Exponentially Better Rotations
All of this touches on maybe 1% of the potential of geometric algebra: <a href="https:&#x2F;&#x2F;bivector.net" rel="nofollow">https:&#x2F;&#x2F;bivector.net</a><p>I would strongly suggest watching some talks about it: <a href="https:&#x2F;&#x2F;bivector.net&#x2F;doc.html" rel="nofollow">https:&#x2F;&#x2F;bivector.net&#x2F;doc.html</a><p>For example a rigid body simulator in n-D in ~40 LOC, 20 of which are boilerplate defining the points of a cube: <a href="https:&#x2F;&#x2F;youtu.be&#x2F;5R2sv9GCwz0?t=371" rel="nofollow">https:&#x2F;&#x2F;youtu.be&#x2F;5R2sv9GCwz0?t=371</a> (in an environment that defines the geometric algebra for you - but only that, it&#x27;s no different than using a library that provides quaternions for you, for example). This is possible because geometric algebra is powerful enough to unify and interpolate not just rotations as one object, but generalize rotations, reflections and translations (and more) as one object (a motor), embedded in an algebra unifying position, velocity, (angular) momentum, torque and force. If it doesn&#x27;t make sense how that works, watch the earlier talks in the series.<p>And it gets more general, the above talk is about PGA which embodies flat (object) Euclidean geometry, but CGA also embeds spheres and circles in the algebra, allowing any conformal transformation to be represented natively in the algebra. This is very powerful for (inverse) kinematics in robots with fixed rotating arms, for example: <a href="https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=j2x_TG8E3_k" rel="nofollow">https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=j2x_TG8E3_k</a>