Happenings – 2012 Feb 11

(This post is also in the “quaternions” category. If that’s what you’re looking for, just get the rest of the post and page down to the section heading.)

Let me begin by saying that the 2nd group theory post was not quite as popular as the 1st one… the blog “only” got 436 hits Monday, and the 2nd group theory post only got 102 hits itself. On Tuesday, as it had the week before, the blog as a whole got more than 300 hits.

I assure you I’m not disappointed in these kinds of numbers.

I hope to put out another post on group theory – about symmetric groups, permutations and cycles, and the quaternion group. On the one hand, the material is not complicated….

On the other hand, that post is at stage II. I’m not sure I’ve ever talked before about a specific post being at stage II. That means that I haven’t even done the mathematics for it! All I have is an outline in my head. We’ll see what happens.

As for the week gone by….
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Rotations: from one Euler angle sequence to another

Let’s suppose we are given an Euler angle sequence ZYX with angles

as = {0.2, 0.25, 0.3}

We have been asked to convert that to ZYZ.

(You probably want to have read the previous quaternion posts.) I will remind us that the notation ZYX means the rotation

Rx Ry Rz,

with the Z-axis rotation first; and I write the angles in the same order, ZYX,, so the given rotation is

m0 = Rx(.3) Ry(.25) Rz(.2).

We do not need to compute it, but we might as well; if for no better reason than that we will be able to check our answer. The rotation matrix for the given Euler angle sequence works out to:

m0 = \left(\begin{array}{ccc} 0.949599 & 0.192493 & -0.247404 \\ -0.118141 & 0.950819 & 0.286333 \\ 0.290353 & -0.242673 & 0.925637\end{array}\right)
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Rotations, especially Euler angle sequences

In this post, I simply want to play with the Euler angle representations of some rotations. This is not a thorough test suite, merely a handful of different cases:

  1. for a random rotation…
  1. ZYX
  2. ZXZ
  • ZXZ for…
    1. x-axis rotation
    2. y-axis rotation
    3. z-axis rotation
  • ZYX for…
    1. x-axis rotation
    2. y-axis rotation
    3. z-axis rotation

    A random orthogonal matrix

    In the previous post, we worked an example starting from a given Euler angle sequence of the form ZYX. This time, let’s begin with a random matrix. By issuing the command
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    Quaternions and Rotations 2: Euler angles etc.

    Let me create a simple example. You may want to have read Quaternions and Rotations 1, and the introductory post as well.

    I will specify an Euler angle sequence, “ZYX”, with angles:

    The notation “ZYX” means that we first rotate about the z-axis, then about y, then about x; and I choose to let the angles be in the same order. This is typically used in aerospace applications. The other Euler angle sequence common in my experience is “ZXZ” for orbits. Altogether, I count 12 kinds: three choices for the first axis, two for the second, and two for the third. There is no point in choosing consecutive axes the same, because consecutive rotations about the same axis are equivalent to one rotation, through the sum of the angles, about that axis.

    The obvious first step is to construct the combined rotation matrix for that Euler angle sequence, but I’m going to go in a different order. I’m going to test all 8 transformations among rotation matrices, angle/axis, quaternion, and Euler angle sequences.

    What the heck? Pictures are good.

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    Quaternions and Rotations 1

    What I’m heading for is to compute the following eight transformations (shown by arrows) among representations of a 3D rotation.

    Let me begin by talking about rotations generally (see rotations 1 for more detail).
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    Quaternions: introduction

    Introduction

    Let me start by listing three references, none of which has everything I would want.

    1. Kuipers’ “Quaternions and Rotation Sequences” is on my bibliographies page.
    2. Kantor and Solodovnikov’s “Hypercomplex Numbers: An Elementary Introduction to Algebras” (Springer 1989, 0 387 96980 2) puts quaternions in the context of number systems.
    3. Pertti Lounesto’s “Clifford Algebras and Spinors” (Cambridge 1997, 0 521 59916 4) had one result I wanted, namely a complex matrix representation. I wouldn’t recommend this book for quaternions, but I find it indispensible for clifford algebras.

    Quaternions were invented by William Rowan Hamilton on October 16, 1843. He may have considered them the greatest achievement of his rather stellar mathematical career. He had been looking for a 3D analog of the complex numbers; we know today that the properties he was hoping to find can only hold in dimensions 1,2,4,8. (They give us the real numbers, the complex numbers, the quaternions, and the octonions.)

    The simplest way to define a quaternion is to write

    q = a + b I + c J + d K,

    where a,b,c,d are real numbers, and the symbols I, J, K have the following properties:

    I^2 = J^2 = K^2 = I\ J\ K = -1\ .

    That should remind you of the complex numbers: z = a + b I, with I^2 = -1\ . We have two additional imaginary units, and there is one additional property relating all three of them.
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