This polar graph draws some amazing shapes on a dry erase board. Part of that is due to the mounting brackets used for the two stepper motors and the stylus. But credit is also due for the code which takes velocity into account in order to plan for the next set of movements.
The Go language is used to translate data into step commands for the two motors. This stream of commands is fed over a serial connection between the RPi board and an Arduino. The Arduino simply pushes the steps to the motor controllers. The inclusion of the RPi provides the horsepower needed to make such smooth designs. This is explained in the second half of [Brandon Green's] post. The technique uses constant acceleration, speed, and deceleration for most cases which prevents any kind of oscillation in the hanging stylus. But there are also contingencies used when there is not enough room to accelerate or decelerate smoothly.
You can catch a very short clip of the hardware drawing a tight spiral in the video embedded after the break.