1 Introduction. 2 The G 4 Algebra. 2.1 Definition of G 4 Algebra

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1 1 Introduction Traditionally translations and rotations have been analysed/computed with either matrices, quaternions, bi-quaternions, dual quaternions or double quaternions depending of ones preference or type of problem. Manipulations of these objects are usually done with 4 4 matrices which obey the relevant algebra. Work is still ongoing to fully understand just how these objects should be manipulated to obtain certain translation and rotation properties. The advantage with the geometric algebra G 4 is that it has the scalars, complex numbers and quaternions contained within it as subalgebras. The G 4 algebra contains all of these structures in one coherent mathematical structure. The actions of rotations and translations are also easily carried out within the G 4 and can be applied to an arbitrarily orientated set of axis. As well as dealing with rotations and translations, geometric algebra has the ability to deal with geometric primitives with ease, to compute intersections of spheres with other spheres, or two different planes. The G 4 algebra can be used to compute this complex geometry and turn it into simple algebra. The G 4 Algebra.1 Definition of G 4 Algebra Let V be a four dimensional inner product space with orthogonal basis elements {e i } 3 i=0, it is possible to define an algebra on the basis elements denoted the outer product, these are denoted a b where a and b are elements of the vector space V. This product is an oriented area of no particular shape) with area given by the area of the parallegram spanned by a and b. The orientation of the element a b is of a followed by b, so the element b a will have the same area as a b but of the opposite orientation, ans so this is written as a b = b a 1) One thing which is immediately concluded is that: a a = 0 The element a b is called a bivector. Using other linearly independent vectors it is possible to construct hypervolumes via the outer product, for example a b c is called a trivector. The highest hypervolume element which can be formed using the vector space V is four. The hyper volumes 1

2 have an orientation and hypervolume defined by the volume given by the hyper-parallepiped spanned by the vectors. It is possible to combine the outer product with the inner product to form another product on the inner product space V called the geometric product defined by: ab = a b + a b ) Note that is is possible to redefine the inner and outer product in terms of the geometric product notes that: ab = a b + a b, ba = b a + b a = a b a b and so: a b = 1 ab + ba) 3) a b = 1 ab ba) 4) To examine the effect of the geometric product on basis vectors e i where i {1,, 3} e i e i = e i = e i e i + e i e i = e i The definition is made so that the e i, 1 i 3 are orthonormal and so e i = e i = 1. Let 0 < ε 1, the square of this element is defined by: e 0 = 1 ε The way that e 0 multiplies the other basis vectors can be found out via the geometric product: e 0 e i = e 0 e i + e 0 e i = e 0 e i = e i e 0 = e i e 0 The algebra can be split into different grades, where the scalars are grade 0 elements, the vectors are grade 1, elements, the bivectors are grade elements, the trivectors are grade 3 elements and the 4-volume is the single grade 4 element. The basis elements for the higher grade elements can by obtained by taking consecutive outer products of the basis elements, the basis elements are: scalars 1 5) vectors e 0, e 1, e, e 3 6)

3 bivectors e 0 e 1, e 0 e, e 0 e 3, e 1 e, e 1 e 3, e e 3 7) trivectors e 0 e 1 e, e 0 e 1 e 3, e 0 e e 3, e 1 e e 3 8) quadvectors e 0 e 1 e e 3 9) The quadvector has a special name, it is called the pseudoscalar and is denoted by e 0 e 1 e e 3 = ω. Definition 1. The algebra G 4 is the vector space spanned by the basis elements in equations 5)-9) equipped with the following algebra: e 0 = 1 ε 10) e i = 1 1 i 3 11) e i e j = e j e i i j 1) Note that for basis elements e i e j = e i e j and the shorthand used is the following: e i e j = e ij, e i e j e k = e ijk 13) Definition. The reverse of a multivector is defined as the following: e i1 i n = e in i 1 14) The reverse is rather useful, there is a lemma associated to the reverse: Lemma 1. AB = B A Let x be a vector and S be a bivector, then SxS is a vector. To see this, examine its reverse: ) SxS = S Sx = SxS = SxS So SxS is equal to its own reverse and therefore is a vector. A typical vector x G 4 will have the form: x = W e 0 + Xe 1 + Y e + Ze 3 15) 3

