Quaternion

Transcript

In mathematics, the quitians are a number system that extends the complex numbers. They were first described by Irish mathematician William Row and Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quitterians is that multiplication of two quitterians is non-commutative. Hamilton defined a quitterian as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors. Quitterians find uses in both theoretical and applied mathematics in particular for calculations involving three-dimensional rotations such as in three-dimensional computer graphics, computer vision, and crystalallographic texture analysis.

In practical applications, they can be used alongside other methods such as oiler angles and rotation matrices or as an alternative to them depending on the application. In modern mathematical language, quitterians form a fourdimensional associative normed division algebra over the real numbers and therefore also a domain. In fact, the quitterians were the first non-commutative division algebra to be discovered. The algebra of quitterians is often denoted by H or in blackboard bold by. It can also be given by the Clifford algebra classifications C 0 2 CO3 0.

The algebra H holds a special place in analysis since according to the Froinius theorem, it is one of only two finite dimensional division rings containing the real numbers as a proper sub ring. The other being the complex numbers. These rings are also uklidian herwitz algebbras of which quitians are the largest associative algebra. The unit quitians can therefore be thought of as a choice of a group structure on the three sphere s3 that gives the group spin which is isomorphisu and also to the universal cover of so history. Quitian algebra was introduced by Hamilton in 1843.

Important precursors to this work included Oiler's four square identity and Olinda Rodriguez's parameterization of general rotations by four parameters. But neither of these writers treated the four parameter rotations as an algebra. Carl Friedrich Gaus had also discovered quturnians 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 Quernians finally came on Monday the 16th of 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 tow path of the Royal Canal with his wife, the concepts behind Quitians were taking shape in his mind. When the answer dawned on him, Hamilton could not resist the urge to carve the formula for the quitterians I2= J2= K2= I J K= -1 into the stone of Brewan Bridge as he paused on it.

On the following day, Hamilton wrote a letter to his friend and fellow mathematician John T. Graves describing the train of thought that led to his discovery. This letter was later published in the London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, Volume XXV pp489 to 95. In the letter, Hamilton states, "And here there dawned on me the notion that we must admit in some sense a fourth dimension of space for the purpose of calculating with triples. An electric circuit seemed to close and a spark flashed forth." Hamilton called a quadruple with these rules of multiplication a quatnian, and he devoted most of the remainder of his life to studying and teaching them.

Hamilton's treatment is more geometric than the modern approach which emphasizes quitians algebraic properties. He founded a school of quitterianists and he tried to popularize quitians in several books. The last and longest of his books elements of quitterians was 800 pages long. It was published shortly after his death. After Hamilton's death, his student Peter Tate continued promoting Quernians.

At this time, Quernians were a mandatory examination topic in Dublin. Topics in physics and geometry that would now be described using vectors such as kinematics in space and Maxwell's equations were described entirely in terms of quitterians. There was even a professional research association, the Quitterian Society, devoted to the study of Quernians and other hyper complex number systems. From the mid 1880s, Quernians began to be displaced by vector analysis which had been developed by Josiah Willard Gibbs, Oliver Heavyside, and Herman von Helmhaltz. Vector analysis described the same phenomena as Quitians.

So it borrowed some ideas and terminology liberally from the literature of quitterians. However, vector analysis was conceptually simpler and notationally cleaner and eventually quitians were relegated to a minor role in mathematics and physics. A side effect of this transition is that Hamilton's work is difficult to comprehend for many modern readers. Hamilton's original definitions are unfamiliar and his writing style was worthy and difficult to understand. However, quitians have had a revival since the late 20th century primarily due to their utility in describing spatial rotations.

The representations of rotations by quitians are more compact and quicker to compute than the representations by matrices. In addition, unlike oiler angles, they are not susceptible to jimble lock. For this reason, quitians are used in computer graphics, computer vision, robotics, control theory, signal processing, attitude control, physics, biioinformatics, molecular dynamics, computer simulations, and orbital mechanics. For example, it is common for the attitude control systems of spacecraft to be commanded in terms of quturnians. Quitterians have received another boost from number theory because of their relationships with the quadratic forms.

Since 1989, the department of mathematics of the National University of Ireland, Mayouth has organized a pilgrimage where scientists take a walk from Dunc Observatory to the Royal Canal Bridge. Hamilton's carving is no longer visible. Historical impact on physics PR Gerard's essay, The Quitian Group and Modern Physics, discusses some roles of Quitterians in physics. It shows how various physical co-variance groups. So the Lawrence group, the general relativity group, the Clifford algebrau and the conformal group can be readily related to the quatian group in modern algebra.

Gerard began by discussing group representations and by representing some space groups of crystalallography. He proceeded to kinematics of rigid body motion. Next, he used complex quitterians to represent the Lawrence group of special relativity, including the Thomas procession. He cited five authors, beginning with Ludvik Silberstein, who used a potential function of one quitterian variable to express Maxwell's equations in a single differential equation. Concerning general relativity, he expressed the run lens vector.

