Questions tagged [rotations]
This tag is for questions about *rotations*: a type of rigid motion in a space.
3,353 questions
2 votes
1 answer
31 views
Finding coordinates of a point on a parabola given a rotated triangle condition
I am working on a geometry problem involving a parabola and coordinate transformations. I have solved the preliminary parts, but I am looking for a more elegant or geometric solution for the final ...
-3 votes
1 answer
62 views
What is the involution that scales and rotates the right side to the left of $\frac23$ and vice-versa in the interval $\frac12,1$? [closed]
Consider the function $f$ on the dyadic rationals in the interval $[\frac12,1]$ shown here in blue. Let $h(x)=\frac{x+2}{4}$. Now function $f$ satisfies the identity $f(h(x))=f(x)$. This is seen a) ...
- 9,762
1 vote
1 answer
84 views
Factorization of a rotation matrix into the product to three rotation matrices
Suppose you're given a rotation matrix $R$. You want to decompose $R$ and write it as the product of three rotation matrices, i.e. you want to express $R$ as $$ R = R_3 R_2 R_1 $$ with the condition ...
- 258
1 vote
2 answers
109 views
Rotating a unit vector to another vector using two consecutive axes
Given a unit vector $u$ and another unit vector $v$, I want to rotate $u$ into $v$ in two stages. In the first stage, I rotate $u$ about a given axis $a_1$ (by an unknown angle) to produce a vector $...
- 258
0 votes
0 answers
52 views
Error in derivation of quaternion derivative
Before I get to the equation, I want to get some notational things clear. Consider 2 frames A and B. The quaternion used to rotate frame A into frame B is notated by $q_{A\to B}^F$ where the F ...
- 323
2 votes
2 answers
116 views
Rotate vector within plane by given angle
Let two 3D unit vectors $V, V'$ be given. Derive vector $W$ created by clockwise rotating $V'$ by angle $\theta'$ around the origin within the plane with normal proportional to $V \times V'$. I tried ...
- 359
0 votes
0 answers
91 views
Constraining a rotation angle in an arbitrary three-axis decomposition of SO(3)
Let $\mathbf{R}_{\mathbf{d}}(\omega) \in SO(3)$ denote a rotation by angle $\omega$ about a fixed unit axis vector $\mathbf{d}$. Consider a rotation $\mathbf{R}$ defined by the sequence:$$\mathbf{R} = ...
- 538
0 votes
0 answers
61 views
Rotating a line in space to align it with another line
In my previous problem, I asked about rotating a plane into another plane. In this question, I am given two lines in 3D space: $P_1(t) = r_1 + t v_1$ , $P_2(s) = r_2 + s v_2$. I am interested in ...
user1480757
2 votes
3 answers
129 views
Rotating a plane into another plane
I am given two planes $n_1 \cdot (r - r_1) = 0 $ and $n_2 \cdot ( r - r_2 ) = 0 $ where $ r = (x, y, z), r_1 = (x_1, y_1, z_1) $ is a point on the first plane, and $r_2 = (x_2, y_2, z_2) $ is a point ...
user1480757
1 vote
1 answer
68 views
When rotating a coordinate frame in the negative angle direction, how do you implement the minus sign?
Rephrased question: Edit: @David K, thank you for your answer. I think I understand what you mean. At the same time, I think the way I phrased my question I am not asking what I wanted to ask. Please ...
- 11
4 votes
1 answer
165 views
Proof of Chasles theorem (Kinematics)
I have been trying to prove Chasles theorem using linear algebra. I am especially doubtful about whether the matrix can be inverted in the plane $\Pi$. And does this theorem also hold for ...
- 461
0 votes
0 answers
63 views
Rotation of a vector in 3D
The cosine of an angle between two vectors X and Y is defined as follows: \begin{equation} \cos \angle (X,Y) = \frac{X \cdot Y}{\lVert X \rVert \lVert Y \rVert} (1) \end{equation} Let us consider an ...
- 1
2 votes
1 answer
78 views
3D coordinate transform that turns rotation into translation
Is there a 3D coordinate transform which turns rotation in cartesian coordinates into translation in the transformed coordinate system? It would be sufficient if the transformation has the desired ...
- 29
1 vote
2 answers
196 views
How the angle between the projection of final y-axis and the initial z-axis is $\gamma$ in XZX (intrinsic) convention of Euler Angles?
I was working with rotating a frame to another frame recently, and got to know about Euler Angles. Since I had to find the specific rotation angles for a given final set of $X$-$Y$-$Z$ axes, I got ...
- 261
0 votes
1 answer
48 views
rotation of 3D circle arc: confusing outputs
I have a 3D circle arc defined by $r\begin{bmatrix} \cos(\theta)\cos(\phi)\\ \sin(\theta)\cos(\phi)\\ \sin(\phi) \end{bmatrix} $ where $r$ is the radius of the circle, $\theta$ the azimuth angle, and $...
- 37