I'm trying to parameterize a path to the summit and a path along the level surface $x^2+y^2=1$. Let's say i start from the point $A=(-1,0,0)$ and i want to end up at the point $B=(1,0,0)$ by taking the two different paths.
If i let $f(x,y,z)=1-x^2 -y^2-z$ and take the direction vector to be the gradient of $f$ at A $\vec\nabla{f}(-1,0,0)$ and by taking the path $\vec{r}(t)=\vec{AB}+t\cdot\vec\nabla{f}(-1,0,0)$,will this get me the desired path that goes through the summit?
For the path along the level surface I was thinking of doing the same thing but taking a direction vector that is perpendicular to the vector $\vec\nabla{f}(-1,0,0)$.Is this also correct?
Choosing the parameterization
$$ x = r \cos t\\ y = r \sin t $$
over the level surface $x^2+y^2 = 1$ you should use $r = 1$ and $ 0 \le t \le \pi$ etc
To the summit taking $t = t_0$ and making $r \rightarrow 0$