Lecture 8 Quaternions

Home , Quaternion, Rotation (mathematics), Square (algebra), Unit sphere

Overview, motivation

Background

Deﬁnition and Lecture 8 properties Rotation using unit Quaternions quaternions Intuition

Using quaternions to represent Matthew T. Mason rotations Why we love quaternions.

Mechanics of Manipulation Lecture 8 Today’s outline Quaternions

Overview, Overview, motivation motivation Background

Deﬁnition and Background properties Rotation using unit quaternions

Deﬁnition and properties Intuition

Using quaternions to represent Rotation using unit quaternions rotations Why we love quaternions. Intuition

Using quaternions to represent rotations

Why we love quaternions. Lecture 8 Motivation Quaternions

Overview, motivation

Background

Deﬁnition and Motivation properties Rotation using unit quaternions I Quaternions have nice geometrical interpretation. Intuition

I Quaternions have advantages in representing Using quaternions to represent rotation. rotations I Quaternions are cool. Even if you don’t want to use Why we love quaternions. them, you might need to defend yourself from quaternion fanatics. 3 Lecture 8 Why can’t we invert vectors in R ? Quaternions

Overview, motivation

Background 1 Deﬁnition and I We can invert reals. x × x = 1. properties 2 Rotation using unit I We can invert elements of R using complex quaternions ∗ 2 ∗ numbers. z × z /|z| = 1, where is complex Intuition

conjugate. Using quaternions 3 to represent I Can we invert v ∈ R ? rotations

I No. Why we love quaternions. 4 I How about v ∈ R ? 4 I Yes! Hamilton’s quaternions are to R what complex numbers are to R2. Lecture 8 Complex numbers versus quaternions Quaternions Deﬁnition (Complex numbers) Overview, motivation

I Define basis elements 1 and i; Background Definition and I Define complex numbers as a vector space over properties reals: elements have the form x + iy; Rotation using unit quaternions 2 I One more axiom required: i = −1. Intuition

Using quaternions to represent Deﬁnition (Quaternions) rotations Why we love quaternions. I Deﬁne basis elements 1, i, j, k;

I Deﬁne quaternions as a vector space over reals: elements have the form q0 + q1i + q2j + q3k; I One more axiom:

i2 = j2 = k 2 = ijk = −1 Lecture 8 Basis element multiplication Quaternions

Overview, From that one axiom, we can derive other products: motivation Background ijk = −1 Deﬁnition and properties

i(ijk) = i(−1) Rotation using unit quaternions − = − jk i Intuition jk = i Using quaternions to represent rotations

Writing them all down: Why we love quaternions. ij = k, ji = −k jk = i, kj = −i ki = j, ik = −j

Quaternion products of i, j, k behave like cross product. Lecture 8 Quaternion notation Quaternions

Overview, motivation

Background

We can write a quaternion several ways: Deﬁnition and properties q = q0 + q1i + q2j + q3k Rotation using unit quaternions q = (q0, q1, q2, q3) Intuition q = q + q Using quaternions 0 to represent rotations

Why we love Deﬁnition (Scalar part; vector part) quaternions.

For quaternion q0 + q, q0 is the scalar part and q is the vector part Lecture 8 Quaternion product Quaternions

Overview, We can write a quaternion product several ways: motivation Background

Deﬁnition and pq = (p0 + p1i + p2j + p3k)(q0 + q1i + q2j + q3k) properties = (p0q0 − p1q1 − p2q2 − p3q3) + ... i + ... j + ... k Rotation using unit quaternions pq = (p0 + p)(q0 + q) Intuition = (p q + p q + q p + pq) Using quaternions 0 0 0 0 to represent rotations The last product includes many different kinds of product: Why we love quaternions. product of two reals, scalar product of vectors. But what is pq? Cross product? Dot product? Both! Cross product minus dot product!

pq = (p0q0 − p · q + p0q + q0p + p × q) Lecture 8 Conjugate, length Quaternions Deﬁnition (Conjugate) Overview, motivation

∗ Background q = q0 − q1i − q2j − q3k Deﬁnition and properties

Rotation using unit Note that quaternions Intuition ∗ qq = (q0 + q)(q0 − q) Using quaternions to represent 2 rotations = q0 + q0q − q0q − qq 2 Why we love = q0 + q · q − q × q quaternions. 2 2 2 2 = q0 + q1 + q2 + q3

