I am looking at the following exercise:
A helicoid is the surface swept out by an aeroplane propeller, when both the aeroplane and its propeller move at constant speed.
If the aeroplane is flying along the $z$-axis, show that the helicoid can be parametrized as $$\sigma (u, v)=(v \cos u, v \sin u, \lambda u)$$ where $\lambda$ is a constant.
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Could you give me some hints how we could show that?
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EDIT:
After that I have to show that the cotangent of the angle that the standard unit normal of $\sigma$ at a point $p$ makes with the $z$-axis is proportional to the distance of $p$ from the $z$-axis.
I have found that the standard unit normal of $\sigma$ is $$\textbf{N}=\frac{(-\lambda \sin u, \lambda \cos u, -v)}{\sqrt{\lambda^2+v^2}}$$
The unit vector of the $z$-axis is $Z=(0,0,1)$, right?
So we have $$Z\cdot \textbf{N}=|Z||\textbf{N}|\cos (Z,\textbf{N}) \Rightarrow \frac{-v}{\sqrt{\lambda^2+v^2}}=\cos (Z,\textbf{N})$$ and $$|Z\times \textbf{N}\|=|Z||\textbf{N}|\sin (Z,\textbf{N}) \Rightarrow \frac{\lambda}{\sqrt{\lambda^2+v^2}}=\sin (Z,\textbf{N})$$ So $$\cot (Z,\textbf{N})=-\frac{v}{\lambda}$$
Which is the distance of $p$ from the $z$-axis?
Best Answer
Define a patch $\sigma(u,v)$ for $(u,v)\in (0,\infty)\times(0,\infty)$ to be the position at time $u$ of a point along the propellor blade that lies at distance $v$ from the propellor's center. If we project the patch onto the $xy$-plane (only look at the $x$ and $y$ coordinates), then we are parametrizing a disc, hence $(v\cos u,v\sin u)$ is the right choice for the first two coordinates of the patch. And, if we project the patch onto the $z$-axis, we are ignoring what part of the propellor we're looking at and just seeing the plane's progress along the $z$ axis, hence $\lambda u$ should be the patch's $z$-coordinate, where $\lambda$ is the plane's speed.