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quaternion

A quaternion is best defined as a symbol of the type q = Za s e s = aoeo + ales = ale, + a3e3, where eo, ...

00Thus every quaternion may be written in the form q = Sq+Vq, where either Sq or Vq may separately vanish; so that ordinary algebraic quantities (or scalars, as we shall call them) and pure vectors may each be regarded as special cases of quaternions.

00The equations q'+x = q and y+q' = q are satisfied by the same quaternion, which is denoted by q - q'.

00In the applications of the calculus the co-ordinates of a quaternion are usually assumed to be numerical; when they are complex, the quaternion is further distinguished by Hamilton as a biquaternion.

00The outer and inner products of two extensive quantities A, B, are in many ways analogous to the quaternion symbols Vab and Sab respectively.

00These may be compared and contrasted with such quaternion formulae as S(VabVcd) =SadSbc-SacSbd dSabc = aSbcd - bScda+cSadb where a, b, c, d denote arbitrary vectors.

00The word "quaternion " properly means " a set of four."

00From the purely geometrical point of view, a quaternion may be regarded as the quotient of two directed lines in space - or, what comes to the same thing, as the factor, or operator, which changes one directed line into another.

00We may state, in passing, that every quaternion can be represented as a (cos 0+ 7 sin 9), - where a is a real number, 6 a real angle, and it a directed unit line whose square is - 1.

00He had now three distinct space-units, i, j, k; and the following conditions regulated their combination by multiplication: - I T = 12 '=' 2 = _ 1, ij= - ji=k, jk= - kj=i, ki= - ik =j.3 And now the product of two quaternions could be at once expressed as a third quaternion, thus (a+ib+jc+kd) (a'+ib'+jc'+kd') = A+iB+jC+kD, where A=aa' - bb' - cc' - dd', B = ab'+ba'+cd' - dc', C = ac'+ca'+db' - bd', D =ad' +da'+bc' - cb'.

00And now a directed line in space came to be represented as ix+jy+kz, while the product of two lines is the quaternion - + yy ' +2z') +i (yz ' - zy') +j (zx' - xz') +k (x y ' - yx').

00Hence, and in this lies the main element of the symmetry and simplicity of the quaternion calculus, all systems of three mutually rectangular unit lines in space have the same properties as the fundamental system i, j, k.

00This fundamental system, therefore, becomes unnecessary; and the quaternion method, in every case, takes its reference lines solely from the problem to which it is applied.

00Hamilton seems never to have been quite satisfied with the apparent heterogeneity of a quaternion, depending as it does on a numerical and a directed part.

00The year after the first publication of the quaternion method, there appeared a work of great originality, by Grassmann," in which results closely analogous to some of those of Hamilton were given.

00The results of these two kinds of multiplication correspond respectively to the numerical and the directed parts of Hamilton's quaternion product.

00Hamilton had geometrical application as his main object; when he realized the quaternion system, he felt that his object was gained, and thenceforth confined himself to the development of his method.

00But his claims, however great they may be, can in no way conflict with those of Hamilton, whose mode of multiplying couples (in which the " inner " and " outer " multiplication are essentially involved) was produced in 1833, and whose quaternion system was completed and published before Grassmann had elaborated for press even the rudimentary portions of his own system, in which the veritable difficulty of the whole subject, the application to angles in space, had not even been attacked.

00Any quaternion may now be expressed in numerous simple forms. Thus we may regard it as the sum of a number and a line, a+a, or as the product, fly, or the quotient, be-', of two directed lines, &c., while, in many cases, we may represent it, so far as it is required, by a single letter such as q, r, &c.

00But every quaternion formula is a proposition in spherical (sometimes degrading to plane) trigonometry, and has the full advantage of the symmetry of the method.

00And one of Hamilton's earliest advances in the study of his system (an advance independently made, only a few months later, by Arthur Cayley) was the interpretation of the singular operator q()q1, where q is a quaternion.

00Here, then, is a case specially adapted to the isotropy of the quaternion system; and Hamilton easily saw that the expression i d x +j - + k dz could be, like ix+jy+ kz, effectively expressed by a single letter.

