I know how to find the normal line at surface $S$ given a vector and a point. How should I find the normal line if I was only able to compute for a set of parametric equations for a surface of revolution in terms $x,y,z$ and $\theta$ given that I will need variables $u$ and $v$ to do so?
For example, the parametric equation is in the form:
$$
x=x, y=f(x)\cos(\theta), z=f(x)\sin(\theta).
$$
How should I express the set of parametric equations to $u$ and $v$ for me to find the normal line at a given point $(a,b,c)$?
Best Answer
$r(x, \theta) = (x, f(x)\cos\theta, f(x)\sin\theta)$
Take partial derivative wrt $\theta$ and $x$.
$r'_\theta = (0, - f(x) \sin\theta, f(x) \cos \theta)$
$r'_x = (1, f'(x) \cos\theta, f'(x) \sin \theta)$
So $ \ r'_\theta \times r'_x = (- f(x) f'(x), f(x) \cos\theta, f(x) \sin\theta)$
Now use this to find normal vector at a given point $P$ on the surface and using position vector of point $P$ and the normal vector, write the equation of the normal line.