4 There is a map f : G 4 R 3 defined as follows: X fw e 0 + Xe 1 + Y e + Ze 3 ) = W, Y W, Z ) W If W 0, then this will quite happily define a point in R 3. if W = 0, then x is represents a point at infinity. It can be seen that individual points in R 3 have multiple representations. Points with W = 1 will be called normalised.. Subalgebras of G 4 The G 4 algebra contains a number of different subalgebras as well. An examination of equation 5) shows that it contains the real numbers as a subalgebra. Consider the bivector e 1 e = e 1, squaring this bivector shows that: e 1 e ) = e 1 = e 1 e e 1 e = e 1e = 1 So we can make a subalgebra of the elements a + be 1 which will be the complex numbers. Likewise it is possible to define: I = e 1, J = e 31, K = e 3 A simple calculation shows that IJ = K and that I = J = K = 1, so the quaternions are a subalgebra of the G 4 algebra. 3 Operators on G Rotations Let a and b be scalars and let e ij with i, j 0 and define the even grade element R by: R = a + be ij 16) The first thing to note that RR is a scalar: RR = a + be ji )a + be ij ) = a + b + abe ij + e ji ) = a + b It is of use to normalise this by setting a + b = 1, so it is possible to set a = cosθ/) and b = sinθ/). As e ij = 1 it is possible to writeas in Euler s relation for complex numbers): cos θ + sin θ ) e θeij ij = exp 17) 4

5 The mapping x RxR is of interest, as noted before, this mapping is an endomorphism of vectors. To see what it does on vectors take as a concrete example R = cosθ/)+sinθ/)e 1 and take the vector e 1 and θ = π/. Note that as R is made up of elements from purely the basis elements e 1, e 31, e 3 it will commute with e 0 and therefore the rotation operation will have no effect on e 0. So Re 0 + e i )R = e 0 + Re i R. The action of R on e 1 is then: Re 1 R = cos π 4 + sin π ) 4 e 1 e 1 cos π 4 + sin π ) e 1 ) ) = = cos π 4 e 1 + sin π 4 e cos π 4 π sin 4 = cos π e 1 + sin π e = e cos π 4 + sin π 4 e 1 ) e 1 + sin π 4 cos π 4 e So we can see this this is a rotation from e 1 to e. In general it can be shown that any even grade element of the form R = cosθ/) + sinθ/)e ij will be a rotation by an angle θ in the ij-plane. The map x RxR preserves angles and lengths. Let a = RaR and b = RbR, then: a b = 1 a b + b a ) = 1 RaRRbR + RbRRaR) = 1 RabR + RbaR) = Ra br = a b Taking a = b shows that the operator R also preserves lengths. There is an inverse to the rotation operation, the inverse operator is RxR as: Ra R = RRaRR = a So the inverse operator to R is simply R. There are a number of different names for R, they are rotor, spinor or simply even-grade element. In terms of the exponential the rotation operation can be written: RxR = exp θe ij ) x exp θeij ) 18) 5

6 3. Translations Consider the even grade element: T = 1 + εe 0a 19) where a = αe 1 + βe + γe 3 is a general vector in R 3. As before T T will be a scalar: T T = 1 + εa 4 This poses the question, what does this mean? Going back to the definition of the G 4 algebra the square of e 0 was ε 1 was meant to represent infinity and the idea was that this should give rise to finite terms. So the answer is treated as an asymptotic series and the only part of the series of use in the O1) portion. The operator T acts in the same way as R did. To get an idea of what this operator does, take a form example of T = 1 + εe 0 / acting on the vector e 0 + e 1 and compute: T e 0 + e 1 )T = 1 + εe ) 0 e 0 + e 1 ) 1 + εe ) 0 = e 0 + e 1 + e + εe ) εe ) 0 = e 0 + e 1 + e + ε 4 e 1 e 0 ) Now extract the O1) part of the expression to obtain e 0 + e 1 + e and it can be seen that the operator T is a translation operator is applied in the appropriate way. In general for the operator defined in 19) 3.3 Combinations of Rotors T e 0 + x)t = e 0 + x + a + Oε) 0) Given the two operators R and T, is it possible to combine them. Let R 1 and R denote two general even grade elements, Apply R 1 and then apply R to obtain: R R 1 xr 1 R = R 1 R xr 1 R So the application of even grade elements one after the other forms a group as seen above. So it can be seen that the whole of rigid body mechanics can just be carried out with these operations in the unified framework. 6

