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On Application of Algebra of Quaternions I International Journal of Scientific and Research Publications, Volume 11, Issue 2, February 2021 333 ISSN 2250-3153 On application of algebra of quaternions I Emran Sasi Althabit *, Ismail.M.Mohamed ** * Higher Institute for water Affairs ** Academic of Engineering and Medical Sciences DOI: 10.29322/IJSRP.11.02.2021.p11040 http://dx.doi.org/10.29322/IJSRP.11.02.2021.p11040 Abstract- We introduced in this paper the quaternions concepts and quaternions algebra, Representation of quaternions, transformation of quaternion in space exactly rotation and as application : Quaternions in space. Key words: Quaternion, Complex quaternions , Interpolation of quaternions . and I. INTRODUCTION i × j = k, j × k = i, k × i = j (1.5) he important of quaternions, appeared in a physics and a Which are very similar to the cross product of two unit Cartesian T Computer programs. vectors. * In physic s , quaternions are correlated to nature of the universe at the level of quantum mechanics. They lead to elegant expression of the Lorentz transformations, which form the basis of the modern theory of relativity.[6] * In a computer programs transformation of quaternions use for Animation and Simulation, The Quaternion Fourier Transform (QFT) is a powerful tool , its application include face Fig.2 A one-to-one correspondence between 3 recognition and voice recognition in and robot kinematics as a pure quaternions and vectors in ℝ [3] simulation of nature of the universe. 3 Octonions 1. Hamilton's Quaternions In the same year in which Hamilton discover the Definition1. 1:A quaternion is a four-dimensional complex quaternions wrote to his friend John Graves about his discovery number that can be used to represent the orientation of a rigid body Three months later on December 26th Graves wrote to or coordinate frame in three-dimensional space. The Hamilton's Hamilton about his discovery of a kind of ‘double-quaternion’ quaternions ℍ can be generalized to allow coefficients in ℝ as that he called ‘octives’, today are they are known as octonions. ℍ = {푄 = 푤 + 푥푖 + 푦푗 + 푧푘: 푤, 푥, 푦, 푧 ∈ ℝ } Like quaternions octonions form a division algebra, but unlike Or ℍ = {(푤, 푥, 푦, 푧): 푤, 푥, 푦, 푧 ∈ ℝ } (1.1) quaternions they are not associative. They are related to So that quaternion is the sum of a scalar part 푤 and a vectorial geometries in 7 and 8 dimensions.[6] 3 part (푥, 푦, 푧) ∈ ℝ i.e.,if 푟⃗ = 푥푖 + 푦푗 + 푧푘 is position vector in 푂 = 푠 + 푒1푥1 + 푒2푥2 + 푒3푥3 + 푒4푥4 + 푒5푥5 + 푒6푥6 space quaternions is define as + 푒7푥7 (1.6) 7 푤 + 푥푖 + 푦푗 + 푧푘 ↔ 푤 + 푟⃗ = [푤, 푟⃗] (1.2) where (푥1, 푥2, 푥3, 푥4, 푥5푥6, 푥7) ∈ ℝ in simply where 푤 real part 푅푒(푞), and 푟⃗ represent imaginary part= 퐼푚(푞)-vector in Mathematically, octonions are the largest normed division 3-dimession space, If = 0 , the quaternion algebras. This means that they satisfy |푎푏|2 = |푎|2|푏|2. 