## JClock VIII

Interpreting the hour hand on a clock as a two-dimensional object on a plane, the hand’s tip can be seen as a complex number.
This clock converts the hour hand’s position into a complex number, sets the number’s length to the current minutes and displays it in the form $a + b \cdot i$.
The angle $\phi$ is determined by the hours passed ($\frac{2 \cdot \pi \cdot \text{hour}}{12} = \frac{\pi \cdot \text{hour}}{6}$) but has to be slightly modified because a complex number starts at the horizontal axis and turns anti-clockwise whilst an hour hand starts at the vertical axis and turns — as the name implies — clockwise.
Thus $\phi = (2 \cdot \pi - \frac{\pi \cdot \text{hour}}{6}) + \frac{\pi}{2} = (\frac{15 - \text{hour}}{6}) \cdot \pi$.
The complex number’s length is simply determined by the minutes passed. Because the length must not be equal to $0$, I simply add 1. $|z| = k = \text{minute} + 1$.
Lastly, to convert a complex number in the form $k \cdot e^{\phi \cdot i}$ into the form $a + b \cdot i$, I use the formula $k \cdot (\cos{\phi} + \sin{\phi} \cdot i) = a + b \cdot i$.

# Python 2.7.7 Code
# Pygame 1.9.1 (for Python 2.7.7)
# Jonathan Frech 29th of July, 2016