With reference to Figure Q3, write parametric functions $x (u), y (u), u \in [0, 1]$ defining this spiral curve.

Question

I am having a hard time trying to figure this out: U domain is [0,1] []1

Answer 1

The shape is an Archimedean spiral, where the radius is continuously increasing to $0.5$ linearly.

Think about the simple spiral shown below, defined as $x=Ru\cos(2k\pi u), y=Ru\sin(2k\pi u)$, where $R$ is the final radius and $k$ is the total revolutions.

Now compare the shape of the question and the shape above, the former is rotated by $\pi/2$ and the initial radius starts from $0.1$ instead of $0$.

Hence the hint would be $x=f(u)\cos(\phi(u))$, where $f(u)$, a linear function, represents the radius, and $\phi(u)$ affects the revolution, rotate direction (clockwise and the rotation of the whole shape).