푞 = 푥푖 + 푦푗 + 푧푘 (1.3) is called pure quaternion i.e., A 3D vector is a pure quaternion When Hamilton discovered quaternions had to give up whose real part is zero. commutative property in order to preserve the norm; now the where 푖 , 푗 and 푘 are the standard orthonormal basis in ℝ3. associative property has to be given up in order to preserve the The multiplication for the primitive elements 푖, 푗 and 푘 is defined norm for the octonions. by 푖푗 = −푗푖 = 푘 A multiplication table for octonions can be made like 푗푘 = −푘푗 = 푖 quaternions as follows: (1.4) 푘푖 = −푘푖 = 푗 푖2 = 푗2 = 푘2 = 푖푗푘 = −1 This publication is licensed under Creative Commons Attribution CC BY. http://dx.doi.org/10.29322/IJSRP.11.02.2021.p11040 www.ijsrp.org International Journal of Scientific and Research Publications, Volume 11, Issue 2, February 2021 334 ISSN 2250-3153 The square of a quaternion is given by 푄 = [푤, 푟⃗], then 푄2 = [푤, 푟⃗][푤, 푟⃗] = [푤2 − 푟⃗. 푟⃗, 2푤푟⃗ + 푟⃗ × 푟⃗] = [푤2 − 푟⃗. 푟⃗, 2푤푟⃗] 푄2 = [푤2 − 푥2 − 푦2 − 푧2, 2푤(푥푖 + 푦푗 + 푧푘)] (2.4) The square of a pure quaternion is 푄2 = [0, 푟⃗][0, 푟⃗] = [0 − 푟⃗. 푟⃗, 푟⃗ × 푟⃗] Some of the properties of octonions as they relate to quaternions = [−푟⃗. 푟⃗, 0] = [−(푥2 + 푦2 + 푧2), 0] are the multiplication table for octonions[6] 3- The quotient We represent the quotient 푄1/푄2 as 푄 푄∗ 3 푄 = 푄 푄−1 = 1 2 (2.5) 3 1 2 | |2 푄2 Algebraic properties 2.2: (1) Norm: ퟐ ퟐ ퟐ ퟐ |푄| = √풘 + 풙 + 풚 + 풛 (2.6) and for two quaternions p , q satisfies the norm of product |퐐푷| = |퐐||푷| (2) Conjugate: ∗ ∗ 푄 = (풘 + 풙풊 + 풚풋 + 풛풌) = 풘 − 풙풊 − 풚풋 − 풛풌 (2.7) For example: 푒푖, 푒푗, and 푒푘 can be represented as follows: and for two quaternions p , q satisfies the conjugate of product(푸푷)∗ = 푷∗푸∗ 푒푖 푒푗 = 푒푘; 푒푗푒푖 = − 푒푘 . (1.7) Let 1 be the multiplicative identity. (3) Multiplicative inverse: 푄∗ Let each circle in the diagram be 푸−ퟏ = (2.8) 2 |푄| represented by the relationship 푒푖 = −1[6] 2. Quaternions Algebras and for two quaternions P , Q satisfies the inverse of product −ퟏ −ퟏ −ퟏ The Operations on quaternions 2.1: (퐏푸) = 푸 퐏 Let 푄 and 푄 be two quaternions , given as (4) Real part of Q 1 2 푸 + 푸∗ 푄1 = 푤1 + 푥1푖 + 푦1푗 + 푧1푘 , then 풘 = (2.9) 푄2 = 푤2 + 푥2푖 + 푦2푗 + 푧2푘 ퟐ 1- The addition and subtraction are define as (5) The dot product 푄 ± 푄 = (푤 + 푥 푖 + 푦 푗 + 푧 푘) ± (푤 + 푥 푖 + 푦 푗 + 푧 푘) 푄ퟏ ∙ 푄ퟐ = 풘ퟏ풘ퟐ + 풙ퟏ풙ퟐ + 풚ퟏ풚ퟐ + 풛ퟏ풛ퟐ 1 2 1 1 1 1 2 2 2 2 ∗ = (푤 ± 푤 ) + (푥 ± 푥 )푖 + (푦 ± 푦 )푗 + (푧 ± 푧 )푘 (2.1) = 푄ퟏ ∙ 푄ퟐ (2.10) 1 2 1 2 1 2 1 2 ퟐ ퟐ ퟐ (6) The unit quaternion: has norm 1, i.e., |푄| = 풘 + 풙 + 풚 + 2- The multiplication 풚ퟐ = ퟏ (2.11) From multiplication of the primitive elements 푖, 푗 and 푘 is and inverse of unit quaternion 푄−ퟏ = 푄∗ defined multiplication of two quaternions 푄1 and 푄1as and it can be represented by 푄1푄2 = (푤1 + 푥1푖 + 푦1푗 + 푧1푘)(푤푡2 + 푥2푖 + 푦2푗 + 푧2푘) 푄 = 퐜퐨퐬 휽 + 퐮̂ 퐬퐢퐧 휽 (2.12) (푤1푤2 − 푥1푥2 − 푦1푦2 − 푧1푧2) Where 퐮̂ is the vector has length 1. However, observe that the +(푤푡 푥 + 푥 푤 + 푦 푧 − 푧 푦 )푖 quaternion product 퐮̂퐮̂ = −ퟏ = 1 