Rotations Definition and 188 Threads

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...

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...

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...

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...

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...

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...

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 .

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...

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 -...

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

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...

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 ...

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...

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...

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...

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...

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...

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)...

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!

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...

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)...

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...

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...

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: [FONT="Times New Roman"]xi is the ith component of a vector, and the length of a vector is determined by the metric ηij...

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...

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...

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...

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...

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?

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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)...

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...

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

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...

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...

Homework Statement A bicycle drives on a straight road with a velocity v = 18 km/h. How many turns (rotations) of peddles per second must be carried out by cyclists, if the radius of the wheels r = 35 cm, and the ratio of the radii of front and rear gear wheels is 2:1? Homework Equations I...

Hello, in order to numerically solve a physics problem I think I need to add 2 (infinitesimal) rotations of one and the same segment each around a different point in 2D space in one iteration of numeric approximization. How does this addition work out? Is it the sum of the vectors connecting...

I want to know the difference between the 2 types of rotations shown in the picture attached . planetary orbiting are similar to the second figure in the picture attached , all i want to know what is the difference between the 2 figures .