He mentioned the Clifford Biquitians as an instance of Clifford algebra. Finally invoking a reciprocal of Abiquitian, Gerard described conformal maps on spaceime. Among the 50 references, Gerard included Alexander Mcfarlan in his bulletin of the Quernian Society. In 1999, he showed how Einstein's equations of general relativity could be formulated within a Clifford algebra that is directly linked to quitians. A more personal view of quitterians was written by Yoim Lambbeck in 1995.

He wrote in his essay if Hamilton had prevailed. Quitians in physics. My own interest as a graduate student was raised by the inspiring book by Silverstein. He concluded by stating I firmly believe that quitians can supply a shortcut for pure mathematicians who wish to familiarize themselves with certain aspects of theoretical physics. Definition as a set the quturnians H are equal to R4 a fourdimensional vector space over the real numbers H has three operations addition scaler multiplication and quturnium multiplication.

The sum of two elements of H is defined to be their sum as elements of R4. Similarly, the product of an element of H by a real number is defined to be the same as the product by a scalar in R4. To define the product of two elements in H requires a choice of basis for R4. The elements of this basis are customarily denoted as 1 I, J and K. Every element of H can be uniquely written as a linear combination of these basis elements that is as A1 + C J + D K where A, B, C and D are real numbers.

The basis element one will be the identity element of H. Meaning that multiplication by one does nothing. And for this reason elements of H are usually written plus by plus CJ + DK suppressing the basis element one. Given this basis associative quanium multiplication is defined by first defining the products of basis elements and then defining all other products using the distributive law. Multiplication of basis elements.

The identities where if J and K are basis elements of H. Determine all the possible products of I, J and K. For example, write multiplying both sides of minus1 equals i j k by k gives all the other possible products can be determined by similar methods resulting in which can be expressed as a table whose rows represent the left factor of the product and whose columns represent the right factor as shown at the top of this article. Non-commutivity of multiplication unlike multiplication of real or complex numbers. Multiplication of quitterians is not commutative.

For example, I j= k while g= minus k. The non-commutivity of multiplication has some unexpected consequences. Among them that polomial equations over the quturnians can have more distinct solutions than the degree of the polomial. The equation z 2 + 1 = 0 for instance has infinitely many quitterian solutions z= y + cj + dk with b2 + c2 + d2 = 1. So that these solutions lie on the two-dimensional surface of a sphere centered on zero in the three-dimensional subspace of quturnians with zero real part.

This sphere intersects the complex plane at two points I and minus I. The fact that quitium multiplication is not commutative makes the quturnians an often cited example of a strictly skew field Hamilton product for two elements A1 + B1 I + C1 J + D1 K and A2 + B2 I + C2 J + D2 K. The product called the Hamilton product is determined by the product of the basis elements and the distributive law. The distributive law makes it possible to expand the product so that it is a sum of products of basis elements. This gives the following expression.

Now the basis elements can be multiplied using the rules given above to get the product of two rotation quturnians will be equivalent to the rotation a1 + b1 i + c1 j + d1 k followed by the rotation a2 + b 2 i + c2 j + d2 k. Ordered list form using the basis 1 i j k of h makes it possible to write h as a set of quadruples. Then the basis elements are and the formulas for addition and multiplication are scalar and vector parts a number of the form plus 0 i + 0 j + 0 k where a is a real number is called real and a number of the form 0 + c j + d k where b c and d are real numbers and at least one of b c or d is non zero is called pure imaginary. If a plus by + cj + dk is an equitian then a is called its scalar part and by + cj + dk is called its vector part. The scalar part of a quitterian is always real and the vector part is always pure imaginary.

Even though every quturnian can be viewed as a vector in a fourdimensional vector space, it is common to define a vector to mean a pure imaginary quturnian. With this convention, a vector is the same as an element of the vector space R3. It is important to note, however, that the vector part of a quturnian is in truth and axial vector or suda vector, not an ordinary or polar vector, as was formally proven by SL Alman in CH 12 of his 1986 book rotations, quturnians, and double groups. A polar vector can be represented in calculations by a pure imaginary quturnian with no loss of information. But the two should not be confused.

The axis of a binary rotation quitterian corresponds to the direction of the represented polar vector. In such a case, Hamilton called pure imaginary quturnians right quitterians and real numbers scalar quaterians. If equitian is divided up into a scalar part and a vector part i.e. then the formulas for addition and multiplication are where is the dotproduct and times is the crossroduct conjugation the norm and reciprocal conjugation of quturnians is analogous to conjugation of complex numbers and to transposition of elements of clifford algebbras. To define it let be a quitterian.

