What is Rotations: Definition and 193 Discussions

A rotation is a circular movement of an object around a center (or point) of rotation. The geometric plane along which the rotation occurs is called the rotation plane, and the imaginary line extending from the center and perpendicular to the rotation plane is called the rotation axis ( AK-seez). A three-dimensional object can always be rotated about an infinite number of rotation axes.
If the rotation axis passes internally through the body's own center of mass, then the body is said to be autorotating or spinning, and the surface intersection of the axis can be called a pole. A rotation around a completely external axis, e.g. the planet Earth around the Sun, is called revolving or orbiting, typically when it is produced by gravity, and the ends of the rotation axis can be called the orbital poles.

View More On Wikipedia.org
Recent contents Top users
  1. Z

    Combine two pitch, yaw, roll rotations

    Firstly, I apologise if this is the wrong section, but as far as I could tell this was the most relevant. I have a pair of triplets (p1, y1, r1) and (p2, y2, r2). Each one describes a set of roll, yaw, and pitch rotations in that order, by the angles given by each component in respective order...
  2. P

    Provide some rotations to obtain common shapes?

    I am attempting to use integration to determine the formula for the volume of three dimensional shapes for the sake of practice. My issue is I really lack the skills to determine what type of two-dimensional rotations will obtain a shape. The only one I have been able to do so far has been...
  3. 1

    How to find angle after two rotations

    I have coordinate system A with bases a, b, c. Say I rotate the whole system 30 degrees, so that the angle between a and a' is 30 degrees. Then I make another rotation so that this plane of rotation is perpendicular to that of the old one. What is the angle between a and a' now? I...
  4. Fredrik

    Distance-preserving maps are compositions of rotations and translations

    How do you prove that a bijection ##f:\mathbb R^n\to\mathbb R^n## such that ##d(f(x),f(y))=d(x,y)## for all ##x,y\in\mathbb R^n## is a composition of a rotation and a translation? (d=Euclidean metric). My first thought is to define ##g=f-f(0)##, prove that g is a bijection that preserves...
  5. D

    Rotations from Fourier Transforms? 4 times gets you back to original.

    Today in my circuits class, we were talking about Fourier transforms and my professor briefly said something about how a Fourier transform is a rotation in infinite dimensional space. I would ask him more about it but since it's beyond our course I'd rather not bug him. Where can I learn more...
  6. D

    Axis angle rotations and changing rotation values

    I have two 3d applications and when an object(a cube for example) is transferred between them, the rotation values of the cube change(the object stay at the same location. translation and scale values stay the same) and I can't find why that occurs and it's driving me crazy. app 1 rotation...
  7. P

    Commutativity of Lorentz Boosts & Rotations

    As per group property, one could make a product of gr members e.g. Lorentz boost (B) and rotation R, as they Commutativity is not valid, R.B or B.R, what should be considered and which order should be preferred? Generally it is known R.B1= B2.R .
  8. C

    Can Dust Particles Alter the Dynamics of Planar Rotations in Space?

    I've been looking at planar rotations, in a volumetric medium, for a while and came up with a rather strange solution I'd like you to take a look at. Taking a planar rotation where the tangential velocity at n is greater than the tangential velocity at n+1, you always get a singularity at r=0...
  9. C

    Can unitary operators on hilbert space behaive like rotations?

    Homework Statement unitary operators on hilbert space Homework Equations is there a unitary operator on a (finite or infinite) Hilbert space so that cU(x)=y, for some constant (real or complex), where x and y are fixed non-zero elements in H ? The Attempt at a Solution I know the...
  10. M

    Generator of Rotations and commutation relationships

    Suppose we have [J_i,J_j] = \sum_k \epsilon_{ijk} J_k and [L_i,L_j] = \sum_k \epsilon_{ijk} L_k 1st question, I am right in thinking that J represents Eingavalues for spin 1/2 particles... next... Computing the commutation relations, I find that \sum_k \epsilon_{ijk} (J_K + L_K - L_k -...
  11. R

    Using quaternions to perform rotations

    Hi, I have been looking at quaternions to perform rotations, however I have come across two slightly different equations to do this: v' = q^{-1}vq v' = qvq^{-1} What is the difference between these two? Thanks, Ryan
  12. K