Deﬁnition (Length)

q p ∗ 2 2 2 2 |q| = qq = q0 + q1 + q2 + q3 Lecture 8 Quaternion inverse Quaternions

Overview, motivation Every quaternion except 0 has an inverse: Background q∗ Definition and −1 = properties q 2 |q| Rotation using unit quaternions Without commutativity, quaternions are a division ring, or Intuition Using quaternions a non-commutative field, or a skew field. to represent Just as complex numbers are an extension of the reals, rotations Why we love quaternions are an extension of the complex numbers quaternions. (and of the reals). If 1D numbers are the reals, and 2D numbers are the complex numbers, then 4D numbers are quaternions, and that’s all there is. (Frobenius) (Octonions are not associative.) Lecture 8 Rotation using unit quaternions Quaternions

Overview, motivation

Background

Deﬁnition and I Let q be a unit quaternion, i.e. |q| = 1. properties Rotation using unit I It can be expressed as quaternions θ θ Intuition q = cos + sin nˆ Using quaternions 2 2 to represent rotations

I Let x = 0 + x be a “pure vector”. Why we love 0 ∗ quaternions. I Let x = qxq . 0 I Then x is the pure vector rot(θ, nˆ)x!!! Lecture 8 Proof that unit quaternions work Quaternions

Overview, motivation

Background

Deﬁnition and properties ∗ Rotation using unit I Expand the product qxq ; quaternions

I Apply half angle formulas; Intuition Using quaternions I Simplify; to represent rotations

I Compare with Rodrigues’s formula. Why we love quaternions. Sadly, not all proofs confer insight. ∗ Lecture 8 Why θ/2? Why qxq instead of qx? Quaternions

Overview, In analogy with complex numbers, why not use motivation Background p = cos θ + nˆ sin θ Deﬁnition and properties 0 x = px Rotation using unit quaternions

Intuition To explore that idea, deﬁne a map Lp(q) = pq. Note that Using quaternions Lp(q) can be written: to represent rotations

    Why we love p0 −p1 −p2 −p3 q0 quaternions.  p1 p0 −p3 p2   q1  Lp(q) =      p2 p3 p0 −p1   q2  p3 −p2 p1 p0 q3

Note that the matrix above is orthonormal. Lp is a rotation of Euclidean 4 space! Lecture 8 Geometrical explanation Quaternions

Although Lp(q) rotates the 4D space of quaternions, it is Overview, not a rotation of the 3D subspace of pure vectors. Some motivation Background of the 3D subspace leaks into the fourth dimension. Deﬁnition and Consider an example using p = i. Is it a rotation about i properties Rotation using unit of π/2? quaternions Intuition

Using quaternions to represent Li qiq iqiLR q Ri qqi i i rotations i iiWhy we love no quaternions. 1- i plane 2 2 rotation

1 1 1

k k k j- k plane 2 2

j jj Lecture 8 What do we do with a representation? Quaternions

∗ Rotate a point: qxq . Overview, motivation Compose two rotations: Background Deﬁnition and q(pxp∗)q∗ = (qp)x(qp)∗ properties Rotation using unit quaternions

Convert to other representations: Intuition I From axis-angle to quaternion: Using quaternions to represent rotations θ θ q = cos + sin nˆ Why we love 2 2 quaternions.

I From quaternion to axis-angle:

−1 θ = 2 tan (|q|, q0) nˆ = q/|q|

assuming θ is nonzero. Lecture 8 From quaternion to rotation matrix Quaternions

Overview, motivation

Background

Deﬁnition and properties Just expand the product Rotation using unit quaternions qxq∗ = Intuition  2 2 2 2  Using quaternions q0 + q1 − q2 − q3 2(q1q2 − q0q3) 2(q1q3 + q0q2) to represent 2 2 2 2 rotations  2(q1q2 + q0q3) q0 − q1 + q2 − q3 2(q2q3 − q0q1)  x 2 2 2 2 Why we love 2(q1q3 − q0q2) 2(q2q3 + q0q1) q0 − q1 − q2 + q3 quaternions. Lecture 8 From rotation matrix to quaternion Quaternions

Overview, Given R = (rij ), solve expression on previous slide for motivation quaternion elements qi Background Deﬁnition and Linear combinations of diagonal elements seem to solve properties the problem: Rotation using unit quaternions