00No better testimony to the value of the quaternion method could be desired than the constant use made of its notation by mathematicians like Clifford (in his Kinematic) and by physicists like ClerkMaxwell (in his Electricity and Magnetism).

00Joly's projective geometrical applications starting from the interpretation of the quaternion as a point-symbol;' these applications may be said to require no addition to the quaternion algebra; (b) W.

00Shaw, in America, independently of Joly, has interpreted the quaternion as a point-symbol.

00This is the basis of a method parallel throughout to the quaternion method; in the specification of rotors and motors it is independent of the origin which for these purposes the quaternion method, pure and simple, requires.

00(Note that the z here occurring is only required to ensure harmony with tri-quaternions of which our present biquaternions, as also octonions, are particular cases.) The point whose position vector is Vrq i is on the axis and may be called the centre of the bi-quaternion; it is the centre of a sphere of radius Srq i with reference to which the point and plane are in the proper quaternion sense polar reciprocals, that is, the position vector of the point relative to the centre is Srg i.

00If Q= Ep+nq+wr and we put Q= (I +Zwt)(Ep-i-nq)X (1 Zwt) -1 we find that the quaternion t must be 2f (r) /f (q - p), where f(r)=rq - Kpr.

00When Su = o, (I + 2wu) () (1 + zwu) -1 is an operator which shifts (without further change) the tri - quaternion operand an amount given by u in direction and distance.

00Hamilton, of quaternion fame.

00An error in the computation of inverses in quaternion algebras with non-standard parameters.

00So it remains an intriguing question, whether e.g. gage field theory is more naturally quantised using quaternion Hilbert space, H.

00quaternion algebras with non-standard parameters.

00quaternion matrix that is unique except for its sign.

00quaternion math routines and contributed the BeOS port.

00quaternion parameters.

00quaternion methods.

00A unit quaternion has m = 1. det( Q R )= m 4.

00A quaternion is best defined as a symbol of the type q = Za s e s = aoeo + ales = ale, + a3e3, where eo, ...

00Thus every quaternion may be written in the form q = Sq+Vq, where either Sq or Vq may separately vanish; so that ordinary algebraic quantities (or scalars, as we shall call them) and pure vectors may each be regarded as special cases of quaternions.

00The equations q'+x = q and y+q' = q are satisfied by the same quaternion, which is denoted by q - q'.

00In the applications of the calculus the co-ordinates of a quaternion are usually assumed to be numerical; when they are complex, the quaternion is further distinguished by Hamilton as a biquaternion.

00The outer and inner products of two extensive quantities A, B, are in many ways analogous to the quaternion symbols Vab and Sab respectively.

00These may be compared and contrasted with such quaternion formulae as S(VabVcd) =SadSbc-SacSbd dSabc = aSbcd - bScda+cSadb where a, b, c, d denote arbitrary vectors.

00The word "quaternion " properly means " a set of four."

00From the purely geometrical point of view, a quaternion may be regarded as the quotient of two directed lines in space - or, what comes to the same thing, as the factor, or operator, which changes one directed line into another.

00We may state, in passing, that every quaternion can be represented as a (cos 0+ 7 sin 9), - where a is a real number, 6 a real angle, and it a directed unit line whose square is - 1.

00He had now three distinct space-units, i, j, k; and the following conditions regulated their combination by multiplication: - I T = 12 '=' 2 = _ 1, ij= - ji=k, jk= - kj=i, ki= - ik =j.3 And now the product of two quaternions could be at once expressed as a third quaternion, thus (a+ib+jc+kd) (a'+ib'+jc'+kd') = A+iB+jC+kD, where A=aa' - bb' - cc' - dd', B = ab'+ba'+cd' - dc', C = ac'+ca'+db' - bd', D =ad' +da'+bc' - cb'.

00And now a directed line in space came to be represented as ix+jy+kz, while the product of two lines is the quaternion - + yy ' +2z') +i (yz ' - zy') +j (zx' - xz') +k (x y ' - yx').

00Hence, and in this lies the main element of the symmetry and simplicity of the quaternion calculus, all systems of three mutually rectangular unit lines in space have the same properties as the fundamental system i, j, k.

