Here we initialize a stack of 3 Parameters. Creating a rotation matrix in NumPy The two dimensional rotation matrix which rotates points in the x y plane anti-clockwise through an angle about the origin is R = ( cos sin sin cos ). Warum verwendet das ffentliche Stromnetz 50-60 Hz und 100-240 V? Can a character use 'Paragon Surge' to gain a feat they temporarily qualify for? Non-anthropic, universal units of time for active SETI, Math papers where the only issue is that someone else could've done it but didn't. Initialize from Modified Rodrigues Parameters (MRPs). Thanks for contributing an answer to Stack Overflow! Warum entsprechen die Generatoren der Rotation im 4-dimensionalen euklidischen Raum den Rotationen in einer Ebene? Wenn Sie der Meinung sind, dass dieser Beitrag fr Sie ntzlich war, mchten wir Sie bitten, ihn mit anderen Junioren zu teilen, damit Sie uns helfen knnen, diese Informationen zu verbreiten. Rotation() is not supposed to be instantiated directly. The underlying object is independent of the How can we create psychedelic experiences for healthy people without drugs? For the ZYZ convection, the Euler angles have a particular nomenclature: Each of the three rotations can be represented mathematically by a rotation matrix. Some coworkers are committing to work overtime for a 1% bonus. The simplest example of application of what we have already seen in the article is the rotation of a point located in a coordinate space (X, Y, Z). Once installed, in python you may create the orientation object which can rotate vectors, or be part of transform objects. for working with other libraries like the affine_transform functions in scipy). major functionalities. Lets see the result of our rotation by plotting with Cartesian axes the position of the vector (which describes the point) before and after the rotation. Why are only 2 out of the 3 boosters on Falcon Heavy reused? To learn more, see our tips on writing great answers. For example, if we rotate 90 degrees around the X axis, the Y axis will collapse on the Z axis. It can also be solved using quaternion theory: Disclaimer: I am the author of this package. converted to any of the others. All these systems are based on calculations and mathematical concepts capable of calculating every single movement in three-dimensional space, most of which were developed by the famous mathematician Euler (1707-1784). ], [ 2.24452282, 0.78093109, 2.89002836]]). It has certain special operators, such as * (matrix multiplication) and ** (matrix power). Dieses Problem kann auf verschiedene Weise behandelt werden, wir zeigen Ihnen jedoch, was fr uns die vollstndigste Lsung ist. This From here you can then move on to the rotation of lines, geometric figures and three-dimensional objects. Each of these will have a different convention for expressing Euler angles. From what I understand your Nx3x3 is the extracted rotation matrices, right? This class provides an interface to initialize from and represent rotations Replacing outdoor electrical box at end of conduit. To create and apply a rotation matrix using python, a solution is to use numpy: \begin{equation} \left( \begin{array}{cc} \cos(\theta) & -\sin . Stack Overflow for Teams is moving to its own domain! rev2022.11.3.43004. Here is a very ugly implementation which seems to work. For further clarification of how quaternion multiplication etc. Code solving the single-point case appears below: What I need is a function that behaves exactly as the above function, but instead of accepting a single angle and a single direction, it accepts an angles array of shape (npts, ) and a directions array of shape (npts, 3). Hmm, I see your point @PaulPanzer but I'm still confused because each pair of 3d points defines a unique plane, which defines a unique normal vector, and there is a unique rotation angle about that normal vector that rotates p into q. Since the condition is different, the mathematical expressions to derive them are also different. Rotates the matrix by 90, 180 degrees as per requirement. This is called the right hand rule, let a be the unit vector along axis, i.e. The rotation should appear to be counter clockwise for an observer to whom the axis vector is pointing. MATLAB command "fourier"only applicable for continous time signals or is it also applicable for discrete time signals? However, in practice, a more complex but more effective mathematical model is often used, the Hamilton quaternions. I have another such list, q. I want to calculate an ndarray U of shape (N, 3, 3) storing the rotation matrices that rotate each point in p to the corresponding point q. import numpy as np def euler_rotation_matrix(alpha,beta,gamma): """ Generate a full three-dimensional rotation matrix from euler angles Input :param alpha: The roll angle (radians . array([[-2. , 1. , 3. For example, the most intuitive is that which is obtained first by performing a rotation on the X axis by an angle , then on the Y axis by an angle and finally on the Z axis by an angle . But the XYZ rotation sequence is only one of 12 possible combinations. The quaternion is in Hamiltonian convetion, i.e., (qw, qx, qy, qz). The technique we have seen is based on the use of