    Nested rotations in a fluid: Implications and Visualization

    A fluid visibly spinning clockwise doesn't necessarily have a net angular momentum. I figured that this might be the case by considering that the fluid rotations within might be counter-cyclical. It would appear that you could apply this relationship many-fold times, where a fluid consists of...
  13. D

    Calculating Rotations for a Space Station

    Homework Statement One design for orbiting space stations has a structure that is very much like a large bicycle wheel. The astronauts live on the inside of this wheel where the spinning provides an acceleration similar to Earth's gravity. Suppose the space station has a diameter of 125m ...
  14. Z

    Fermion Rotations: Exploring the Mystery of 720 Degree Symmetry

    It is interesting that our elementary fermions have 1/2 spin, meaning it takes a full 720 degree rotation to bring them back to their original state and these fermions constitute ordinary matter, eg. quarks, and electrons. Classical nature, however, does not have a 720 degree symmetry, but...
  15. T

    Decompose rotations of a vector

    Hello guys, I'm programming a class in C++ that generates a circular signal. The signal consists of a sin and cos in perpendicular directions. The user has to input the norm to the surface, and the program generates the sine and cosine in 2 perpendicular directions to that norm to generate the...
  16. V

    Understanding spin, spinors, and rotations

    There's something I don't think I quite understand about spin and how it acts a generator of rotations. I'll start with quickly going over what I do understand. Suppose you want to do an infinitesimal rotation around the z-axis on some state: \def\ket#1{\left | #1 \right \rangle} \ket{\psi...
  17. V

    Solving for Euler angles and 3-D coordinate Rotations.

    Hi, (attachment with visuals is included) I have a 3-D vector dataset that is measured in a reference frame (measurement reference frame) that is oriented relative to a horizontal coordinate system. In this dataset I have x-y- and z-component data for the vectors relative to a coordinate...
  18. D

    Finding moment of inertia of a flywheel using rotations, with friction.

    ive recently done a experiment which involved a flywheel which was rotated by the force of gravity on a mass which was wrapped around the flywheel axis. I have tried to calculate functions which will predict the behaviour as well as get the moment on inertia of the flywheel via torques of...
  19. Demon117

    The set of Lorentz boosts and space rotations form a group

    Ok. I understand that the set of Lorentz boosts and space rotations is equivalent to the set of Lorentz transformations. I understand that they form a group, but what I cannot seem to grasp is this. What the explicit form of such 4x4 matrices? One needs to know this in order to show that the...
  20. T

    Composition of Quaternions as rotations

    Homework Statement Hi, I am having problems in showing that in practise the composition of two rotations represented by quaternions is still a rotation. The example I have constructed is: Rotate (1,1,0) by 45 degrees about the z axis. The quaternion to use is thus q = cos(22.5)+ksin(22.5)...
  21. C

    Non-invariance under 2-Pi rotations?

    I have heard that quantum systems (and therefor all physical systems) are not truly invariant under 2∏ rotations. Something to do w/ the wave function changing sign. Is this true? Can someone point me to an on-line primer on this? Thanks!
  22. T

    Rotations with Quaternions and their exponential.

    Homework Statement Hello, I'm trying to get my head around the various properties of quaternions that all seem very similar, but I can't quite understand the underlying differences between them. I would like to know the differences between unit quaternions, purely imaginary quaternions...
  23. L

    Determining the orientation of ellipsoids after 2 rotations

    I am writing some code in which I am working with ellipsoids. The ellipsoids can be rotated in its body frame with three angles (First rotation is about the z-axis in the body frame, second is about the y-axis in the body frame, and finally another rotation about the z-axis in the body frame)...
  24. S

    Orthogonal Matricies and rotations.

