Quaternion Visualizer
Dec 4, 2019

I was reading about quaternions, so I decided to make a simple Quaternion library and visualizer in C++ to better understand them.

If you know what a complex numer is, a quaternion is fairly similar. It is essentially a complex number with 3 imaginary components instead of one. I wanted to plot quaternions on a 2D graph to make visualizing quaternion equations easier to understand. Quaternions are often used to model 3D rotations. I first heard about them a few years ago when I was volunteering in a biophysics lab. One of the “killer features” is that they prevent gimbal lock, a major problem with modelling 3D rotations.

## Using Complex Numbers for 2D Rotation

If you don’t know how complex numbers can be used to model 2D rotations, here is a quick recap. Consider the number $$\frac{\sqrt{2}}{2} (1 + i)$$. The argument is $$\phi = 45^{\circ}$$. If you square it, $$(\frac{\sqrt{2}}{2} (1 + i))^2 = i$$, which has $$\phi = 90^{\circ}$$.

In summary:

$$\frac{\sqrt{2}}{2} (1 + i), \phi = 45^{\circ}$$ $$(\frac{\sqrt{2}}{2} (1 + i))^2, \phi = 90^{\circ}$$ $$(\frac{\sqrt{2}}{2} (1 + i))^3, \phi = 130^{\circ}$$ $$(\frac{\sqrt{2}}{2} (1 + i))^4, \phi = 180^{\circ}$$

Similarly, dividing will rotate clockwise, instead of anticlockwise.

Multiplying or dividing any complex number $$x$$ by a number $$z, |z| = 1$$, will rotate $$x$$ by the argument of $$z$$.

## Using Quaternions for 3D Rotation

Quaternions work similarly for 3D rotations as complex numbers do for 2D rotations. Multiplying a quaternion will rotate an object, and dividing will rotate it back to the initial position.

## Visualizations

In this project, I plot a vector of quaternions on a two dimensional grid. The X axis reperesents the element in the quaternion vector. The Y axis represents the versor value of the real component of each quaternion. For the red, green, and blue values of each pixel, I used the values of the imaginary components of the versors. for instance, the unit quaternion would be gray (2552 for each imaginary component). Negative values go to 0, positive values go to 255.

In the picture below, I started with the quaternion $$-1 - 1k$$, and added $$0.02i$$ for every element in the size 400 quaternion vector.
This one is more interesting - I am rotating the $$-1 + 0.5k$$ quaternion by the $$0.999550 + 0.029987i$$ quaternion. Notice that the rotation quaternion is a versor (it has a norm of 1). Since it’s nearly entirely a real versor, it rotates slowly around both the real and complex components, which is why the below image is sinusoidal in the Y axis and in color space (it rotates through colours) !