00This fundamental system, therefore, becomes unnecessary; and the quaternion method, in every case, takes its reference lines solely from the problem to which it is applied.

00Hamilton seems never to have been quite satisfied with the apparent heterogeneity of a quaternion, depending as it does on a numerical and a directed part.

00He indulged in a great deal of speculation as to the existence of an extra-spatial unit, which was to furnish the raison d'etre of the numerical part, and render the quaternion homogeneous as well as linear.

00The year after the first publication of the quaternion method, there appeared a work of great originality, by Grassmann," in which results closely analogous to some of those of Hamilton were given.

00The results of these two kinds of multiplication correspond respectively to the numerical and the directed parts of Hamilton's quaternion product.

00Hamilton had geometrical application as his main object; when he realized the quaternion system, he felt that his object was gained, and thenceforth confined himself to the development of his method.

00But his claims, however great they may be, can in no way conflict with those of Hamilton, whose mode of multiplying couples (in which the " inner " and " outer " multiplication are essentially involved) was produced in 1833, and whose quaternion system was completed and published before Grassmann had elaborated for press even the rudimentary portions of his own system, in which the veritable difficulty of the whole subject, the application to angles in space, had not even been attacked.

00Any quaternion may now be expressed in numerous simple forms. Thus we may regard it as the sum of a number and a line, a+a, or as the product, fly, or the quotient, be-', of two directed lines, &c., while, in many cases, we may represent it, so far as it is required, by a single letter such as q, r, &c.

00But every quaternion formula is a proposition in spherical (sometimes degrading to plane) trigonometry, and has the full advantage of the symmetry of the method.

00And one of Hamilton's earliest advances in the study of his system (an advance independently made, only a few months later, by Arthur Cayley) was the interpretation of the singular operator q()q1, where q is a quaternion.

00Here, then, is a case specially adapted to the isotropy of the quaternion system; and Hamilton easily saw that the expression i d x +j - + k dz could be, like ix+jy+ kz, effectively expressed by a single letter.

00No better testimony to the value of the quaternion method could be desired than the constant use made of its notation by mathematicians like Clifford (in his Kinematic) and by physicists like ClerkMaxwell (in his Electricity and Magnetism).

00Joly's projective geometrical applications starting from the interpretation of the quaternion as a point-symbol;' these applications may be said to require no addition to the quaternion algebra; (b) W.

00Shaw, in America, independently of Joly, has interpreted the quaternion as a point-symbol.

00This is the basis of a method parallel throughout to the quaternion method; in the specification of rotors and motors it is independent of the origin which for these purposes the quaternion method, pure and simple, requires.

00(Note that the z here occurring is only required to ensure harmony with tri-quaternions of which our present biquaternions, as also octonions, are particular cases.) The point whose position vector is Vrq i is on the axis and may be called the centre of the bi-quaternion; it is the centre of a sphere of radius Srq i with reference to which the point and plane are in the proper quaternion sense polar reciprocals, that is, the position vector of the point relative to the centre is Srg i.

00If Q= Ep+nq+wr and we put Q= (I +Zwt)(Ep-i-nq)X (1 Zwt) -1 we find that the quaternion t must be 2f (r) /f (q - p), where f(r)=rq - Kpr.

00When Su = o, (I + 2wu) () (1 + zwu) -1 is an operator which shifts (without further change) the tri - quaternion operand an amount given by u in direction and distance.

00Hamilton, of quaternion fame.

00So it remains an intriguing question, whether e.g. gage field theory is more naturally quantised using quaternion Hilbert space, H.

00Every 3#3 rotation matrix corresponds to a quaternion matrix that is unique except for its sign.

00Jason Wilkins (fenix at io dot com). Wrote the quaternion math routines and contributed the BeOS port.

00Block diagrams illustrate the conversion of an airplane motion parameters from body to earth axis for Euler angles and for the quaternion parameters.

00His work was a major factor in the development of vector analysis, in opposition to quaternion methods.

00A unit quaternion has m = 1. det( Q R)= m 4.

00

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