Finding Normal Vectors to Surfaces of Revolution

[Kalidor]

Suppose you have a single variable differentiable function r: R -> R restricted to an interval, like [0,1] for simplicity, if you want. Consider the surface of revolution obtained by rotating the graph of r around the x axis. How do I find the NORMAL VECTOR to the surface at each of its points (x,y,z)? I guess I could find the normal vector to the graph of r in the xy plane and then rotating it with the same rotation I use to generate the surface of revolution. Somehow though this process seems to become unfeasible at some point so I must be missing something or doing it the wrong way altogether.

 

[HallsofIvy]

To be more specific, let's assume you have y= f(x) rotated around the x-axis. That is, of course, a two dimensional surface and can be written in terms of two parameters, x and the angle, \theta, of rotation. At each x, the point (x,y) rotates around a circle of radius y= f(x) and so we have x= x, y= f(x)cos(\theta), and z= f(x)sin(\theta).

We can write that as a vector function: \vec{r}(x,\theta)= x\vec{i}+ f(x)cos(\theta)\vec{j}+ f(x)sin(\theta)\vec{k}.

The derivatives of that vector function, with respect to the two variables:
\vec{r}_x= \vec{i}+ f'(x)cos(\theta)\vec{j}+ f'(x)sin(\theta)\vec{k}
\vec{r}_\theta= -f(x)sin(\theta)\vec{j}+ f(x)cos(\theta)\vec{k}
are in the tangent plane.

Their cross product
\begin{bmatrix}\vec{i} & \vec{j} & \vec{k}\\ 0 & -f sin(\theta) & f cos(\theta) \\ 1 & f' cos(\theta) & f' sin(\theta) \end{bmatrix}= -ff'\vec{i}+ f cos(\theta)\vec{j}+ f sin(\theta)\vec{k}
is the outward normal at each point.

 

[Kalidor]

Hi HallsofIvy, this vector is something similar to what I got (with inverted signs though, I probably did the cross product wrong). I don't see why it should have norm 1 in general, though and i need it to have norm 1 (Sorry, I just found out I should have written UNIT normal vector in the original post). But then again I guess I could just divide by sqrt[(rr')^2 + r^2] and I'd get a vector with norm 1. So it's fine I hope.

 

[HallsofIvy]

There are, of course, an infinite number of vectors perpendicular to a surface at a point, of differing lengths and directions. I chose the order of cross product to get the "outward pointing" normal (away from the x-axis which lies in the interior of the surface). You didn't say anything about a "unit vector" in your original post. If you want a unit vector, divide by the length of this vector.