Questions tagged [quaternions]

For questions about the quaternions: a noncommutative four dimensional division algebra over the real numbers. Also for questions about quaternion algebras.

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Isomorphism of real quaternion algebra with the Hamiltonian quaternion.

For a given quaternion algebra $A=\left(\frac{a,b}{\mathbb{R}}\right)$ where $a,b<0$, we know that it is isomorphic to $\mathbb{H}$. Can I check if the following is the appropriate isomorphism ...
yx12321's user avatar
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Effect of quaternion normalization when differentiating

I'm interested in optimizing across rotations which are represented as quaternions. You can either differentiate and then normalize after the update or you can include the normalizing term in the ...
maxical's user avatar
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Solving the quaternion sum of four squares/reduced norm

Let $K$ be an algebraic totally real number field, and let $O_K$ its ring of integers denote by $Q$ the quaternion algebra over $K$ I am looking for references on the resolution of the reduced norm ...
Don Freecs's user avatar
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$3\times3$ real matrix decomposition to SVD using two unit quaternions and scale vector

I've been trying to search about doing $3\times3$ real matrix SVD, but instead of decomposing it into matrices, represent the two rotations as unit quaternions with the singular values as separate ...
Venom's user avatar
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Calculating an Inverse Matrix of a Matrix with variables

I am trying to understand a part of an article regarding quaternions spline interpolation, where the situation folded into the equation: $$ (\vec{a}\cdot\hat{e})\hat{e}+\frac{\sin\Delta\theta}{\Delta\...
BlueRevel 's user avatar
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Quaternion Integration resources

Recently, in a course of projective geometry, I was introduced to the topic of Quaternions. Since I am just out of Complex Analysis, it seemed natural to wonder how integration would be handled. In ...
Clueless Student's user avatar
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Rotating a quaternion so that one unit vector moves to a specified unit vector via the smallest rotation [closed]

I am given a quaternion $q$ and a unit vector $e_1$. From the quaternion, I am building its rotation matrix $Q \in \mathbb{R}^{3 \times 3}$ and taking its column vectors $Q_1, Q_2$ and $Q_3$. Now, I ...
Daniel Benedí's user avatar
2 votes
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Show that a linear fractional transform is a rotation

I have a linear fractional transformation given by: $$F_K (z) = \frac{\cos{\theta}z + i\sin{\theta}}{i\sin{\theta}z + \cos{\theta}}$$ And I am supposed to find the fixed points and then verify that ...
Alex Lott's user avatar
7 votes
1 answer
171 views

Mysterious Coordinates on $S^4$ involving Quaternions

Let $U$ and $U'$ be $S^4 - x_N$ and $S^4 - x_S$, respectively, where $x_N$ is the North pole and $x_S$ is the South pole. The usual stereographic projection maps $U$ into $R^4$ and $U'$ into $R^4$. If ...
User175a23's user avatar
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Real usage of “pure” quaternions in stereometry?

There are two major categories of the "quaternions". It is well-known that a (nonzero) versor represents a three-dimensional rotation operator. A versor is a unit quaternion or a normalized ...
user688486's user avatar
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conditions for quaternions to be valid

I have begun reading about quaternions after a long time. However I should answer this soon to other members of my work group. My understanding of quaternions is composed of four values w,x,y and z ...
KansaiRobot's user avatar
2 votes
5 answers
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What is the relationship between the Laplace equation and the Wave equation? [closed]

What is the relationship between the Laplace equation: $$ (\delta^2_x + \delta^2_y)\phi = 0 $$ and the Wave equation: $$ (\delta^2_x - \delta^2_y)\phi = 0 $$ What is the relationship between the ...
Edgar Mueller's user avatar
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2 answers
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What is the meaning of this notation $\mathbb C_2$ in Jacobson's Basic Algebra I?

I was looking at Chapter 2 (Rings) in Basic Algebra I by Jacobson (1974). (Section 2.1 is Definition and Elementary Properties, 2.2 is Types of Rings, 2.3 is Matrix Rings (the ring $M_n(R)$ of $n\...
J. W. Tanner's user avatar
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Inverse of a 2x2 quaternion matrix: whence this formula?

