For questions about the quaternions: a noncommutative four dimensional division algebra over the real numbers.
37
votes
5answers
784 views
Why are the solutions of polynomial equations so unconstrained over the quaternions?
An $n$th-degree polynomial has at most $n$ distinct zeroes in the complex numbers. But it may have an uncountable set of zeroes in the quaternions. For example, $x^2+1$ has two zeroes in $\mathbb ...
20
votes
2answers
780 views
Why are the only division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?
Why are the only (associative) division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?
Here a division algebra is an associative algebra where every ...
9
votes
2answers
402 views
How do quaternions represent rotations?
I wonder how $qvq^{-1}$ gives the rotated vector of $v$.
Is there any easy-to-understand proof for it?
I was on Wikipedia, but I could not understand the proof there
because of the conversions.
Why ...
5
votes
2answers
1k views
Euler angles and gimbal lock
Can someone show mathematically how gimbal lock happens when doing matrix rotation with Euler angles for yaw, pitch, roll? I'm having a hard time understanding what is going on even after reading ...
13
votes
7answers
1k views
How can one intuitively think about quaternions?
Quaternions came up while I was interning not too long ago and it seemed like no one really know how they worked. While eventually certain people were tracked down and were able to help with the ...
4
votes
1answer
5k views
How do you rotate a vector by a unit quaternion?
Given a 3-variable right-handed vector v that is a translation measured in local space and a unit quaternion representing an orientation from local to world space, how do you use the quaternion to ...
13
votes
9answers
1k views
Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$ [duplicate]
Possible Duplicate:
Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$
On Wikipedia, it says that:
Matrix ...
3
votes
1answer
64 views
Does anyone know any resources for Quaternions for truly understanding them?
I've been studying Quaternions for a week, on my own. I've learned various facts about them but I still don't understand them. My goal is to understand rotation quaternions specifically. I don't want ...
6
votes
3answers
174 views
4 dimensional numbers
I've tought using split complex and complex numbers toghether for building a 3 dimensional space (related to my previous question). I then found out using both together, we can have trouble on the ...
4
votes
2answers
215 views
Square roots of $-1$ in quaternion ring
In this Wikipedia page it is said that the square roots of -1 in the quaternion ring are the elements of the imaginary sphere. I don't understand why this is so. I don't understand the system that's ...
13
votes
1answer
355 views
What lies beyond the Sedenions
In the construction of types of numbers, we have the following sequence:
$$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S}$$
or:
$$2^0 \mathrm{-ions} \subset ...
5
votes
1answer
171 views
Uniform distributions on the space of rotations in 3D
I believe on moral grounds that the following three definitions are equivalent, and determine "the" uniform distribution on rotations in three dimensions.
The Haar measure on $SO(3)$.
The uniform ...
3
votes
1answer
174 views
The Join of Two Copies of $S^1$
So I know the fact that the join of $S^1$ and $S^1$ is homeomorphic to the 3-sphere, but I'm having trouble "seeing" this. I'd prefer something that appeals to geometric intuition, but more formal ...
4
votes
3answers
89 views
How do you construct the quaternion and the multiplication rules, like Hamilton did?
So, I understand complex number multiplication, and how it represents $2D$ rotations.
What I don't understand is, how you add two more imaginary numbers $j$ and $k$, and get $3D$ rotations. I believe ...
2
votes
1answer
187 views
Octonionic Hopf fibration and $\mathbb HP^3$
Real, complex and quaternionic versions of Hopf fibration ($S^0\to S^n\to\mathbb RP^n$, $S^1\to S^{2n+1}\to\mathbb CP^n$ and $S^3\to S^{4n+3}\to\mathbb HP^n$) give rise to spherical fibrations ...
3
votes
2answers
43 views
Combing rotation quaternions
If I combine 2 rotation quaternions by multiplying them, lets say one represents some rotation around x axis and other represents some rotation around some arbitrary axis.
The the order of rotation ...
3
votes
3answers
171 views
How to get a part of a quaternion? e.g. get half of the rotation of a quaternion?
if I have a quaternion which describes an arbitrary rotation, how can I get for example only the half rotation or something like 30% of this rotation?
Thanks in advance!
3
votes
2answers
193 views
Quaternions as roots
So, I StumbledUpon this really cool site and the last picture looked almost as if it had 3D structure. This reminded me of another website where I saw pictures of the order-8 Mandelbulb. I got to ...
2
votes
1answer
103 views
Unit elements in Hurwitz quaternions
Hurwitz quaternions are defined as:
$$H_u = \left\{a+bi+cj+dk\mid (a,b,c,d) \in\mathbb{Z}\mbox{ or }(a,b,c,d) \in\mathbb{Z}+\frac{1}{2}\right\}$$
(that is, all integer or half integer quaternions).
...
1
vote
1answer
180 views
Quaternions and Rotations
Two of the interesting achievements in Mathematics are Classification of platonic solids, and also classification of finite groups acting on the unit sphere in $\mathbb{R}^3$, and they are very nicely ...
0
votes
2answers
510 views
How would I create a rotation matrix that rotates X by a, Y by b, and Z by c?
How would I create a rotation matrix that rotates X by a, Y by b, and Z by c?
I need to formulas, unless you're using the ardor3d api's functions/methods.
Matrix is set up like this
...
