# What is Euler angles: Definition and 45 Discussions

The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.They can also represent the orientation of a mobile frame of reference in physics or the orientation of a general basis in 3-dimensional linear algebra. Alternative forms were later introduced by Peter Guthrie Tait and George H. Bryan intended for use in aeronautics and engineering.

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1. ### I Game Design: Set rigid body into a position (Euler angles) using angular velocity

I apologize for messy title, but that's as short as I could make it to say all that I wanted. I also apologize if I'm not using correct math phrases, but I'll try to picture a problem the best I can. I've been pulling my hair out trying to solve a problem I'm having for couple of months. The...
2. ### I How can Euler angles be visualized using a polar plot?

Dear Forum, say I am projecting an ellipsoid along the z-axis to the xy-Plane. The resulting ellipsis is rotated around the z-axis by the angle gamma until the principal axes coincide with the x- and y axis. Now before projecting, I rotate the ellipsoid first around the z- and then around the...
3. ### A Invariance of ##SO(3)## Lie group when expressed via Euler angles

I am trying to understand the properties of the ##SO(3)## Lie Group but when expressed via Euler angles instead of rotation matrix or quaternions. I am building an Invariant Extended Kalman Filter (IEKF), which exploits the invariance property of ##SO(3)## dynamics ##\mathbf{\dot{R}} =...
4. ### Angular velocity in terms of Euler angles

In Chapter 4, derivation 15 of Goldstein reads: "Show that the components of the angular velocity along the space set of axes are given in terms of the Euler angles by \omega_x = \dot{\theta} \cos \phi + \dot{\psi} \sin \theta \sin \phi, \omega_y = \dot{\theta} \sin \phi - \dot{\psi} \sin...
5. ### A Algebraic proof that Euler angles define a proper rotation matrix

I have asked this question twice and each time, while the answers are OK, I am left dissatisfied. However, now I can state my question properly (due to the last few responses). Go to this page and scroll down to the matrix for sixth row of the proper Euler angles...
6. ### Difference between gyroscope angular displacement and Euler angles

Hi guys, I'm trying to understand between gyroscope angular displacement and euler angles? for example { Δx = Δx + h * Rx * SCx);} this is gyroscope output about anguler displacement.This value can be used to determine angle that device created.Why we should euler angles to fly.(I know...
7. ### How can I rotate Euler angles through a specific angle?

Summary: I found an old topic on this forum that describes a similar problem to what I am facing currently. I want to rotate an object with euler angle values, but the rotation has to be translated based around a specific angle. However, I do not fully understand the solution provided in that...
8. ### A Book to understand Euler angles (and Tait–Bryan )

Hi I am trying to understand Euler angles and why they work. What are some great books to use as a resource to build a deeper understanding of them? I want to know why 3-2-1 Euler angles are so commonly used, and proofs regarding the mathematical properties of this method..
9. ### I Understanding 3D Coordinate Rotations with Euler Angles

I'm trying to wrap my head around the concept. we use 3 rotations to transfer our regular cartesian coordinates (3 x,y,z unit vectors) to other 3 unit vectors. each rotation is associated with an angle. so far I'm good. but now I saw in Landau's and Lifshitz's "mechanics" book this thing...
10. ### A How do I KNOW that Euler angles are sufficient?

Hello Before I "phrase" my question (and that may be my problem), may I first state what I do know. I understand that a Rotation matrix (a member of SO(3)) has nine elements. I also understand that orthogonality imposes constraints, leaving only three free parameters (a sub-manifold) I also...
11. ### Euler angles in torque free precession of a symmetric top

Is calculating the Euler angles analitically possible? I am trying to obtain the angles to transform the body-fixed reference frame to the inertial reference frame. I can get them without problems with numerical methods. But I would to validate them analitically, if possible. I followed the...
12. ### Rigid body orientation using Euler angles confusion

Hello, Homework Statement I'm given the following exercise: "A rod with neglected thickness exists. What is the relation between the α,β angles to Euler angles of orientation? α is defined as the angle between the rod and its projection on the XY plane. β is defined as the angle between the...
13. ### A About the nodal line in defining Euler angles

I hope someone can explain this to me. In the definition of Euler angles, the body-fixed-azimuthal angel γ is measured from the nodal line that defines the intersection of the body-fixed-XY plane and the space fixed-xy plane to the body-fixed-Y axis. This is the green line in this image from...
14. ### A How do you KNOW the Euler (Tait) angles cover orienations

Well, that question just about states my issue. We have a body and we rotate about, say, the 3-axis of its body frame. Then, we must do the next rotation about the 1 or 2 axis. Let me say we choose the 1-axis Then we have a choice: continue on to the 2 axis or repeat the 3 axis. One set is...
15. ### I How does the angle γ change under inversion in Euler angles?