The conjugate of Q is the quitian. It is denoted by Q Q or conjugation is an involution meaning that it is its own inverse. So conjugating an element twice returns the original element. The conjugate of a product of two quturnians is the product of the conjugates in the reverse order. That is if P and Q are quturnians then equals QP not PQ.

Unlike the situation in the complex plane, the conjugation of a quitterian can be expressed entirely with multiplication and addition. Conjugation can be used to extract the scalar and vector parts of a quturnian. The scalar part of P is two and the vector part of P is two. The square root of the product of a quitterian with its conjugate is called its norm and is denoted Q. In formula, this is expressed as follows.

This is always a non-gative real number and it is the same as the ukidian norm on h considered as the vector space R4. Multiplying a quitterian by a real number scales its norm by the absolute value of the number. That is if alpha is real then this is a special case of the fact that the norm is multiplicative. Meaning that for any two quturnians P and Q, multiplicativeity is a consequence of the formula for the conjugate of a product. Alternatively, it follows from the identity and hence from the multiplicative property of determinants of square matrices.

This norm makes it possible to define the distance D between P and Q as the norm of their difference. This makes H into a metric space. Addition and multiplication are continuous in the metric topology. Indeed for any scalar positive I it holds the continuity follows for vanishing a. Similarly for the multiplication unit quaterian a unit quturnian is a quturnian of norm one.

Dividing a nonzero quturnian q by its norm produces a unit quturnian q called the verer of q. Every quitian has a polar decomposition q= q uq using conjugation and the norm makes it possible to define the reciprocal of a nonzero quturnian. The product of a quitterian with its reciprocal should equal one and the considerations above imply that the product of n is one. So the reciprocal of q is defined to be this makes it possible to divide two quturnians p and q in two different ways. That is the quotient can be either PQ minus1 or Q minus1 P.

The notation P Q is ambiguous because it does not specify whether Q divides on the left or the right. Algebraic properties. The set H of all quitterians is a vector space over the real numbers with dimension four. Multiplication of quitterians for example is associative in distributes over vector addition but it is not commutative. Therefore the quturnians h are a non-commutative associative algebra over the real numbers.

Even though h contains copies of the complex numbers it is not an associative algebra over the complex numbers. Because it is possible to divide quitians they form a division algebra. This is a structure similar to a field except for the non-commutivity of multiplication. Finite dimensional associative division algebbras over the real numbers are very rare. The froinius theorem states that there are exactly three r c and h.

The norm makes the quturnians into a normed algebra and norm division algebbras over the rails are also very rare. Herwitz's theorem says that there are only four R, C, H, and O. The quitians are also an example of a composition algebra and of a unit orbanic algebra because the product of any two basis vectors is plus or minus another basis vector. The set plus or minus1 plus or minus I plus or minus J plus or minus K forms a group under multiplication. This group is called the quattonian group and is denoted Q8.

The real group ring of Q8 is a ring R. Q8 which is also an eightdimensional vector space over R. It has one basis vector for each element of Q8. The quturnians are the quotient ring of R, Q8 by the ideal generated by the elements 1 + I + J + and K plus. Here the first term in each of the differences is one of the basis elements 1, i, j and k.

And the second term is one of basis elements minus one minus i, minus j and minus k, not the additive inverses of 1, i, j, and k. Quotians and the geometry of R3. Because the vector part of a quturnian is a vector in R3. The geometry of R3 is reflected in the algebraic structure of the quturnians. Many operations on vectors can be defined in terms of quitterians and this makes it possible to apply quturnian techniques wherever spatial vectors arise.

For instance, this is true in electronamics and 3D computer graphics. For the remainder of this section, I J and K will denote both imaginary basis vectors of H and a basis for R3. Notice that replacing I by minus I, J by minus J and K by minus K sends a vector to its additive inverse. So the additive inverse of a vector is the same as its conjugate as a quturnian. For this reason, conjugation is sometimes called the spatial inverse.

Choose two imaginary quitterians. P= B1 I + C1 J + D1 K and Q= B2 I + C2 J + D2 K. Their dot product is this is equal to the scalar parts of PQ, QP, PQ and QP. It also has the formulas the crossroduct of P and Q relative to the orientation determined by the ordered basis I, J and K. is this is equal to the vector part of the product pq as well as the vector part of minus qp.

It also has the formula for the commutator pq= pq minus qp of two imaginary quturnians. one obtains in general let P and Q be quitians and write where P S and Qs are the scalar parts and another the vector parts of P and Q then we have the formula this shows that the non-commutativity of quitterian multiplication comes from the multiplication of pure imaginary quturnians. It also shows that two quturnians commute if and only if the vector parts are collinear. For further elaboration on modeling three-dimensional vectors using quturnians, see quturnians and spatial rotation. A possible visualization was introduced by Andrew J.

Hansen.