From here, the inverse of a $2\times2$ quaternion matrix is given by $$ \begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1} =\frac{1}{\mathrm{Nrd}} \begin{bmatrix} |d|^2\overline{a}-...
coiso's user avatar
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Is this statement about quaternion function true?

Let $\mathbb{H}=\{q=t+xi+yj+zk:t,x,y,z\in\mathbb{R}\}$ be the quaternions and $f:\mathbb{H}\to\mathbb{H}$ be a function satisfying $\overline{\partial_c}f=0,$ where $$\overline{\partial_c}=\frac{1}{2}\...
Summer Daiy Stay summer night's user avatar
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2 answers
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Finding a point on $SO(3)$ that is equivalent (up to diffeomorphism) to a point on $\mathbb{RP}^3$

We know that topologically speaking, $$\text{SO}(3) \simeq \frac{D^3}{\sim_A}\simeq \frac{\mathbb{S}^3}{\sim_A}\simeq \mathbb{RP}^3 $$ where $\sim_A$ is the equivalence relation identifying antipodal ...
Luk'yan Vilshansky's user avatar
2 votes
1 answer
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How to convert from 2D points to 3D points on a plane [closed]

I have some 3D coplanar points. The plane is defined by normal vector and constant. I need to work with the points in 2D and then convert them back to 3D. In order to convert the points to 2D I made a ...
capr's user avatar
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Triple Cross Product Identity for imaginary quaternions $\mathbb{H}_0$

Considering $x,y,z \in \mathbb{H}_0$, $x,y,z=\alpha$i $+ \beta $j $+ \gamma $k, prove the Triple Cross Product Identity: \begin{equation} (x \times y) \times z = y(x \bullet z) - x(y \bullet z) \end{...
Alex A.G.'s user avatar
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converting pose (which is a quaternion & a vector) from a coordinate system to another

The question is about a world and a camera that is defined in this world. I want to transform the pose (which is a rotation and a translation) of the camera given in the world coordinate system (...
Hanaa's user avatar
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Geometric explanation of Fueter-Sce-Qian theorem and similar situations

In Clifford analysis there is a fundamental theorem due to Fueter and extended by Sce and Qian that says (in modern terminology) that the given a slice regular function $f:\mathbb{R}^{m+1}\to\mathbb{R}...
Giulio Binosi's user avatar
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1 answer
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representation for elements in $\pi_3(S^3)$ [closed]

I want to know if the map between $\mathbb{Z}$ and $\pi_3(S^3) $ given by $a \mapsto (x \mapsto x^a)$, where $S^3$ is seen as a subset of $\mathbb{H}$ with its multiplication, gives an isomorphism ...
Levi's user avatar
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What is this field in $\mathbb{R}^4$ that contains both the real and complex numbers called?

Note: this question is wrong – this is not a field, though it is not obvious why it wouldn't be. So, I (first year undergraduate mathematics student) was looking around the internet and found an ...
jkan5855's user avatar
2 votes
1 answer
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Fixed points of automorphism over quaternions

I know that for any $y\in\mathbb{H}\setminus\{0\}$, the map $\rho_y:\mathbb{H}\to\mathbb{H}$, $\rho_y(x):=y\overline{x}y^{-1}$ is an automorphism. I would like to know, for fixed $y$, which are the ...
Giulio Binosi's user avatar
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How to apply Yaw and Pitch Quaternions to another Quaternion representing an objects rotation

Let’s say I have a rectangle object. I have access to the objects rotation. I have have two axisAngle quaternions for yaw and pitch. How can i apply the rotations independently of each other to the ...
mmaghei's user avatar
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How do I get a Unit Vector Representing the Facing of a Quaternion?