The well known Euler angles (αβγ) are defined as in the image It is easy to see that under inversion α → π+α β → π-β but I cannot figure out how γ transforms under inversion. actually I am stuck at the question whether I should measure it from the same intersection line ON (thence γ →π+γ) or...
16. ### I Body rates from Euler angles....

Referring to slides 3-4 (page 2) of this link: https://www.princeton.edu/~stengel/MAE331Lecture9.pdf The author states the relationship between body rates [p q r] and Euler angle rates [φ_dot θ_dot ψ_dot]. I want to verify this but have been failing miserably... My reasoning: 1) p, q, and r...
17. ### Mathematica Rotation of 3D Plot using Euler angles

So, I'm trying to plot a 3D "dipole" (an arrow with a small torus around it basically) in mathematica, and want to rotate it according to Euler angles... I use this code for the rotation matrix: rot[a, b, g] := RotationMatrix[g, {1, 0, 0}].RotationMatrix[b, {0, 1, 0}].RotationMatrix[a, {0, 0...
18. ### Euler angles in latitude longitude space

In most physics introductions Euler angles(pitch, roll, yaw) are defined with respect to Cartesian coordinate system. If I chose not to use a Cartersian coordinate system but instead use a latitude, longitude and a proprietary vertical coordinate(and no back transformations to Cartersian...
19. ### 3D Coordinate transformation and Euler Angles

Hello, I'm running a galaxy formation simulation. The output specifies the coordinates in (x, y, z) of all the particles in a galaxy, which usually fall in a disk. The orientation of the disk depends on the initial conditions, but it is generally not aligned with any of the coordinate axes...
20. ### Diagonalization of 8x8 matrix with Euler angles

I am trying to diagonalize the following matrix: M = \left( \begin{array}{cccc} 0 & 0 & 0 & a \\ 0 & 0 & -a & 0 \\ 0 & -a & 0 & -A \\ a & 0 & -A & 0 \end{array} \right) a and A are themselves 2x2 symmetric matrices: a = \left( \begin{array}{cc} a_{11} & a_{12}\\ a_{12} & a_{22}...
21. ### Euler Angles/ Rigid Body application

I apologize upfront, as I have no experience with math to begin with. However, I have a real life problem I am trying to figure out here at work and I would appreciate some help. Here is my situation: Homework Statement In a 3 dimensional plane, I have an artillery piece pointing at...
22. ### Modeling Rigid Body - Unsure about Euler angles and angular velocity

I'm modeling a single 3D rigid body in preparation for some more complicated modeling in order to gain a better understanding of Euler angles, the angular velocity vector and the rotating coordinate system. The body is rotated in inertial frame by an intrinsic ZXZ rotation, with respective...
23. ### Finding Euler angles from rotation about arbitrary axis

Homework Statement An object is rotated 45 degrees about an axis whose + direction is that of (i-k). Find zxz Euler angles (that is, Euler angles as introduced by Goldstein) for a set of three active rotations that gives the same net motion of the object. Homework Equations...
24. ### Angle projections to Euler angles

Consider a vector in 3D. Its projections on two planes, say YX and YZ planes, makes some angle with the vertical axis ( the y-axis in this case). I know these two angles (I call them projected angles). This is the only information I have about the vector. I need Euler angles which when...
25. ### Euler angles equivalence with a single rotation

imagine we rotate a vector centered at the origin with euler angles alpha,beta,gamma. now the question is, can we do this rotation by the means of defining a vector N(which its length is 1).and rotating the vector zeta radians counter clockwise around N? I think it must be possible and I want...
26. ### Euler Angles - Why Post Multiplication

In robotics, you have a co-ordinate frame which is at the base & another which is at the next joint. You want to describe the position and orientation of the 2nd frame with respect to the first in terms of position and orientation. For the position, you use a translation matrix. The...
27. ### Euler Angles - Why Post Multiplication

Normally with column vectors, we premultiply rotation matrices if the angles are with respect to fixed axis. Why then do we post multiply if the angles are Euler Angles, angles with respect to the mobile axis?
28. ### How to Compute Euler Angles for Rotating Reference Frames

I know this is rather trivial question but it's not homework! I need this as part of a bigger project. What I have to do is rotate my reference frame to another one. I want my new z-axis to be a vector Z=(Z1,Z2,Z3) Following the notation of this wikipage...
29. ### Euler angles and angular velocity

How do you prove that angular velocity is just the time derivative of each Euler angle times the basis vector of its respective frame? I remember it used to be perfectly clear to me a while back, but now I don't remember how the result was derived, and I couldn't find it in any of my books I...
30. ### 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...
31. ### Euler angles. Quantum Mechanics Question

