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Quaternions by Wasinee Siewrichol

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Quaternions by Wasinee Siewrichol Quaternions Wasinee Siewsrichol Historical Context Complex numbers was a very popular subject in the early eighteenth hundreds. People knew how to multiply two numbers, but they did not know how to multiply three numbers. This question puzzled William Rowan Hamilton for a very long time. Hamilton wrote to his son: "Every morning in the early part of the above-cited month [Oct. 1843] on my coming down to breakfast, your brother William Edwin and yourself used to ask me, 'Well, Papa, can you multiply triplets?' Whereto I was always obliged to reply, with a sad shake of the head, 'No, I can only add and subtract them.'" Hamilton eventually found the solution, but in the fourth dimension. The concept of quaternions was first invented by the Irish mathematician William Rowan Hamilton on Monday, October 16th 1843 in Dublin, Ireland. Hamilton was on his way to the Royal Irish Academy with his wife and as he was passing over the Royal Canal on the Brougham Bridge. He made a dramatic realization that he immediately carved into the stone on the bridge. It is hard to tell, but on the bottom it writes i2 = j2 = k2 = ijk = −1 The first thing that Hamilton did was set the fourth dimension equal to zero. Then he spent the rest of his life trying to find a practical use for quaternions, but he did not find anything. By the early nineteenth century, Professor Josiah Willard Gibbs of Yale came up with the vector dot products and the cross product. Vectors are huge in physics for finding the velocity, acceleration, and force of an object. Then Albert Einstein found a use for the fourth dimension by working on problems in space and time, special relativity. Other uses for quaternions are for coordinates for rotations and orientations. SIGGRAPH is an international community of esearchers, artists, developers, filmmakers, scientists, and business professionals who share an interest in computer graphics and interactive techniques. In 1985, SIGGRAPH used quaternion curve methods to facilitate roation animation in computers. Quaternions were then used in physically based modeling, constraint systems, and user interfaces. There are four main types of number systems and they are the real, complex, quaternion, and cayley number systems where the complex has no order, the quaternion is not commutative, and cayley is not associative. Background Knowledge In order to understand quaternions, one must understand complex numbers. The Complex Number System introduces imaginary numbers. Imaginary numbers were invented for equations with no solutions. x2 + 1 = 0 x2 = −1 There is a problem here. If we square any number, positive or negative, the answer will always be positive. p p x2 = −1 = i p x = −1 = i Similarly, we cannot take the square root of a negative number. That is why imaginary numbers were invented. They are in the form of i2 = −1 The set of imaginary numbers is represented by the symbol I The set of complex numbers is represented by the symbol C. The complex numbers set includes real and imaginary numbers z = a + bi where a and b are real numbers. In other words, real numbers are complex numbers where b = 0 z = a + (0)i z = a Alternatively, imaginary numbers are complex numbers where a = 0 z = 0 + bi z = bi Adding and Subtracting Complex Numbers The imaginary number is treated like a variable by adding or subtracting all of the real numbers and imaginary numbers separately. Addition: (a1 + b1i) + (a2 + b2i) = (a1 + a2) + (b1 + b2)i where a1; b1; a2; b2 are real numbers Subtraction: (a1 + b1i) − (a2 + b2i) = (a1 − a2) + (b1 − b2)i 2 Multiplying Complex Numbers c(a + bi) = ca + cbi where a, b, c are real numbers Product of Complex Numbers z1 = (a1 + b1i) z2 = (a2 + b2i) z1z2 = (a1 + b1i)(a2 + b2i) 2 = a1a2 + a1b2i + b1a2i + b1b2i since i2 = −1 therefore = a1a2 + a1b2i + b1a2i − b1b2 Squaring Complex Numbers z = a + bi z2 = (a + bi)(a + bi) = a2 + abi + abi + b2i2 = a2 + 2abi − b2 Complex Conjugate A conjugate number is where we change the sign in the middle of the two terms. z = (a + bi) the conjugate of z is z∗ = (a − bi) zz∗ = (a + bi)(a − bi) = a2 − abi + abi − b2i2 = a2 + b2 3 Quotient of Two Complex Numbers z1 = a1 + b1i z2 = a2 + b2i z1=z2 = (a1 + b1i)=(a2 + b2i) In order to solve this equation, we multiply the numerator and denominator by the