The explanation of "What's that extra 1 for?" in the column representation of 3-d coordinates (x y z 1) could benefit from mentioning that translation—moving things—is not a linear transformation (the origin is not mapped to itself) but an affine transformation. Therefore, you cannot represent translation in 3-d space with a 3x3 matrix. What you can do, though, is embed that 3-d space within a 4-d space fixed at some coordinate on its 4th dimension, typically w=1. Then, a translation in the original 3-d space can be represented as a linear transformation in the 4-d space and thus can also be represented by a 4x4 matrix multiplication. So the extra 1 is actually what allows all common 3-d operations, including translation, to be done via linear algebra and thereby harness the brutal power of matrix multiplication on modern computing devices.
Is that why quaternions are interesting for computer graphics? I studied mathematics and computer science but never understood (or even looked so hard into) the connection.
Kind of! The scalar allows vectors and scalars to be expressed together as one object, which has a lot of computational and mathematical niceties.
The killer feature is that you can put the two together for a rotation axis (~ vector) and angle (~ scalar).
With just three gimbals (rotating circles), if gimbals A and B are aligned, you only have two degrees of freedom (rotating A is the same as rotating B). Because of this, interpolating angles is unwieldy in vector space. 'Gimbal lock' confounds animation (in hilarious but unrealistic ways) but also aerospace (four hours before 'one small step for man', just after landing, Collins joked he would like a fourth gimbal for Christmas).
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The killer feature is that you can put the two together for a rotation axis (~ vector) and angle (~ scalar).
With just three gimbals (rotating circles), if gimbals A and B are aligned, you only have two degrees of freedom (rotating A is the same as rotating B). Because of this, interpolating angles is unwieldy in vector space. 'Gimbal lock' confounds animation (in hilarious but unrealistic ways) but also aerospace (four hours before 'one small step for man', just after landing, Collins joked he would like a fourth gimbal for Christmas).