7 3.4 Rigid Body Motion This section describes the typical physicists understanding of rigid body motion and how it is governed by the angular velocity as the primary piece of information for which the rotor is then derived from. Let {f i } 3 i=1 be a set of orthonormal basis vectors for R 3 and let {e i } 3 i=1 denote the standard basis for R 3. The usual way of connecting f i and e i is via: f i = Rt)e i Rt) 1) Then if ω denotes the angular velocity of a particular motion then the relationship between the who basis is given by: f i = ω f i ) where the vector cross product is used. In terms of geometric products this is: f i = Iω f i where I = e 13. Note that: So f i can be written as: b Ia) = 1 bia Iab) = 1 Iba ab) = Ib a = Ia b Ia) b = 1 Iab bia) = 1 Iab ba) = Ia b = a b f i = f i Iω) = Iω) f i The notation used for Iω will be Ω. Differentiate 1) to obtain the equation: f = Ṙe ir + Re i Ṙ 3) From equation 1), we can multiply on the right by R and on the left by R to obtain: f i R = Re i, Rf i = e i R 4) 7

8 Inserting 4) into 3) yields: f i = ṘRf i + f i RṘ 5) Note that we also have the equation RR = 1, so we can differentiate this to obtain: ṘR = RṘ 6) Inserting equation 6) into equation 5) shows that: f i = f i RṘ RṘf i = f i RṘ) 7) Equation the two expressions for f 1, shows that RṘ = Ω, multiplying on the left by R gives the rotor equation Note: Ṙ = 1 RΩ 8) RṘ) = ṘR = RṘ So RṘ is equal to the negative of itself which shows that and hence Ω) it is a pure bivector. Equation 8) encodes all the kinematics of the motion. 4 Bézier Curves and G 4 It was shown earlier that given two different even grade elements, R 1, R, it is possible to obtain another even grade element from multiplication, R 3 = R 1 R. It is also possible to simple add even grade elements together to obtain another even grade element. One has to be careful to avoid pathological cases, for example adding R and R together will not give another even grade element because 0x0 = 0. Let S 1 and S define two unit even grade elements, that is S 1 S 1 = S S = 1. These are maps into R 3 which define two different poses at two different positions. It is possible to apply as linear interpolation between the poses: St) = 1 t)s 1 + S t 0 t 1 9) This will have S0) = S 1 and S1) = S. Note that: Let p be a vector, then: S = SS 1 S 1 = [1 t) + ts S 1 ]S 1 = P S 1 SpS = P S 1 pp S 1 = S 1 P pp S 1 = S 1 P pp )S 1 8

9 So the path of p as t varies is the transform of the action of S 1 of the path given by P pp. The transform preserves geometric properties it is convenient to take S 1 = 1 and take: S = 1 t + tp = 1 t + trt 30) where R denotes a rotation about a given axis and T denotes a translation along the axis of rotation, it is assumed that RR = T T = 1. Now let a be a point on that axis, it is obviously invariant under rotation: To see how S acts on a: RaR = a ar = Ra SpS = 1 t + trt )a1 t + trt ) = 1 t)a + T T Ra)1 t + trt ) = 1 t) a + t1 t) [art + T Ra] +t T at }{{} b The operation T represents a translation along the axis, and therefore the point a will remain on the axis. It is clear that b must be a vector, and so b = b which implies art + T Ra = T Ra + art. The result of the previous calculation contained three terms, the first and last are clearly on the axis, so it remains to show that b is on the axis. RbR = RaRT + T Ra)R = RaRT R + RT RaR) = at R + T Ra = b = b So b remains on the axis and the whole transform remain on the axis. 5 Conclusions The G 4 algebra has a representation of R 3 via a projection of the original vector space. There is a common form of operations on vectors throughout the geometric algebra, namely x SxS for translations and rotations which applies to the whole of rigid body transforms. All of the mathematical structure set up can be used to define poses for the instruments and combinations of these poses can be written down in a general form which extends in a natural way to free forms curves and potentially surfaces also. 9