a sequence of elementary rotations referring to one of the Cartesian axes at a time. We use cookies to ensure that we give you the best experience on our website. Represent as Modified Rodrigues Parameters (MRPs). And there are a ton of different ways of representing a rotation as three numbers and a few ways to represent it as 4 numbers. But it is also possible to perform the reverse operation. I have two vectors as Python lists and an angle. I prefer women who cook good food, who speak three languages, and who go mountain hiking - what if it is a woman who only has one of the attributes? MATLAB command "fourier"only applicable for continous time signals or is it also applicable for discrete time signals? Can I spend multiple charges of my Blood Fury Tattoo at once? A matrix is a specialized 2-D array that retains its 2-D nature through operations. Each of these sequences gives a triplet of Euler angles with different values, as we have also verified above. Not the answer you're looking for? Rotation around an axis is positive if it meets the right hand rule. Each elementary rotation can be transcribed as a 33 matrix (homogeneous transformation). My application will have N be very large, 1e7 or greater, so a vectorized function that keeps all the relevant axes aligned is necessary for performance reasons. [0. , 0.38268343, 0. , 0.92387953], [0.39190384, 0.36042341, 0.43967974, 0.72331741]]). Now use the concatenate function and store them into the result variable.In Python, the concatenate method will help the user to join two or . I do not really understand how your solution extracts the multiple rotation matrices. It relies on the quaternion extension to numpy found here. I've checked that your rotation_matrices function outputs agree exactly with the method outlined in the comments. Explanations: To construct a rotation matrix . To learn more, see our tips on writing great answers. I can open a new question for this if you want. The coordinates of the point in space after the rotation described by R will correspond to the values of the vector v2. Whether this instance represents a single rotation. Quaternion Theory: Then the rotation matrix and the inverse formula will change accordingly. array([[0. , 0.38268343, 0. , 0.92387953], array([-2. , -1.41421356, 2.82842712]), scipy.spatial.transform.Rotation.from_quat, scipy.spatial.transform.Rotation.from_matrix, scipy.spatial.transform.Rotation.from_rotvec, scipy.spatial.transform.Rotation.from_mrp, scipy.spatial.transform.Rotation.from_euler, scipy.spatial.transform.Rotation.as_matrix, scipy.spatial.transform.Rotation.as_rotvec, scipy.spatial.transform.Rotation.as_euler, scipy.spatial.transform.Rotation.concatenate, scipy.spatial.transform.Rotation.magnitude, scipy.spatial.transform.Rotation.create_group, scipy.spatial.transform.Rotation.__getitem__, scipy.spatial.transform.Rotation.identity, scipy.spatial.transform.Rotation.align_vectors. Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. But even though the declarations of np.array objects from np.matrix look very similar, their behavior can be very different in many contexts. The package is on github (mgen) and can be installed via pip: Note that the matrices are just regular numpy arrays, so no new data-structures are introduced when using this package. E.g. What is a good way to make an abstract board game truly alien? Do US public school students have a First Amendment right to be able to perform sacred music? In other words, if we consider two Cartesian reference systems, one (X0,Y0,Z0) and the other (X1,Y1,Z1) which have the same origin point O, but different orientation, there will always be a single axis of rotation with which the first system will assume the same configuration as the second system. This corresponds to the following quaternion (in scalar-last format): How can I find a lens locking screw if I have lost the original one? rot90 will be used which is a built-in function. That is, knowing the general rotation matrix, obtain the three Euler angles. Generate uniformly distributed rotations. Irene is an engineered-person, so why does she have a heart problem? Now Imagine moving by the polar angle theta away from the z axis (as in spherical coord. Note that this method is particularly efficient if you have to rotate a vector through many sequential rotations, as the quaternion product can just be calculated as q = q1 * q2 * q3 * q4 * * qn and then the vector is only rotated by 'q' at the very end using v' = q * v * conj(q). Euler transformations with their relative angles are a wonderful tool for applying rotations of points in space. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. there are different possible combinations of three elementary rotations, such as ZYX, ZYZ, XYX, etc. File ended while scanning use of \verbatim@start". Why is proving something is NP-complete useful, and where can I use it? It may be possible to do what you wish, but I have no immediate idea about how to mix in an analytical function with math3d. Another drawback is that the angles depend on the sequence of the three rotations around the Cartesian axes and reported as the name of the convention: ZYZ, XZX, YXZ, etc. There are several more ways to use Rotation based on what data you have about the rotation: from_dcm Initialized from direction cosine matrices. Browse other questions tagged, Where developers & technologists share private knowledge with coworkers, Reach developers & technologists worldwide. See here for why half the angle is required. Python rotation matrix around axis code beispiel, Numpy matrix multiplication code beispiel, Wie erstellt man eine diagonale Matrix in Python Numpy Code, Wie man einer Numpy-Matrix Polsterungen hinzufgt, Kann man jeden Eigenwert einer unendlichen Matrix durch. expm (code here) computes the taylor series of the exponential: above x2: screenshots from here. Here is the rotation matrix that takes care of rotation of a robot in 3D about the global z-axis: Return to Table of Contents. Finally, the rotation of the vector is calculated by the following operation. I'll check this as the accepted answer since you took the time to write it up. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. In this situation, a degree of freedom is lost as the rotations around the Y, Z axes become equivalent. If you continue to use this site we will assume that you are happy with it. it supports. It also provides a helper function rotate(A,theta,B) if you don't want to call the method on A. http://vpython.org/contents/docs/visual/vector.html. Also in robotics, in particular with robotic arms, systems are used that are able to calculate a certain movement, establishing how much the individual motors that compose them must rotate. Consider a counter-clockwise rotation of 90 degrees about the z-axis. Extract rotation(s) at given index(es) from object. The function would imply, I guess, an analytical function, whereas math3d works better with point sets. E.g. Represent as rotation matrix. How can I best opt out of this? array([[ 2.22044605e-16, -1.00000000e+00, 0.00000000e+00]. For example, imagine projecting a normal distribution on the xy plane from the z axis. But the rotation around an axis can occur in two opposite directions. I just wanted to mention that if speed is required, wrapping unutbu's code in scipy's weave.inline and passing an already existing matrix as a parameter yields a 20-fold decrease in the running time. corresponds to the following quaternion (in scalar-last format): The rotation can be expressed in any of the other formats: The same rotation can be initialized using a rotation matrix: The rotation vector corresponding to this rotation is given by: The from_euler method is quite flexible in the range of input formats A Rotation instance can be initialized in any of the above formats and What are the differences between numpy arrays and matrices? Here we initialize a single rotation about a single axis: Again, the object is representation independent and can be converted to any the cross product of the identity matrix with a. then M = exp( A) is the rotation matrix. I find it very surprising that there is no functionality for this in SciPy (or similar easily accessible package); vector rotation isn't that exotic. This theorem was formulated by Euler in 1775. Each movement of a rigid body in three-dimensional space, with a point that remains fixed, is equivalent to a single rotation of the body around an axis passing through the fixed point. Asking for help, clarification, or responding to other answers. Making statements based on opinion; back them up with references or personal experience. Rotation direction is from the first towards the second axis. """ axis = np.asarray (axis) axis = axis / math.sqrt (np.dot (axis, axis)) a = math.cos (theta / 2.0) b, c, d = -axis * math.sin (theta / 2.0) aa, bb, cc, dd = systems capable of performing calculations that simulate the movement and rotation of objects in a three-dimensional system. QGIS pan map in layout, simultaneously with items on top, Finding features that intersect QgsRectangle but are not equal to themselves using PyQGIS. Can you elaborate a bit more. Which one should I use? In fact, it can be tempting to use the more common np.array. When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. note: A nicer looking and correct answer will still get accepted, thanks! Nice answer, @DanielF. To create a rotation matrix as a NumPy array for = 30 , it is simplest to initialize it with as follows: I've read on page 27 here that a 3x3 transform matrix can be just the nine dot products - thank you U. Auckland's prof. Kelly!. Number of rotations contained in this object. In C, why limit || and && to evaluate to booleans? The axis angle representation is then constructed by normalizing then multiplying by half the desired angle theta. rev2022.11.3.43004. We start by converting your axis and angle to a quaternion whose imaginary dimensions are given by your axis of rotation, and whose magnitude is given by half the angle of rotation in radians. Using levi-cevita symbols from @jaime's answer here and the matrix form of the Rodrigues formula here and some algebra based on k = (a x b)/sin(theta), Dropping another solution for bulk rotation of a Nx3x3 matrix. It still does not use Cython, but relies heavily on the efficiency of numpy. dataarray_like or string. What is