I have a point that has a rotation, the quaternion $(.5, .5i, .5j, -.5k)$. If I place this at the origin of a unit sphere, how can I project a vector from the origin to the point on the unit sphere ...
UpTide's user avatar
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Basis for quaternion analytic functions

If I have a 3D quaternion function: $$f(x,y,z)=f_0(x,y,z)+f_1(x,y,z)i+f_2(x,y,z)j+f_3(x,y,z)k$$ Where $f_1,f_2,f_3:\mathbb{R}^3\rightarrow\mathbb{R}$, and $f(x,y,z)$ is analytic (or more formally left ...
QPhysl's user avatar
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2 answers
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What is the math required to do 4d rotations?

I recently wondered how I could calculate 4d rotations. I know that matrices and quaternions could be used to do so, but I haven't really found anything clear and specific about what I want. Basically,...
Pierre Carlier's user avatar
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Quaternion derivation and integration for attitude representation in common simulators

I have been checking the derivation and integration of quaternions to implement in my simple drone simulator. The problem is that I'm not sure if I can use a simplification for my case or not. The ...
Aurelie Navir's user avatar
1 vote
1 answer
83 views

Hyperkähler structure of Quaternion

I think there was an issue with the question I asked earlier. I want to prove that the quaternion is a hyperkahler manifold. I know that there is a natural metric $\rho$ on that given by $\rho(a,a)=a·...
ymm's user avatar
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Proving a remark of Gauss on quaternions and spherical triangles in a more transparent way.

In the fragment "Rotations of Space" (1819) in which Gauss outlined the general properties of a quaternions algebra, Gauss stated the following: Given three consecutive scales with the ...
user2554's user avatar
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Geometric proof that there's only way to make a power of four out of four odd squares

There's a straightforward elementary number theory proof that if four odd squares sum to a power of four, then the squares are $1$ and the power is $4$. Is there a geometric proof? I am ultimately ...
Greg Nisbet's user avatar
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Rate of change of norm of a quaternion

I am new to quaternions and currently trying to learn it. Here is my doubt. Let $\bf{q}$ be a quaternion whose norm is defined as $\|\bf{q}\| = \sqrt{q\otimes q^*}$. As you see, this norm is a scalar ...
user146290's user avatar
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Would some of the solutions to this function be considered hypercomplex numbers?

Consider the following function; $$f(x,y) = \sqrt{x} + \sqrt{y}$$ If this function were to be plotted onto a 3-dimensional co-ordinate space, then the x and y axes would be orthogonal to each other. ...
Craig Sanders's user avatar
3 votes
1 answer
41 views

Calculating the class number of a maximal order from a Quaternion Algebra

I am currently reading Quaternion Algebras from John Voight and saw how he calculated the class number of a maximal order at example 17.6.3. His quaternion algebra is $B' = \left( \frac{-1,-23}{\...
Enzor's user avatar
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Finding all ideals of a prescribed norm in a quaternion order

This question comes from Examples 17.6.3 and 41.1.2 in John Voight's Quaternion Algebras. The two exercises both work with the definite quaternion algebra $B := \left(\frac{-1,-23}{\mathbb{Q}}\right)$,...
stillconfused's user avatar
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1 answer
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An example of local and global containments in quaternion algebras

Let $p$ be a prime. Let $B_{p,\infty}$ be a (unique) quaternion algebra ramified at exactly at $p$ and $\infty$ with a standard basis $1,i,j, k=ij=-ji$. Let $K=\mathbb{Q}(i) \subseteq B_{p,\infty}$ be ...
Jason Dil's user avatar
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2 votes
0 answers
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A formula for $S$-units in quaternion algebras

Let $D=\left(\frac{a,b}{\mathbb{Q}}\right)$ be a quaternion algebra over $\mathbb{Q}$. Suppose $D$ is ramified at a finite set $S$ of places and $\infty\in S$. It is known that the $S$-units (the unit ...
Jun's user avatar
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2 answers
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Exponential Quaternions is Crazy [closed]

A quaternion $q \in \mathbb{H}$ can be write like $$q = a_0 + a_1 i + a_2j + a_3 k; ~ a_\iota \in \mathbb{R}$$ , where: $$i^2 = j^2 = k^2 = ijk = -1$$ $$ij = -ji = k$$ $$jk = -kj = i$$ $$ki= -ik = j$$ ...
Gabriel Fanini's user avatar
3 votes
1 answer
91 views

What is the name of the quaternion-like group generated by rotations and multiplications?