    Hi all, I have been trying to gain a deeper insight into quadratic forms and have realized that my textbook makes the assumption that an orthogonal matrix corresponds to either a rotation and/or reflection when viewed as a linear transformation. The textbook outlines a proof that demonstrates...
  25. F

    Heading as a function of wheel rotations

    Hello, I have a three robots which consists of two motorized wheels and a skid wheel. The two motorized wheels are on the front of the robot. I am trying to find the change in heading of the robot based on the previous state. I have found out that ΔHeading = [(ΔDL-ΔDR)*radius of...
  26. A

    Rotations in Quantum Mechanics Question

    Homework Statement This is more maths than QM I think, but it's at the beginning of my Quantum Questions. Basically, it's about rotations preserving length: xi is the ith component of a vector, and the length of a vector is determined by the metric ηij according to the equation: l2...
  27. M

    Are these compositions of linear transformations reflections or rotations?

    Homework Statement if Sa: R2 -> R2 is a rotation by angle a counter-clockwise if Tb: R2 -> R2 is a reflection in the line that has angle b with + x-axis Are the below compositions rotations or reflections and what is the angle? a) Sa ○ Tb b) Ta ○ Tb Homework Equations I don't...
  28. O

    Are Rotations Exponential? Understanding the Concepts Behind Euler Identity

    This isn't a homework question, but I'm having trouble understanding something about rotations conceptually. While reading about the Euler Identity online, I keep running into a few things that I can't wrap my head around and never come with an explanation. Here are the concepts I can't...
  29. M

    Rotations in spherical coordinates

    I have a few questions about rotations. First off if i have two vectors r_{a,b}=(1,\theta_{a,b},\phi_{a,b}) And i define \Delta\theta=\theta_b-\theta_a and \Delta\phi=\phi_b-\phi_a. Then take the map T(1,\theta,\phi)=(1,\theta+\Delta\theta,\phi+\Delta\phi). Is T a rotation? I would...
  30. P

    Group of rigid rotations of cube

    I'm having trouble visualizing some of the rotations that compose this group. Clearly the group has 24 elements by argument any of 6 faces can be up, and then cube can assume 4 different positions for each upwards face. My book describes the rotations as follows: 3 subgroups of order 4...
  31. D

    Generator of Rotations: J_x, J_y, J_z Relationship

    My textbook derives the relationship [J_x, J_y] = 2pi*iJ_z by considering the J's as the generators of the Euclidean rotation operator. Why does this result hold when J is the generator of other rotations, such as rotations in a quantum mechanical spin-state space?
  32. B

    Solving Rotations Homework: Double Speed vs Halving Radius

    Homework Statement A truck was driven around a circular track. Which would have a greater effect on the magnitude of its acceleration: moving to a track with half the radius or doubling the speed. Show proof/sample calculations. Homework Equations I'm not quite sure but I went ahead and...
  33. C

    What Are Embedded Axis Frames in Euler Equations?

    my book says that it is actually difficult to get the true motion of a body by using these equations because it says that euler equations are written in embedded axis frame ... what is an embedded axis frame?where is it different from normal frames that i used in before?after solving euler...
  34. L

    Quaternion Rotations: Show R2∘R1 Is a Rotation

    Homework Statement 3. (a) Show that every quaternion z of length 1 can be written in the form z = cos(\alpha/2) + sin(\alpha/2)n, for some number α and some vector n, |n| = 1. (b) Consider two rotations of the 3d space: the rotation R_1 through \alpha_1 around the vector n_1 and the rotation...
  35. L

    Boost generator transforms as vector under rotations

    Hi, I've read quite a few times now in group theory and QFT books that [X_i,Y_j]=i\epsilon_{ijk}Y_k can be regarded as saying that \vec{Y} , the vector of boost generators transforms as a vector under rotations (where X are SO(3) generators). I don't really understand why this implies...
  36. H

    Understanding the Relationship of Sphere Coordinates and Rotations

    hi , every one! I have a problem with a sphere rolling on a fixed sphere. My problem is to find relationship between coordinate of center of sphere (X,Y,Z) and orientations (alpha, beta, gamma) or Euler angles of sphere. as we know a sphere has 6 DOF in space (3 coordiantes and 3 rotation) when...
  37. B

    Euler rotations in galactic plane to change to equatorial

    HI there, I am having problems understanding something. If I have an axis pointing towards the galactic north pole, and I rotate it using an Euler rotations how can I can I establish the rotation angles needed so that it will be pointing to the equatorial North pole. I am looking for...
  38. S

