Left-handed Compliment for Quaternion Multiplication

If you understand Electromagnetism, then the concept of the “Right hand Rule” is quite familiar.  However, for those who are foreign to this concept, please bear with us.  In mathematics and physics, the right-hand rule is a common mnemonic for understanding notation conventions for vectors in 3 dimensions. It was invented for use in electromagnetism by British physicist John Ambrose Fleming in the late 19th century.

The right-hand rule in relation to a three dimensional coordinate axis system is described as follows. The thumb is orientated along the Z-Axis, the first or index finger is orientated to the X-Axis, and the second finger is orientated to the Y-Axis. The finger-tips of the three digits denote the positive direction of a linear translation or displacement. A positive rotation of the Z-Axis, clockwise, is indicated by the third and fourth fingers. Positive rotations of the X and Y-Axis are also clockwise.

Right-hand rule for cross products

The cross product of two vectors is often encountered in physics and engineering. For example, in statics and dynamics, torque is the cross product of lever length and force, and angular momentum is the cross product of linear momentum and distance from an origin. In electricity and magnetism, the force exerted on a moving charged particle when moving in a magnetic field B is given by:

Fb = qv  X  B

Magnetic force on a moving charged particle

The direction of the cross product may be found by application of the right hand rule as follows: Using your right hand,

•             Point your index finger in the direction of the first vector A.

•             Point your middle finger in the direction of the second vector B.

•             Your thumb will point in the direction of the cross product C.

For example, for a positively-charged particle moving to the North, in a region where the magnetic field points West, the resultant force will point up.

Quaternion Left-hand rule

Now let’s talk about “The Left Hand Rule”.  What, you haven’t heard about the “Left Hand Rule”?  No not Fleming’s Left hand rule for motor effect.  This is the general rule for Quaternion Multiplication. In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space.

A feature of quaternions is that multiplication of two quaternions is no commutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors.

Essentially, take your left hand and when you multiply one vector to another you just let your fingers of your open hand to align with the first vector and move to the second vector and the thumb will align with the product of the two, viola!

So why is this significant?

Quaternion algebra was introduced by Hamilton in 1843.  Important precursors to this work included Euler's four-square identity (1748) and Olinde Rodrigues' parameterization of general rotations by four parameters (1840), but neither of these writers treated the four-parameter rotations as an algebra.  Carl Friedrich Gauss had also discovered quaternions in 1819, but this work was not published until 1900. 

Hamilton knew that the complex numbers could be interpreted as points in a plane, and he was looking for a way to do the same for points in three-dimensional space. Points in space can be represented by their coordinates, which are triples of numbers, and for many years he had known how to add and subtract triples of numbers. However, Hamilton had been stuck on the problem of multiplication and division for a long time. He could not figure out how to calculate the quotient of the coordinates of two points in space.

The great breakthrough in quaternions finally came on Monday 16 October 1843 in Dublin, when Hamilton was on his way to the Royal Irish Academy where he was going to preside at a council meeting. As he walked along the towpath of the Royal Canal with his wife, the concepts behind quaternions were taking shape in his mind. When the answer dawned on him, Hamilton could not resist the urge to carve the formula for the quaternions,

i2 = j2 = k2 = ijk = −1

into the stone of Brougham Bridge as he paused on it.

Significance Today to the everyday man

Well now that we have laid out the significance to mathematicians and physics students.  Why is this significant to anyone outside this field?  One word, Christmas.

Yes, Christmas where literally billions are spent on one of the leading entertainment items nearly 10 years straight, video games.  So, how does this equate? Primarily Quaternion math is crucial to video game programmers. 

The main reason games use quaternions is because they represent rotations almost as space-efficiently as Euler angles, without suffering from Gimbal lock. Gimbal Lock begins when any Euler angle reaches a rotation of 90 degrees around any axis: you immediately lose a degree of freedom. Quaternions address this issue by adding a fourth dimension. If you stuck with Euler angles, you'd have to restrict one axis to never rotating more than ~89 degrees.   In layments language movement is no longer robotic as in the early 90s.

So folks enjoy your new found knowledge and rest assured that 3D motion shall be free flowing due to your left hand.  I would say thanks to Hamilton for providing the single most significant breakthrough in mathematics in the 19th century, but that would kinda be a left handed compliment.

Source(s):

• http://gamedev.stackexchange.com/questions/2029/what-math-should-all-game-programmers-know
• http://en.wikipedia.org/wiki/Right-hand_rule
• http://3dgep.com/understanding-quaternions/
• http://www.physicsforums.com/showthread.php?t=707052
• http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/

So “Once more unto the breach, dear friends, once more;”

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About Rick Ricker

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