Quaternions

• • A quaternion is made up of a scalar w and a vector component.

  • A quaternion p can be listed as [w, v] or [ w, (x, y, z) ].

  • It's math is based on complex numbers and makes no sense for anyone.

  • To negate a quaternion we negate each component in it. -q =) [-w, -n]

• The identity quaternion i = [1, 0].

  • The magnitude of a quaternion is ||q|| = sqrt(w² + ||v||²).

  • The conjugate of q is obtained by negating the vector portion of the quaternion. q* = [w, -v]

  • The inverse of q is the conjugate divided by its magnitude. q^-1 = q* / ||q||.

• If only unit vectors are used then the inverse is the same as the conjugate.

Axis Angle Pair:

• • With n the axis of rotation and θ the angle around it then q = [ cos(θ/2), sin(θ/2)n, ) ].

Reference

• • Rotating a single vector using a quaternion - 2019

  • Visualizing quaternions - An explorable video series - 2018

  • Quaternion half/double angle and Cayley transforms - 2016

• Showing the Correctness of Quaternion Rotation

  • Fantastic Quaternions - Numberphile - 2016

  • Approximating slerp - 2015

  • Why Do Quaternions Double-Cover? - 2014

  • Animate Your Way to Glory - Part II - 2013

  • On Vector Math Libraries - 2013

  • Understanding Quaternions - 2013

  • Quaternions and Dual Quaternion Skinning - 2012

  • Understanding Quaternions - 2012

  • Nested Complex Numbers - 2012

  • Quaternions part 1 - 2012

  • Points, Vertices and Vectors - 2011

  • Hacking Quaternions - 2002

  • Representing Rotations in Quaternion Arithmetic - 2002

  • Inverse Kinematics with Quaternion Joint Limits - 2002

  • Using Quaterions for Animation in OpenGL - 1998

  • Using Quaterions for Animation in OpenGL Part 2, Interpolation - 1998

  • Rotating Objects Using Quaternions - 1998

• F*cking Quaternions: How Do They Work?

• Noel Hughes

• Optimizing slerp

Rotating a single vector using a quaternion

https://fgiesen.wordpress.com/2019/02/09/rotating-a-single-vector-using-a-quaternion/