Homework Statement Let U = e^{iG_{3}\alpha}e^{iG_{2}\beta}e^{iG_{3}\gamma} where ( \alpha, \beta, \gamma ) are the Eulerian angles. In order that U represent a rotation ( \alpha, \beta, \gamma ) , what are the commutation rules satisfied by the G_{k} ?? Relate G to the angular...
32. ### Most Efficient way to solve for Euler Angles

Hi guys, could you please help me out? Essentially, a laser is pointing at a certain point in 3 dimensional space. There is a fixed target that the laser is specifically supposed to point to. My job is to find the most efficient way to solve for the euler angles(yaw pitch and roll) in order...
33. ### Calculating Euler Angles from Two Frames of Reference

Fairly straight forward question. If you have a set of three vectors specifying a frame of reference and a second set of 3 vectors stating another frame of reference. How do you get the Euler angles associated with that rotation? More generally I am considering the relative orientation of one...
34. ### What is the uniqueness of Euler angles?

Hi.. The wikipedia article on euler angles claims that the Euler angles in zxz convention are unique if we constrain the range they are allowed to take (except in the case of the gimbal lock). This seems reasonable. But can someone give me a reference... a book or a paper where this is...
35. ### Robotics Yaw Pitch Roll Euler Angles

I have multiple robot cells that i would like to move fixtures from robot to robot with the out come of the part to be perfect. in order to do that i need to make sure my work object is in the same spot each time. so i need to find the yaw pitch and roll and convert them into Euler angles. On...
36. ### Finding Components of Rotated Ellipsoid using Euler Angles

Hello guys, I'm trying to find an algorithm to solve an overlap problem between many polyhedra in space, so here this question appears. Imagine an Ellipsoid rotated by Euler angles in 3D space. This Ellipsoid could be characterised by its 3 radii Rx,Ry and Rz, and by 3 Euler angles in...
37. ### Euler angles of rotation about x=y=z

What would be the euler angles of rotation 2pi/3 about the line x=y=z? If something were in the xy plane and it underwent that rotation, would it end up in the yz plane?
38. ### Convert from unit vector to (maybe) special euler angles on a machine tool

Hello, I am trying to extract the 3 angles from a unit vector to position a part on a 5 axis milling machine: This machine has 3 linear axes (x,y,z), direct coordinate system It has 2 rotary axes: A turns about the x (called 'tilt') C is carried on the A axis (called 'rotary') See...
39. ### Euler angles and symmetric top

Homework Statement Check out problem 5.7 part a I want to express the exterior gravitational potential in terms of the Euler angles so that I might eventually use (dV/dB)B=B0 = 0 - the condition for equilibrium. I am therefore expecting the Lagrangian to be cyclic in terms of the other two...
40. ### Euler Angles between two versors

Dear All, Suppose you have two versors x and y. You would like to calculate the 3 Euler angles between them. How would you do that? You know that x=Ry where R is the rotation matrix (see e..g Goldstein). Now the point is that I know both x and y and want to get phi, theta and psi (no...
41. ### Givens rotations versus Euler angles

Trying to implement QR decomposition using Givens rotations, I calculate G1 to zero n32 of original matrix A, then G2 to zero n31 of G1 * A, then G2 to zero n21 of G2 * G1 * A. Residual matrix, R = G3 * G2 * G1 * A comes upper triangular as expected, so I believe my code is correct. Looking...
42. ### Can someone explain Euler angles?

Can someone explain "Euler" angles? From what I read, "Euler" rotations are composed out of matrices like * * 0 1 0 0 * * 0 * * 0 0 * * * * 0 0 0 1 0 * * 0 0 1 which is pretty distinctive in that they rotate around same axis twice, and makes sense for devices like this...
43. ### Translation of euler angles into rotation around arbitrary axis

i have an orientation of a 3d object in space given by theeta, si and phi i.e. angles which the objects makes with respect to three axis. Now i want to translate the problem such that i get an arbitrary axis rotation about which to some calculated degrees would produce same orientation...
44. ### Vector rotation (possibly Euler angles)

I'm looking for a method to rotate a 3D vector, and place it at an arbitary 3D point (x,y,z) without changing the vectors magnitude. I have briefly investigated eulers angles (mainly through wikipedia links etc), but don't fully understand the process yet. As an example, given the vector ...
45. ### Earth's Euler angles - nutation?

Hi, I would like to see the resemblance between planet Earth and a spherical top. I draw here the x-convention Euler angles as I know them: 24 hours to complete one complete one revolution around it's own z' axes. And the 23.5° Earth axes revolves around the "sun's z axes" - the line...