are compositions of rotations and/or reflections. But I can't clearly state the axis of rotation or the plane of reflection. For example, the first one seems to be a reflection of a point across the xy plane and reflecting that point across xz plane. Though I'm not sure. I have absolutely no idea about the third one.
Any help is appreciated.Thanks for your time.