I know that the algebra of quaternions is isomorphic to the division algebra of 2×2 complex matrices which are real multiples of the elements of SU(2). I also know that SU(2) is a double cover of SO(3)...
mma's user avatar
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0 answers
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Conformal Maps of Quaternions

When we use the wolframalpha to plot a complex function, like $$f \left( z \right) = e^z, \text{ } z \in \mathbb{C}$$ , the program show us the conformal map, that take vertical and horizontal lines ...
Gabriel Fanini's user avatar
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1 answer
42 views

Sequence and Limit Definition at Quaternions

A sequence at $\mathbb{C}$ is a map $z_n: \mathbb{N} \to \mathbb{C}$, where $z_n$ converges to a $z_0 \in \mathbb{C}$ if: $$\forall \epsilon > 0, \text{ } \exists N \in \mathbb{N}; \text{ } \forall ...
Gabriel Fanini's user avatar
2 votes
1 answer
45 views

Quaternions Triangular Inequality

Let $z_1, \text{ } z_2 \in \mathbb{C}$. We now that: $$\left| z_1 + z_2 \right| = \left| z_1 \right| + \left| z_2 \right| \iff \frac{z_1}{z_2} > 0 \vee \left( z_1 = 0 \vee z_2 = 0 \right)$$ My ...
Gabriel Fanini's user avatar
1 vote
1 answer
57 views

Quaternion Rotation and Product

I am reading a dissertation where there's a part about quaternion rotation: Let $w = r_1 e^{\theta _1 i} + r_2 e^{\theta _2 i} j \in \mathbb{H}$ and $q = e^{i \alpha}$ So, the book concludes that: $...
Gabriel Fanini's user avatar
3 votes
0 answers
46 views

Shortest paths in SO(3)

In $\mathbb{R}^3$, we have a sense of "shortest path" between a pair of points and can easily project nearby points in $\mathbb{R}^3$ onto this path. Specifically, given $a, b, c\in\mathbb{R}...
Tom's user avatar
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1 vote
1 answer
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Square Root of a Arbitrary Octonion

Let $q = a + bi + cj + dk = a + Q \in \mathbb{H}$ a quaternion. So, we have: $$\sqrt{q} = \sqrt{\frac{|q| + a}{2}} + \frac{Q}{|Q|} \sqrt{\frac{|q| - a}{2}}$$ I have a question: this formula works for ...
Gabriel Fanini's user avatar
2 votes
3 answers
93 views

Factor a 3D rotation as a rotation along an horizontal axis followed by a rotation along the vertical

As the title says, I am trying to find, for any rotation R, a unit vector $u = [u_x, u_y, 0]$, and two angles $\alpha$ and $\beta$, such that R is equal to a rotation of angle $\alpha$ around $u$ and ...
Maxime Lucas's user avatar
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0 answers
46 views

Fast expression to rotate vector by the same rotation that orients one vector to another

The title is a bit convoluted but I don't know a good way to express it. I am wondering whether this expression that I've worked out for my specific task has been discussed in the past since I have ...
Matrefeytontias's user avatar
1 vote
2 answers
91 views

Is there a standard notation for the unit quaternions?

Of course, I mean other than $SU(2)$. I think the standard notation for the entire algebra of quaternions is $\mathbb{H}$, so I would imagine something like $\mathbb{H}^*$ or $\mathbb{H}_1$, etc. Is ...
mma's user avatar
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What is the unitary subset of Lie algebra $su(2)$?

Lie algebra $su(2)$ consists of the $2\times 2$ skew-hermitian complex matrices with addition and multiplication by real numbers as vector space operations and commutator as Lie bracket. The i-...
mma's user avatar
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3 votes
0 answers
71 views

Closed form of product of sedenions

I'm a math student and I'm taking an algebra course. The professor introduced us to the field of quaternions ($\mathbb{Q}$), I became very curious about the topic and I saw that in addition to ...
Efesto's user avatar
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