Dose of STEM: Rotate in 3D with Quaternions

Humans are capable of visualizing only 3 dimensions. What we perceive, surprisingly, is only 2 dimensions. The depth that we think we see, is just a trick - an illusion our brains have learnt after years of evolution. This is what made the discovery that 4 dimensional numbers are required to describe rotation in 3D all the more riveting.

Visualize the rotation of earth as opposed to a spinning frisbee. Each point on the frisbee's circumference moves in the same manner but if you look at the earth, every point on the surface moves with a different velocity based on distance from the equator. The points where the axis of rotation passes through don’t move at all. This is what led to William Hamilton concluding that what were missing, was a whole dimension of numbers to describe this rotation.

The rotations in 3D are just projections of what was happening in the fourth dimension. This is similar to how 3D objects have a 2 dimensional shadow. So rotating the object using quaternions will have a corresponding projection in 3D that we will be able to observe.

Any quaternion q can be represented as q = A + Bi + Cj + Dk where A, B, C, and D are real numbers and i, j and k are unit vectors corresponding to the axes.

Similar to vectors, I times j is k , j times k is I and k times I is j and multiplying in opposite order gives the same magnitude with a minus sign. Also, i^2 = j^2 =k^2= -1= ijk

One crucial difference is that quaternion multiplication is NOT commutative. The conjugate of a quaternion A + Bi + Cj + Dk is given by A - (Bi+Cj+Dk), similar to complex numbers. Thus the rotation of any point x given as Xi + Yj+ Zk is given by x' = q x q' where q is the quaternion that the point is to be rotated by, q' is the conjugate and x is the point To visualize the usage of quaternions and witness their practical application, check out this explorable video.