A Fun 2d Rotation Matrix Derivation

A few weeks ago I joined NVIDIA as part of the graphics dev tech team.

The dev tech team’s scope is pretty wide, but it encompasses nearly all real time graphics programming needs that NVIDIA may have: making code samples for new tech, helping developers optimize their games and make them look prettier, working with graphics researchers, and contributing to graphics research as well.

Sometimes it’s like being a meta engine programmer (making stuff for engine programmers who make stuff for people making games), and other times it’s like working on an early prototype dev kit, where the dev kit is the PC itself.

I’m enjoying it quite a bit so far. It’s a bit surreal to be working along side really respectable folks I’ve seen present at siggraph, or who have written papers I’ve read, but at the same time it feels appropriate because I’m interested in the same areas and am looking forward to mixing in my own contributions and getting to collaborate on cool stuff.

Speaking of which, yes, my first contribution involved blue noise, and I’ll point it out when the thing it’s part of is public!

My new boss is a guy named Rahul, and I could tell things were going to be great, because when we were talking about math, he became visibly dettached from the real world while explaining something. Alternately, he was interested in learning about some obscure topics I had, like dual numbers, or abusing texture sampling to calculate polynomials.

He is the one that showed this logic chain to me.

Act 1 – The 2D Cross Product

So there is this thing that people call the “2d cross product” which is not really a cross product but if you have a 2d vector, it will give you a vector perpendicular to that vector. This is a really valuable tool in the toolbox for game developers, as you are often working in 2d coordinates, even though the world may be 3d, or you can sometimes simplify problems to 2d and bring them back into 3d after solving them.

You calculate the 2d cross product by flipping the x and y components of a vector and flipping the sign of the new x component:

newVec.x = -oldVec.y;
newVec.y = oldVec.x;

You could negate the new y instead of the new x to get the other perpendicular vector (there are 2!) but for this derivation, the way I have it above is important to the result. (I’ll revisit this at the end of the post)

As a simple example, if we take the vector (1,0), flip x and y, and negate the new x, we get (0,1), which is indeed perpendicular.

This works with arbitrary 2d vectors though. Doing it to (1,2) gives you (-2,1) which you can see in the image below is definitely perpendicular.

You can actually express this operation as a matrix:

\begin{bmatrix}0 & 1\\-1 & 0\end{bmatrix}

which you can see results in the same:

\begin{bmatrix}x & y\end{bmatrix} \cdot \begin{bmatrix}0 & 1\\-1 & 0\end{bmatrix} = \begin{bmatrix}-y & x\end{bmatrix}

Act 2 – Complex Numbers

Way back in 2014 I wrote a blog post on how to use imaginary (complex) numbers to do vector rotation: https://blog.demofox.org/2014/12/27/using-imaginary-numbers-to-rotate-2d-vectors/

The key take away relevant here is that if you multiply a vector by the imaginary number i, it’s a 90 degree rotation counter clockwise.

Strangely, that’s exactly what our matrix does too… could this matrix be i? Well, i*i is -1, so let’s see if that happens when we multiply this matrix by itself:

\begin{bmatrix}0 & 1\\-1 & 0\end{bmatrix} \cdot \begin{bmatrix}0 & 1\\-1 & 0\end{bmatrix} = \begin{bmatrix}-1 & 0\\0 & -1\end{bmatrix} = -1 * I

So yeah, that checks out… this matrix, which represents the “2d cross product” is also the imaginary number i, the square root of -1. That’s fun. (the I above is just the identity matrix)

That means that multiplying by this matrix is the same as if we were multiplying a vector by the complex number (0+1i).

We can see this is true by re-doing the example (1,2), and multiplying the values together using FOIL (first, outer, inner, last).

(1 + 2i)*(0+i) = \\ (1*0)+(1*i)+(2i*0)+(2i*i) = \\ 0 + i + 0 -2 = \\ -2 + i = \\ (-2, 1)

Act 3 – Complex Exponentials

Euler’s formula is a way of calculating points on a circle on the complex plane and is given as:

e^{i\theta} = \cos{\theta} + i\sin{\theta}

The value (0+1i) is the just the above formula when theta is 90 degrees, which is the amount of rotation we got when multiplying. We can easily verify that this is 90 degrees by remembering that cosine of 90 is 0, and sine of 90 is 1.

So, let’s replace our multiplication of (0+1i) with the right side of Euler’s formula. This way we can rotate by arbitrary angles, not just 90 degrees.

(x+iy) * (\cos{\theta}+i\sin{\theta}) = \\ (x \cos{\theta}) + (x i\sin{\theta}) + (iy \cos{\theta}) + (iy i\sin{\theta}) = \\ (x \cos{\theta} - y \sin{\theta}) + i(x\sin{\theta}+y\cos{\theta})

From here you can extract it back into matrix form to get:

(x \cos{\theta} - y \sin{\theta}) + i(x\sin{\theta}+y\cos{\theta}) = \\ \begin{bmatrix}x & y\end{bmatrix} \cdot \begin{bmatrix}\cos{\theta} & sin{\theta}\\-\sin{\theta} & \cos{\theta}\end{bmatrix}

Bam, there’s the 2d rotation matrix.

Other Fun Matrices

This post showed the matrix form of the imaginary number i, where i*i=-1.

\begin{bmatrix}0 & 1\\-1 & 0\end{bmatrix}

Dual numbers are another fun type of number where there is an \epsilon that is not zero, but \epsilon \cdot \epsilon is zero.

Dual numbers let you do forward mode automatic differentiation, and you can read some more about them at https://blog.demofox.org/2014/12/30/dual-numbers-automatic-differentiation/

A matrix form for them is:

\begin{bmatrix}0 & 1\\0 & 0\end{bmatrix}

Lastly are hyperbolic numbers (or split complex numbers), where you have an i that is not 1, but i*i is 1. I’m not sure what they are useful for but you can read more here: https://en.wikipedia.org/wiki/Split-complex_number

A matrix form for them is:

\begin{bmatrix}-1 & 0\\0 & 1\end{bmatrix}

By the way, when I introduced the 2d cross product i said flip x and y and negate the new x, but said the other way is possible as well. If you do it the other way, that operation is multiplying by -i, instead of multiplying by i. It isn’t a special case that breaks any of the things in this post, it’s just a different value. It’s a rotation by -90 degrees.


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