    Linear algebra - Rotations in R3

    Homework Statement Given, R_1(x) = [ 1 0 0 0 cos(x) -sin(x) 0 sin(x) cos(x)] R_2(x) = [cos(x) 0 -sin(x) 0 1 0 sin(x) 0 cos(x)] R_3(x) = [cos(x) -sin(x) 0 sin(x) cos(x) 0 0 0 1] Describe the transformations defined by each of these matrices on vectors in R3. Homework...
  39. Z

    Exploring Double Rotations in 4-Space: An Investigation in Group Theory

    I was wondering if anybody could help me understand the "double rotations" in 4-space. These are evidently rotations that fix only a single point--the center of rotation--and that take place in two hyperplanes simultaneously and independently. Beyond that, I have an even more specific...
  40. H

    Rotations in general relativity?

    In special relativity, the mass of an object increases as the speed approaches c. Geometrically, this can be interpreted as a pure result of the relativistic length contraction. As the length (and volume) of a point-like particle becomes smaller, the field lines are forced together and the...
  41. O

    Are Tidal Forces the Opposite of Centripetal Forces in Rotational Dynamics?

    I want to ask here if tidal forces can in a sense be considered as the opposite of the centripetal forces that drive rotations or spins. I'd appreciate any web-accessible references about this. First consider a uniform spherical cloud of non-interacting test masses falling radially toward a...
  42. A

    Basic conceptual/intuitive question on molecular vibrations and rotations.

    Could someone please help me get a very basic intuitive understanding of what rotational and vibrational quantum numbers mean? For simplicity, assume a diatomic molecule. For example, does a higher rotational quantum number mean that the molecule is rotating faster? Is the vibration number a...
  43. C

    Lorentz boost and equivalence with 3d hyperbolic rotations

    I was thinking that if i have for example a boost in the direction of x, then the boost should be equivalent to an hyperbolic rotation of the y and z axes in the other direction. I don't know if it's true or not. Then I want to know if somebody knows this result or why is false? I was...
  44. M

    'Givens' Matrix rotations and QR Factorisation

    Hey there all! I'm a little confused by the concept of Givens rotations and was hoping someone could help elaborate a little bit with the following problem for me - if i could get some help understanding how to approach the problem it would be a really great help to me. 'Let A be an n x n...
  45. J

    Linear Algebra with linear operators and rotations

    Definie linear operators S and T on the x-y plane as follows: S rotates each vector 90 degress counter clockwise, and T reflects each vector though the y axis. If ST = S o T and TS = T o S denote the composition of the linear operators, and I is the indentity map which of the following is true...
  46. M

    Linear transformation: Rotations in R3

    Homework Statement A robot arm in a xyz coordinate system is doing three consecutive rotations, which are as follows: 1) Rotates (Pi/4) rad around the z axis 2) Rotates (Pi/3) rad around the y axis 3) Rotations -(Pi/6) rad around the x axis Find the standard matrix for the (combined)...
  47. C

    Permutations in rotations and reflections

    Hi all, I've been having difficulty with the following question. Let P be a regular pentagon. Let R be the rotation of P by 72degrees anticlockwise and let F be the reflection of P in the vertical line of symmetry. Represent R and F by permutations and hence calculate: F R^2 F R F^3 R^3 F...
  48. P

    Matrix Rotations: Article for Rigorous Description | PF

    where can I find a good article about a riguros description of matrix rotations? I can't find anything on PF...
  49. D

    with problem with matrices, reflections, rotations

    Looking for help with a problem I'm working on: "Show that matrix [0 -1 0] [-1 0 0] [0 0 1] for a reflection about line y=-x is equivalent to a reflection relative to the y-axis followed by a counter-clockwise rotation of 90 degrees." So for my answer, first I have for the reflection...
  50. B

    Rotations in nth Dimensional Space

    Let me start by saying I do not have a lot of background in linear algebra, but I'm not afraid of learning. I am working on a flash animation with action script. That does the following: 1. Start with a point. 2. Add width so it turns into a line. 3. Rotate about the x-y plane 360 deg. 4...
Back
Top