2 1 2 1 2 1 2 (2.2) +(푤 푦 − 푥 푧 + 푦 푤 + 푧 푥 )푗 Note in general : a pure unit quaternion is a square root of −ퟏ, 1 2 1 2 1 2 1 2 such as imaginary units and Euler’s identity of quaternion +(푤푡 푧 + 푥 푦 − 푦 푥 − 푧 푤푡 )푘 1 2 1 2 1 2 1 2 The unit Quaternion is normalised with respect to the Hermitian inner product , where 퐼2 is the 2x2 identity matrix. We have 푄1푄2 ≠ 푄2푄1 i.e., the multiplication of quaternions is not commutative in general . the quaternions is also A quaternion with a unit norm is called a normalised quaternion viewed as 푄 = 푤 + 푟⃗ , where 푄 푄 푟⃗ = 푥푖 + 푦푗 + 푧푘 , we identify with 3D vector (푥, 푦, 푧) so the dot 푄′ = = (2.13) product (∙) and cross product (×) of vectors introduced the |푄| √푤2 + |풓⃗⃗|2 familiar notation and so we define the canonical scalar product on (7) The unit length Euler's identity for quaternions ℍ by 퐞퐱퐩(퐮̂휽) = 퐜퐨퐬 휽 + 퐮̂ 퐬퐢퐧 휽 . (2.14) 푄1푄2 = (푤1 + 푟⃗1)(푤푡2 + 푟⃗2) where the exponential on the left-hand side is evaluated by 풏 풏 = (푤1푤2 − 푟⃗⃗⃗1⃗. 푟⃗⃗⃗2⃗) + 푤1푟⃗⃗⃗2⃗ + 푤2푟⃗⃗⃗1⃗ + 푟⃗⃗⃗1⃗ × 푟⃗⃗⃗2⃗ (2.3) 푄 = (퐜퐨퐬 휽 + 퐮̂ 퐬퐢퐧 휽) = 퐜퐨퐬(풏휽) + 퐮̂ 퐬퐢퐧(풏휽) The multiplication of quaternions is not a commutative = 퐞퐱퐩(퐮̂풏휽) (2.15) over a real numbers so that it is called is a non-commutative (8) It is also possible to define the logarithm of a unit quaternion, algebra. 퐥퐧 푄 = 퐥퐧(퐜퐨퐬 휽 + 퐮̂ 퐬퐢퐧 휽) This publication is licensed under Creative Commons Attribution CC BY. http://dx.doi.org/10.29322/IJSRP.11.02.2021.p11040 www.ijsrp.org International Journal of Scientific and Research Publications, Volume 11, Issue 2, February 2021 335 ISSN 2250-3153 ퟎ ퟏ ퟎ 풊 = 퐥퐧 퐞퐱퐩(퐮̂휽) = 퐮̂휽 (2.16) 풋 = [ ] = [ ] 풊 , The quaternions can be thought of as a choice of a group structure −ퟏ ퟎ −풊 ퟎ ퟎ 풊 ퟎ ퟏ (9) The quaternion polar coordinate representation is 풌 = [ ] = [ ] 풊 푄 = |푄|(퐜퐨퐬 휽 + 퐮̂ 퐬퐢퐧 휽) 풊 ퟎ ퟏ ퟎ so that if we define the matrices훔ퟏ, 훔ퟐ, 훔ퟑsuch that = |푄| 퐞퐱퐩(푢̂휽) (2.17) ퟏ ퟎ ퟎ −풊 ퟎ ퟏ 흈 = [ ] , 흈 = [ ] , 흈 = [ ] (3.3) (10) The multiplication of two vectors ퟏ ퟎ −ퟏ ퟐ 풊 ퟎ ퟑ ퟏ ퟎ If we multiply together two imaginary quaternions 푷, 푄 ∈ ℍ퐩, we can write where ℍ퐩 is set of pure quaternions we obtain a quaternionic 푄 = 풘ퟏ + 풙풊 + 풚풋 + 풛풌 ퟑ version of the scalar product and vector product on ℝ as = 푤ퟏ + 풊(풛흈ퟏ + 풚흈ퟐ + 풙흈ퟑ) (3.4) follows[10] The matrices훔ퟏ, 훔ퟐ, 훔ퟑare called the Pauli spin matrices. 푷푄 = −푷 · 푄 + 푷 ∧ 푄 ∈ ℝ ⊕ ℍ퐩 ≅ ℍ Note that the irtraces are null and that they are Hermitian (recall (2.18) that a complex matrix is Hermitian if it is equal to the transpose ∗ Others algebraic properties of its conjugate, i.e.,푨 = 푨). If we let 풆ퟎ = 풕, The set of quaternions satisfy some other algerbic properties that 풆ퟏ = 풛, 풆ퟐ = 풚and풆ퟑ = 풙, then 풒 can be written as worth mentioning 푄 = 풆ퟎퟏ + 풊(풆ퟏ흈ퟏ + 풆ퟐ흈ퟐ + 풆ퟑ흈ퟑ) (3.5) * The set of quaternions is an abelian group (ℍ, +)under And풆ퟎ, 풆ퟏ, 풆ퟐ, 풆ퟑ are called theEuler parametersof the rotation quaternion addition .
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