conjugate of the denomi- nator. The numerator is the numbers on top of the division sign and the denominator is the numbers at the bottom of the division sign. [(a1 + b1i)=(a2 + b2i)][(a2 − b2i)=(a2 − b2i)] 2 2 2 2 (a1a2 − a1b2i + a2b1i − b1b2i )=(a2 − b2i ) 2 2 (a1a2 − a1b2i + a2b1i + b1b2)=(a2 + b2) Powers of i i0 = 1 i1 = i i2 = −1 i3 = i2i = (−1)(i) = −i i4 = i2i2 = 1 And then we will see a pattern denotated by 1; i; −1; −i; 1; ::: Alternatively, i0 = 1 i−1 = −i i−2 = −1 i−3 = i i−4 = 1 By taking the powers of imaginary numbers, a pattern also emerges. Complex Plane We will show the complex numbers in a 2D grid by putting the real parts on the horizontal axis and the imaginary parts on the vertical axis. For example, we will take a random point p p = 2 + i 4 and we multiply p by i to get a new point q q = pi q = (2 + i)i = 2i + i2 = −1 + 2i multiply q by i to get a new point r r = qi = (−1 + 2i)i = −i + 2i2 = −2 − i multiply r by i to get a new point s s = ri = (−2 − i)i = −2i − i2 = 1 − 2i multiply s by i to get a new point t t = si = (1 − 2i)i = i − 2i2 = 2 + i By plotting points p, q, r, s, t on the 2D complex plane, and connecting the dots, a circle will be made. Quaternions Quaternions can be defined in several different ways. The first way is by defining the general form. The general form of quaternions can be expressed as: q = s + xi + yj + zk in which s, x, y, z are real numbers and i, j, k are imaginary numbers. The concempt of i, j, k is very simialr to the cross poduct rules. ixj = k jxk = i kxi = j jxi = −k kxj = −i ixk = −j 5 where x is the multiplication sign. Commutative property states that numbers can be added or multiplied in any order. This shows that quater- nions multiplication is not communitative because ixj does not equal jxi In conclusion, quaternions is defined as two things. Quaternions have an x, y, and z component which represents the axis about which a rotation will occur. It also has a w component which represents the amount of rotation which will occur about this axis. With these four numbers, a matrix can be built. Quaternions is technically four numbers but three of them are imaginary. Quaternions as an Ordered Pair q = [s; v] where s is an element of the first dimension, the real dimesnion, and v is an element of the third dimension, the imaginary dimension. v can be split into individual parts in which v = xi + yj + zk which makes q = [s; xi + yj + zk] where s, x, y, z are elements of the first dimension Adding and Subtracting Quaternions qa = [sa; a] qb = [sb; b] qa + qb = [sa + sb; a + b] This method is very similar to the addition of vectors in calculus III where w = [a; b; c] v = [d; e; f] w + v = [a + d; b + e; c + f] The x components cannot be added or subtracted from the y or z components. Similarly, the y components cannot be added or subtracted from the x or z components. Quaternion Norm The square root of the product of a quaternion with its conjugate is called its norm and is denoted |q| (Hamilton called this quantity the tensor of q). 6 jqj = p(x2 + y2 + z2 + w2) The norm is always a postive number. The quaternion norm can also be refered to as the magnitude. Quaternion Proof Proposition: Prove the four square identity jfgj = jfjjgj where f is a quaternion and g is a different quaternion. Proof: Let f = a + bi + cj + dk g = x + yi + zj + wk) where a, b, c, d are real numbers and i, j, k are imaginary numbers. By defintion, the norm of f is p jfj = a2 + b2 + c2 + d2 p jgj = x2 + y2 + z2 + w2 So if you do the cross product of f and g, there will be four total group of numbers. (a + bi + cj + dk)(x + yi + zj + wk) = (n0) + i(n1) + j(n2) + k(n3) where n0 represents a group of real numbers and n1; n2; n3 represent groups of real and imaginary numbers. Part 1, For the first group of numbers, in order to multiply them to get real numbers, one must remember the quaternion identity which is i2 = j2 = k2 = −1 2 2 2 (n0) = (ax + byi + czj + dwk ) = (ax − by − cz − dw) Part 2, The second group of numbers, we will use something similar to the cross product to get i. We must remember the rules of cross products in which ixj = k jxk = i kxi = j jxi = −k kxj = −i ixk = −j 7 Therefore, i(n1) = (ayi + bxi + cjxwk + dkxzj) = (ayi + bxi + cwi − dwi) = i(ay + bx + cw − dz) where x is the cross product and x is different from x. Part 3, For the third group of numbers, we will try to use the cross product to get j.
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