For one of our lectures at HAW Hamburg the challenge was to use a 2 dimensional plotting device to draw a figure in the quickest time possible. For simplification the parameters were like this: The plotter has 2 motors. One which could move along the x-axis and another one that could move along the y-axis. Both motors have the same speed and can be driven simultaneously (which would create a diagonal).

So I put some thoughts into how to win this competition. Here my idea: Lets assume you have to draw a square.

And assume that the width is 4cm. There apparently are not many solutions for the minimal time right? Wrong. The fastest way to draw a rectangle with the same width is to draw it at 45° rotated.

So why is that? The reason is simple. If you draw it at 45° both motors move at the same time all the time instead of only 50% of the time. And as Pythagorus is going to confirm you, for one edge each motor has to go less than what he would need to go for the 0° example. Some math a^2+b^2=c^2 right? c in our case is 4 cm and a and b are the same. So the equation is a^2=2*4 so a is about 2.83 or only around 70% of the 4cm.

Unfortunately we did not win the contest because the contest was to draw the “Haus des Nikolaus” which has the same drawing time at 45° as at 0°.