Video: Spherical basis vectors in geometric algebra

August 8, 2023 Geometric Algebra for Electrical Engineers , ,

I’ve made a new manim-based video with a geometric algebra application.

In the video, the geometric algebra form for the spherical unit vectors are derived, then unpacked to find the conventional vector algebra form. We will then use our new tools to find the expression for the kinetic energy of a particle in spherical coordinates.

Prerequisites: calculus (derivatives and chain rule), complex numbers (exponential polar form), and geometric algebra basics (single sided rotations, vector multiplication, vector commutation sign changes, …)

You can find the video on Google’s censorship-tube, or on odysee.

Geometric algebra: a very short video introduction.

August 1, 2023 math and physics play , ,


Here’s another geometric algebra video, weighing in at a massive 2:29 (minutes.)

This video is a very short introduction to geometric algebra, showing the most basic concepts and how to apply them to the 2D geometric algebra of the Euclidean plane. Those concepts aren’t developed further in this video, but the idea is just to show the most basic consequences of the definitions.

Prerequisites: basic vector algebra (basis, vector space, dot product space, arrow representation of vectors, graphical vector addition, …)

If you watched yesterday’s video, don’t both watching this one, since it is extracted from that with no additions.

You can find the video on Google’s censorship-tube, and on odysee.

Video: Circular velocity and acceleration with geometric algebra

July 31, 2023 math and physics play , , , ,

Months ago, I used Manim to create a outline a geometric algebra treatment of the derivation of the circular velocity and acceleration formulas that you would find in a first year undergrad physics course.  I never published it, since overlaying audio and getting the timing of the audio and video right is hard (at least for me.)  I’m also faced with the difficulty of not being able to speak properly when attempting to record myself.
Anyways, I finally finished the audio overlays (it was sitting waiting for me to record the final 10s of audio!), and have posted this little 11 minute video, which includes:
  • A reminder of what circular coordinates are.
  • A brief outline of what is meant by each of the circular basis vectors.
  • A derivation of those basis vectors (just basic geometry, and no GA.)
  • A brief introduction to geometric algebra, and geometric algebra for a plane, including the “imaginary” \( i = \Be_1 \Be_2 \), and it’s use for rotation and polar form.
  • How to express the circular basis vectors in polar form.
  • Application of all the ideas above to compute velocity and acceleration.
  • Circular coordinate examples of velocity and acceleration.
It probably doesn’t actually make sense to try to pack all these ideas into one video, but oh well — that’s what I did.
You can find the video on google’s censorship-tube, and on odysee.

Cleaning the exhaust vents on my NUCs

July 3, 2023 electronics , , , , ,

My NUC needed an SSD brain transplant, which presented a good opportunity to clean the dust out of the fan exhaust vents.  It was almost completely clogged:

I vacuumed it out, which didn’t work too well, since the dust was a little greasy, so I hauled out the carpentry tools (my compressor and blower attachment),

and gave it a good blast.  Check out the after picture for the vents:

Here it is, reassembled

Ready to be my little workhorse once again.  I also blew the dust out of my older non-skull canyon NUC, which I had abandoned (switching to a backup) when it started emitting a strong burnt electronics smell.  That unit is now fully functional again too, and doesn’t smell like burnt electronics anymore!

These NUCs are nice little machines, but are really susceptible to dust clogs, as the vent spacing is really tight.  Once those gets clogged, there’s basically no airflow possible.

Complex-pair representation of GA(2,0) multivectors

June 15, 2023 math and physics play , ,

[Click here for a PDF version of this post]

We found previously that a complex pair representation of a GA(2,0) multivector had a compact geometric product realization. Now that we know the answer, let’s work backwards from that representation to verify that everything matches our expectations.

We are representing a multivector of the form
\begin{equation}\label{eqn:bicomplexCl20:20}
M = a + b \Be_1 \Be_2 + x \Be_1 + y \Be_2,
\end{equation}
as the pair of complex numbers
\begin{equation}\label{eqn:bicomplexCl20:40}
M \sim \lr{ a + i b, x + i y }.
\end{equation}
Given a pair of multivectors with this complex representation
\begin{equation}\label{eqn:bicomplexCl20:60}
\begin{aligned}
M &= \lr{ z_1, z_2 } \\
N &= \lr{ q_1, q_2 },
\end{aligned}
\end{equation}
we found that our geometric product representation was
\begin{equation}\label{eqn:bicomplexCl20:80}
M N \sim
\lr{ z_1 q_1 + z_2^\conj q_2, z_2 q_1 + z_1^\conj q_2 }.
\end{equation}

Our task is now to verify that this is correct. Let’s set
\begin{equation}\label{eqn:bicomplexCl20:100}
\begin{aligned}
z_1 &= a + i b \\
q_1 &= a’ + i b’ \\
z_2 &= x + i y \\
q_2 &= x’ + i y’,
\end{aligned}
\end{equation}
and proceed with an expansion of the even grade components
\begin{equation}\label{eqn:bicomplexCl20:120}
\begin{aligned}
z_1 q_1 + z_2^\conj q_2
&=
\lr{ a + i b } \lr{ a’ + i b’ }
+
\lr{ x – i y } \lr{ x’ + i y’ } \\
&=
a a’ – b b’ + x x’ + y y’
+ i \lr{ b a’ + a b’ + x y’ – y x’ } \\
&=
x x’ + y y’ + i \lr{ x y’ – y x’ } + \quad a a’ – b b’ + i \lr{ b a’ + a b’ }.
\end{aligned}
\end{equation}
The first terms is clearly the geometric product of two vectors
\begin{equation}\label{eqn:bicomplexCl20:140}
\lr{ x \Be_1 + y \Be_2 } \lr{ x’ \Be_1 + y’ \Be_2 }
=
x x’ + y y’ + i \lr{ x y’ – y x’ },
\end{equation}
and we are able to verify that the second parts can be factored too
\begin{equation}\label{eqn:bicomplexCl20:160}
\lr{ a + b i } \lr{ a’ + b’ i }
=
a a’ – b b’ + i \lr{ b a’ + a b’ }.
\end{equation}
This leaves us with
\begin{equation}\label{eqn:bicomplexCl20:180}
\gpgrade{ M N }{0,2} = \gpgradeone{ M } \gpgradeone{ N } + \gpgrade{ M }{0,2} \gpgrade{ N }{0,2},
\end{equation}
as expected. This part of our representation checks out.

Now, let’s look at the vector component of our representation. First note that to convert from our complex representation of our vector \( z = x + i y \) to the standard basis representation of our vector, we need only multiply by \( \Be_1 \) on the left, for example:
\begin{equation}\label{eqn:bicomplexCl20:220}
\Be_1 \lr{ x + i y } = \Be_1 x + \Be_1 \Be_1 \Be_2 y = \Be_1 x + \Be_2 y.
\end{equation}
So, for the vector component of our assumed product representation, we have
\begin{equation}\label{eqn:bicomplexCl20:200}
\begin{aligned}
\Be_1 \lr{ z_2 q_1 + z_1^\conj q_2 }
&=
\Be_1 \lr{ x + i y } \lr{ a’ + i b’ }
+
\Be_1 \lr{ a – i b } \lr{ x’ + i y’ } \\
&=
\Be_1 \lr{ x + i y } \lr{ a’ + i b’ }
+
\lr{ a + i b } \Be_1 \lr{ x’ + i y’ } \\
&=
\gpgradeone{ M } \gpgrade{ N}{0,2}
+ \gpgrade{ M }{0,2} \gpgradeone{ N},
\end{aligned}
\end{equation}
as expected.

Our complex-pair realization of the geometric product checks out.