Quaternions We will now delve into a subject which at first may seem quite

Quaternions we will now delve into a subject which at

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Quaternions: We will now delve into a subject, which at first may seem quite unrelated. But keep the above expression in mind, since it will reappear in most surprising way. This story begins in the early 19th century, when the great mathematician William Rowan Hamilton was searching for a generalization of the complex number system. Imaginary numbers can be thought of as linear combinations of two basis elements, 1 and i , which satisfy the multiplication rules 1 2 = 1, i 2 = - 1 and 1 · i = i · 1 = i . (The interpretation of i = - 1 arises from the second rule.) A complex number a + bi can be thought of as a vector in 2-dimensional space ( a, b ). Two important concepts with complex numbers are the modulus , which is defined to be a 2 + b 2 , and the conjugate , which is defined to be ( a, - b ). In vector terms, the modulus is just the length of the vector and the conjugate is just a vertical reflection about the x -axis. If a complex number is of modulus 1, then it can be expressed as (cos θ, sin θ ). Thus, there is a connection between complex numbers and 2-dimensional rotations. Also, observe that, given such a unit modulus complex number, its conjugate is (cos θ, - sin θ ) = (cos( - θ ) , sin( - θ )). Thus, taking the conjugate is something like negating the associated angle. Hamilton was wondering whether this idea could be extended to three dimensional space. You might reason that, to go from 2D to 3D, you need to replace the single imaginary quantity i with two imaginary quantities, say i and j . Unfortunately, this this idea does not work. After many failed attempts, Hamilton finally came up with the idea of, rather than using two imaginaries, instead using three imaginaries i , j , and k , which behave as follows: i 2 = j 2 = k 2 = ijk = - 1 ij = k, jk = i, ki = j. Combining these, it follows that ji = - k , kj = - i and ik = - j . The skew symmetry of multiplication (e.g., ij = - ji ) was actually a major leap, since multiplication systems up to that time had been commutative.) Hamilton defined a quaternion to be a generalized complex number of the form q = q 0 + q 1 i + q 2 j + q 3 k. Thus, a quaternion can be viewed as a 4-dimensional vector q = ( q 0 , q 1 , q 2 , q 3 ). The first quantity is a scalar, and the last three define a 3-dimensional vector, and so it is a bit more intuitive to express this as q = ( s, u ), where s = q 0 is a scalar and u = ( q 1 , q 2 , q 3 ) is a vector in 3-space. We can define the same concepts as we did with complex numbers: Conjugate: q * = ( s, - u ) Modulus: | q | = p q 2 0 + q 2 1 + q 2 2 + q 2 3 = p s 2 + ( u · u ) Unit Quaternion: q is said to be a unit quaternion if | q | = 1 Quaternion Multiplication: Consider two quaternions q = ( s, u ) and p = ( t, v ): q = ( s, u ) = s + u x i + u y j + u z k p = ( t, v ) = t + v x i + v y j + v z k. Lecture 6 7 Spring 2018
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CMSC 425 Dave Mount & Roger Eastman If we multiply these two together, we’ll get lots of cross-product terms, such as ( u x i )( v y j ), but we can simplify these by using Hamilton’s rules. That is, ( u x i )( v y j ) = u x v h ( ij ) = u x v h k .
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