Intuition 2 1 q0 = (1 + r11 + r22 + r33) Using quaternions 4 to represent rotations 2 1 q1 = (1 + r11 − r22 − r33) Why we love 4 quaternions. 1 q2 = (1 − r + r − r ) 2 4 11 22 33 1 q2 = (1 − r − r + r ) 3 4 11 22 33 so take four square roots and you’re done? You have to ﬁgure the signs out. There is a better way ... Lecture 8 Look at the off-diagonal elements Quaternions

Overview, I motivation 1 Background q0q1 = (r32 − r23) Deﬁnition and 4 properties 1 q q = (r − r ) Rotation using unit 0 2 4 13 31 quaternions 1 Intuition q0q3 = (r21 − r12) Using quaternions 4 to represent 1 rotations q1q2 = (r12 + r21) Why we love 4 quaternions. 1 q q = (r + r ) 1 3 4 13 31 1 q q = (r + r ) 2 3 4 23 32

I Given any one qi , could solve the above for the other three. Lecture 8 The procedure Quaternions

Overview, motivation 1. Use first four equations to find the largest q2. Take its Background i Definition and square root, with either sign. properties Rotation using unit 2. Use the last six equations (well, three of them quaternions anyway) to solve for the other qi . Intuition Using quaternions to represent I That way, only have to worry about getting one sign rotations right. Why we love quaternions. I Actually q and −q represent the same rotation, so no worries about signs.

I Taking the largest square root avoids division by small numbers. Lecture 8 Properties of unit quaternions Quaternions

Overview, motivation

Background

Deﬁnition and properties

Rotation using unit 4 quaternions I Unit quaternions live on the unit sphere in R . Intuition

I Quaternions q and −q represent the same rotation. Using quaternions ∗ to represent I Inverse of rotation q is the conjugate q . rotations

Why we love I Null rotation, the identity, is the quaternion 1. quaternions. Lecture 8 Metric on SO(3) Quaternions

Overview, motivation

Background What is the right metric between two spatial rotations? Deﬁnition and properties I.e. between two points in SO(3)? Rotation using unit I provided an answer in Lecture 2! SO(3) is a quaternions Intuition conﬁguration space. Let R and R be two displacements 1 2 Using quaternions of the sphere S2. Let x by a point on the sphere. Let α be to represent rotations

the usual metric on the sphere—the length of the shortest Why we love great circle arc joining two points. Deﬁne quaternions. d(R1, R2) = maxx∈S2 α(R1(x), R2(x)). That is equivalent −1 to taking the rotation angle of R1 R2. In brief, the angle of the rotation that takes you from R1 to R2. Lecture 8 Metric on sphere of unit quaternions Quaternions

I Consider unit quaternion Overview, motivation θ θ q = cos + sin nˆ Background 2 2 Deﬁnition and properties Shortest path on the unit sphere joining ±q with 1 Rotation using unit quaternions

has length θ/2. Intuition I What is the shortest distance on the unit quaternion Using quaternions 3 to represent sphere S from ±p to ±q? rotations Why we love I Multiplication by a unit quaternion is a rotation of the quaternions. unit quaternion sphere. I.e. distance-preserving. So the distance from ±p to ±q is the same as the distance from ±pq∗ to 1. I.e. α/2, where α is the rotation angle required from p to q.

I So usual metric on the sphere, applied to unit quaternions, is the right metric for SO(3). (Factor of two is irrelevant.) Lecture 8 Topology of SO(3) Quaternions

Overview, motivation

Background

Deﬁnition and properties

Rotation using unit The right metric matters. Our representation is smooth quaternions and one-to-one, apart from the fact that antipodes Intuition represent the same rotation. That means the unit Using quaternions to represent quaternions, with antipodes identiﬁed, have the same rotations topology as SO(3). That’s how we know the topology of Why we love quaternions. SO(3) is Projective 3-space, P3. Lecture 8 Measure, probabilities, sampling Quaternions

I Likewise we can turn our attention to measure. Overview, motivation I Easiest way is to think in terms of probabilities. Background Suppose you are given some probability density Deﬁnition and function on SO(3). How can you decide whether it is properties Rotation using unit unbiased? quaternions I Pick some subset of SO(3), integrate the probability Intuition of that subset. Now rigidly rotate that subset. If the Using quaternions to represent probability is invariant with respect to rotation, the rotations PDF is unbiased. A measure with this property is the Why we love quaternions. Haar measure, and it is uniquely determined up to some scale factor.

I Look at a uniform density function on the unit quaternion sphere S3. It has the right property — it is invariant with respect to rotations. The quaternion representation gives you the right measure, and is an easy way to correctly sample SO(3).