Geometric Algebra Worked Exercises, Chapter 7

$\let\oldcdot\cdot \renewcommand{\cdot}{\kern-.2em\oldcdot\kern-.2em} \let\oldwedge\wedge \renewcommand{\wedge}{\kern-.2em\oldwedge\kern-.2em} \renewcommand{\vec}[1]{\mathbf{#1}} \let\oldtimes\times \renewcommand{\times}{\kern-.2em\oldtimes\kern-.2em} \newcommand{\dvec}[1]{\dot{\vec{#1}}} \newcommand{\ip}[2]{\vec{#1}\cdot\vec{#2}} \newcommand{\op}[2]{\vec{#1}\wedge\vec{#2}} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\hf}{\frac{1}{2}} \newcommand{\rn}{\rangle^\hf_0} $ Previous 7.1 Unit Vector Derivative $\vec{u} = \vec{u}(t)$ a vector valued function, $u = \lvert \vec{u} \rvert$. Show $$ \ddt\left(\frac{\vec{u}}{u}\right) = \frac{\vec{u}(\vec{u}\wedge\dvec{u})}{u^3} = \frac{(\vec{u}\times\dvec{u})\times\vec{u}}{u^3} $$ Two useful intermediate results. Vector Magnitude Squared: $$ u^2 = \vec{u}^2 $$ Proof: $$ u^2 = \left(\langle \vec{u}^\sim\vec{u} \rn\right)^2 $$ $$ = \langle \vec{u}^2 \rangle_0 $$ $$ = \vec{u}^2 $$

Geometric Algebra Worked Exercises, Chapter 4

$\renewcommand{\vec}[1]{\mathbf{#1}} \let\oldtimes\times \renewcommand{\times}{\kern-.2em\oldtimes\kern-.2em} \let\oldcdot\cdot \renewcommand{\cdot}{\kern-.2em\oldcdot\kern-.2em} \let\oldwedge\wedge \renewcommand{\wedge}{\kern-.2em\oldwedge\kern-.2em} \newcommand{\et}{e^{\vec{i}\theta}} \newcommand{\net}{e^{-\vec{i}\theta}} \newcommand{\eti}[1]{e^{#1\vec{i}\theta}} \newcommand{\gt}{\mathcal{G}_3} \newcommand{\hf}{\frac{1}{2}} \newcommand{\rn}{\rangle^\hf_0} $ Previous, Next. TODO: projection/rejection This chapter introduces rotations in the $\vec{i}$-plane where $\vec{i} = \vec{a}\wedge\vec{b}$ is a unit 2-blade. Unit blades have an orthonormal factorization, so for our 2-blade $$\vec{i}^2 = \vec{\sigma_1\sigma_2\sigma_1\sigma_2} = -\vec{\sigma_1\sigma_1\sigma_2\sigma_2} = -1$$ The cosine and the sine of the angle between two unit vectors $\vec{a}$ and $\vec{b}$ are (temporarily) defined as the scalar and bivector components of $\vec{ab}$

Geometric Algebra Worked Exercises, Chapter 3

This continues chapter 2 of NFCM.Next. This time I’ve chosen beauty over clarity; the equations are aligned nicely, but there are no hyperlinks as it seems to be impossible to escape from the align-environment-in-mathjax-in-hugo Inception style nightmare I have created. I think the awful alignment was better. $\let\oldtimes\times \renewcommand{\times}{\kern-.2em\oldtimes\kern-.2em} \let\oldcdot\cdot \renewcommand{\cdot}{\kern-.2em\oldcdot\kern-.2em} \let\oldwedge\wedge \renewcommand{\wedge}{\kern-.2em\oldwedge\kern-.2em} \renewcommand{\vec}[1]{\mathbf{#1}} \newcommand{\e}[1]{(-1)^{#1}} \newcommand{\eh}{(-1)^{r(r-1)/2}} \newcommand{\gt}{\mathcal{G}_3} \newcommand{\hf}{\frac{1}{2}} \newcommand{\rn}{\rangle^\hf_0}$ This chapter is set in familiar 3-space, $\gt$, and sees the introduction of the unit pseudoscalar $i$, a unit trivector with complex unit-like properties.

Geometric Algebra Worked Exercises, Chapter 2

$\let\oldcdot\cdot \renewcommand{\cdot}{\kern-.2em\oldcdot\kern-.2em} \let\oldwedge\wedge \renewcommand{\wedge}{\kern-.2em\oldwedge\kern-.2em} \renewcommand{\vec}[1]{\mathbf{#1}} \newcommand{\e}[1]{(-1)^{#1}} \newcommand{\eh}{(-1)^{r(r-1)/2}}$ I’ve been reading Hestenes’ brilliant New Foundations for Classical Mechanics, and I would love to be able to reach for Geometric Algebra as naturally as complex numbers, but sadly, the exercises in NFCM paint a different picture, so I thought I’d work through some of them here. And with the added yak shaving delights of mathjax in hugo, who could resist? I use the following precedence rules so I can omit parentheses:

WWDC 2015 Predictions

Emboldened by my successful prediction of a new language at WWDC2014 (Swift), I decided to publicly test my clairvoyance with some predictions for WWDC2015. Announcements new, free music streaming service, in which you can donate food to your favourite artists ApplePay launched outside US AppleTV SDK not announced (but nobody cares) OS X 11 Belmont Taptic API UXKit fails to appear New deduplicating file system iOS 9 CoreData deprecated Siri SDK ApplePay tech opened up to 3rd parties for phone/watch as 2FA Inter Process Communication finally comes to iOS, driven in part by watch extensions Xcode 7 Refactoring implemented!

AudioFileClose says wht?

You’re so close you can feel it - you’ve created your m4a file (AudioFileCreateWithURL), you’ve written your lovingly crafted MPEG4 AAC packets to it (AudioFileWritePackets), now all you have to do is close the file (AudioFileClose) and you’re done. Easy. But wait, what’s this? AudioFileClose failed, returning kAudioFileUnspecifiedError = ‘wht?’, a.k.a. 0x7768743F, a.k.a 2003334207. Why? Closing is easy, what could possibly go wrong? Have you forgotten something? Scanning AudioFile.h, the only function that you’ve never used is AudioFileOptimize.