Unit Equations on Quaternions

Some Notes on Unit Quaternions and Rotation

Unit Quaternions and the Bloch Sphere

The spinor representation of spin-1/2 states can equally well be mapped to a single unit quaternion, yielding a new perspective despite the equivalent mathematics. This paper first demonstrates a useable map that allows Bloch-sphere rotations to be represented as quaternionic multiplications, simplifying the form of the dynamical equations. Left-multiplications generally correspond to non-unita...

Closed-form solution of absolute orientation using unit quaternions

Received August 6,1986; accepted November 25,1986 Finding the relationship between two coordinate systems using pairs of measurements of the coordinates of a number of points in both systems is a classic photogrammetric task. It finds applications in stereophotogrammetry and in robotics. I present here a closed-form solution to the least-squares problem for three or more points. Currently vario...

Representing Attitude: Euler Angles, Unit Quaternions, and Rotation Vectors

We present the three main mathematical constructs used to represent the attitude of a rigid body in threedimensional space. These are (1) the rotation matrix, (2) a triple of Euler angles, and (3) the unit quaternion. To these we add a fourth, the rotation vector, which has many of the benefits of both Euler angles and quaternions, but neither the singularities of the former, nor the quadratic ...

Perfect Sequences and Arrays over the Unit Quaternions

The quaternions were discovered by the Irish mathematician Sir William Rowan Hamilton in 1843. Hamilton was interested in the connection between complex numbers and 2-dimensional geometry. He tried in vain to extend the complex numbers to R, only years later would it be discovered that there is no 3-dimensional normed division algebra. Hamilton’s breakthrough came when he extended the complex n...