Complex numbers, Quaternions and Octonions

https://en.wikipedia.org/wiki/Real_number   2^0  = 1 Dimention

https://en.wikipedia.org/wiki/Complex_number  2^1 = 2 Dimentions

https://en.wikipedia.org/wiki/Quaternion  2^2 = 4 Dimentions

https://en.wikipedia.org/wiki/Octonion  2^3 = 8 Dimentions

https://en.wikipedia.org/wiki/Sedenion  2^4  = 16 Dimentions

Trigintaduonions  2^5  = 32 Dimensions

https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction

https://en.wikipedia.org/wiki/Clifford_algebra

In the construction of types of numbers, we have the following sequence:

R⊂C⊂H⊂O⊂S$$R\subset C\subset H\subset O\subset S$$

or:

20−ions⊂21−ions⊂22−ions⊂23−ions⊂24−ions$$$$

or:

"Reals" ⊂$\subset$ "Complex" ⊂$\subset$ "Quaternions" ⊂$\subset$ "Octonions" ⊂$\subset$ "Sedenions"

With the following "properties":

• From R$\mathbb{R}$ to C$\mathbb{C}$ you gain "algebraic-closure"-ness (but you throw away ordering).
• From C$\mathbb{C}$ to H$\mathbb{H}$ we throw away commutativity.
• From H$\mathbb{H}$ to O$\mathbb{O}$ we throw away associativity.
• From O$\mathbb{O}$ to S$\mathbb{S}$ we throw away multiplicative normedness.

Why am I talking about this, Well specifically Quaternions are of interest for Robotics. 

There are many different parameterizations for orientations:

• Euler Angles 
• Angle Axis
• Rotation matrix 
• Quaternions

Euler-Angle, Angle-Axis  have Singularities! 

Rotation Matrix 

• 9 scalars, more complex regularization 
• Concatenation: 27 multiplications
• Rotating a vector: 9 multiplications

Quaternion

• 4 scalars, easy regularization
• Concatenation: 16 multiplications
• Rotating a vector: 18 multiplications 

Compared to Euler angles they are simpler to compose and avoid the problem of gimbal lock. Compared to rotation matrices they are more compact, more numerically stable, and more efficient. ... When used to represent rotation, unit quaternions are also called rotation quaternions.

Quaternions and spatial rotation - Wikipedia

https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation

William Hamilton invented quaternions in 1843 as a method to allow him to multiply and divide vectors, rotating and stretching them.

https://www.allaboutcircuits.com/technical-articles/dont-get-lost-in-deep-space-understanding-quaternions/

Alternative to Euler and Dot Products. http://en.wikipedia.org/wiki/Dot_product

http://en.wikipedia.org/wiki/Quaternion

http://www.tinkerer.eu/Quaternions/

Quaternions are an expansion of compex numbers. A quaternion has three imaginary elements: $i$, $j$ and $k$ and can be written in the form:

   $\tilde{Q} = q_w + q_x i + q_y j + q_z k$

http://en.wikipedia.org/wiki/Axis%E2%80%93angle_representation#Unit_quaternions

http://en.wikipedia.org/wiki/Conversion_between_quaternions_and_Euler_angles

http://www.j3d.org/matrix_faq/matrfaq_latest.html#Q60

OpenSCAD

• https://www.spolearninglab.com/curriculum/lessonPlans/math/quaternions.html

Singularities

One must be aware of singularities in the Euler angle parametrization when the pitch approaches ±90° (north/south pole). These cases must be handled specially. The common name for this situation is gimbal lock.

Code to handle the singularities is derived on this site: www.euclideanspace.com

A sensor fusion algorithm for an integrated angular position estimation with inertial measurement units

• http://www.date-conference.com/proceedings/PAPERS/2011/DATE11/PDFFILES/IP1_06.PDF

A Gyro-Free Quaternion based Attitude Determination system suitable or implementation using low cost sensors.

• https://users.soe.ucsc.edu/~elkaim/Documents/QuatAttitude.pdf

Orientation estimation using a quaternion-based indirect Kalman filter with adaptive estimation of external acceleration

• http://infolab.ulsan.ac.kr/research/orientation/paper.pdf

http://en.wikipedia.org/wiki/Rotation_matrix

http://www.chrobotics.com/library/understanding-quaternions

http://www.euclideanspace.com/maths/geometry/rotations/conversions/quaternionToMatrix/

Videos

https://www.quantamagazine.org/the-strange-numbers-that-birthed-modern-algebra-20180906/

https://youtu.be/d4EgbgTm0Bg What are quaternions, and how do you visualize them? A story of four dimensions.

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https://www.youtube.com/watch?v=dttFiVn0rvc Math for Game Developers - Axis-Angle Rotation

https://www.youtube.com/watch?v=SCbpxiCN0U0 Math for Game Developers - Rotation Quaternions

https://www.youtube.com/watch?v=A6A0rpV9ElA Math for Game Developers - Quaternion Inverse

https://www.youtube.com/watch?v=CRiR2eY5R_s Math for Game Developers - Multiplying Quaternions

https://www.youtube.com/watch?v=Ne3RNhEVSIE Math for Game Developers - Quaternions and Vectors

https://www.youtube.com/watch?v=x1aCcyD0hqE Math for Game Developers - Slerping Quaternions Spherical Linear interpolation.

https://www.youtube.com/watch?v=fRSaaLtYj68 Math for Game Developers - Quaternion Wrapup and Review

https://github.com/BSVino/MathForGameDevelopers

https://twitter.com/VinoBS

http://www.vinoisnotouzo.com/

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https://www.youtube.com/watch?v=dul0mui292Q Math for Game Developers - Perspective Matrix Part 2

https://www.youtube.com/watch?v=jeO_ytN_0kk Math for Game Developers - Perspective Matrix

https://www.youtube.com/watch?v=8gST0He4sdE Hand Calculation of Quaternion Rotation

https://www.youtube.com/watch?v=KdW9ALJMk7s Quaternions Explained by Dan

https://www.youtube.com/watch?v=0_XoZc-A1HU FamousMathProbs13b: The rotation problem and Hamilton's discovery of quaternions (II)

https://www.youtube.com/watch?v=d4EgbgTm0Bg  What are quaternions, and how do you visualize them? A story of four dimensions.