the best way to sponsor the creation of new hyphenation patterns for languages without them? How can i extract files in the directory where they're located with the find command? : What is the best/easiest way to get the resulting vector when rotating the v vector around the axis? rotations using the from_euler method: The other representations also now return a stack of 3 rotations. Returns. Connect and share knowledge within a single location that is structured and easy to search. We reinsert the same three Euler angles and multiply the three elementary rotation matrices in the right sequence, We will get the following rotation matrix. However, it requires installation of my math3d package. What's that vector? how to speed up a vector cross product calculation. Demo of the Rotation Matrix Representation of a 3D Rotation. Beispiel: Eulerwinkel zu Rotationsvektor python - Yngve Moe Mar 4, 2018 at 22:09 Add a comment 0 I think you should consider a "vector" representation for your data, instead of the current "raster" representation. Numpy rotation matrix 3d code beispiel. Take a look at http://vpython.org/contents/docs/visual/VisualIntro.html. 1D array or list: In fact it can be converted to numpy.array: Multiple rotations can be composed using the * operator: Finally, it is also possible to invert rotations: These examples serve as an overview into the Rotation class and highlight Just as a unit complex number 'c' can represent all 2d rotations by c=exp(i * theta), a unit quaternion 'q' can represent all 3d rotations by q=exp(p), where 'p' is a pure imaginary quaternion set by your axis and angle. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. other format: It is also possible to initialize multiple rotations in a single instance rotating coordinate system via a quaternion, Implementing 3D vectors in Python: numpy vs x,y,z fields, How to generate regular points on cylindrical surface, Applying quaternion rotation to a vector time series, How to create an animation using a plotted 3D parametric function in python, How to calculate cartesian coordinates from dihedral angle in python, Matrix rotate a bunch of vectors about a point in Python. representation used for initialization. Pitch (Rotation about the y-axis) . Despite all these drawbacks, Euler angles are widely used today and are a very important reference point for those who work in the field of CAD modeling, 3D video game engines, and robotics and automation in general. This site uses Akismet to reduce spam. Consider a counter-clockwise rotation of 90 degrees about the z-axis. For more thorough examples of the range of input and That is, knowing the rotation matrix, it is possible to derive the three Euler angles. We added the print () function to display the result of the rotation. Rotation matrix - represented as a numpy 3-by-3 matrix R. Unit quaternion - represented as an object q of a custom quaternion class defined in the pyrotation module. I made a fairly complete library of 3D mathematics for Python{2,3}. rotation matrix: [[ 0.8660254 -0.5 ] [ 0.5 0.8660254]] vector v: [0 1] apply the rotation matrix r to x: r*v [-0.5 0.8660254] \begin . Here first, we will create two numpy arrays arr1 and arr2 by using the numpy.array() function. Irene is an engineered-person, so why does she have a heart problem? A Rotation instance can be initialized in any of the above formats and converted to any of the others. ( A)^k Would it be possible to use the math3d library to create 2D projections of 3D functions over an arbitrary axis more easily? Therefore the direction of the three elementary rotations will be the one shown in the following figure. Here is an elegant method using quaternions that are blazingly fast; I can calculate 10 million rotations per second with appropriately vectorised numpy arrays. Should we burninate the [variations] tag? Does either numpy or scipy have a compact vectorized function computing the appropriate rotation matrices in a way that avoids using for loops? By executing the code above, we obtain the values of the three Euler angles, which then correspond precisely to those entered initially. Here, too, you can perform the reverse operation. In addition to the XYZ sequence, another very common one is the one that refers to the convention associated with the ZYZ angles characterized by the following operations: The order of the elementary rotations changes the final result. This method gives you a seamless transformation between axis angle <---> 3d rotation operator simply by exp and log functions (yes log(q) just returns the axis-angle representation!). Find centralized, trusted content and collaborate around the technologies you use most. Rotates the matrix in Clockwise and Counterclockwise as per requirement. Clockwise & Counterclockwise Rotation of matrix using Numpy Library Clockwise & Counterclockwise Rotation of a matrix using Numpy Library. But which of the two is the positive one? from_euler Initialized from Euler angles. In C, why limit || and && to evaluate to booleans? By applying these rotations in sequence it can happen that one of the reference axes can collapse into another. Confusion between numpy, scipy, matplotlib and pylab. Meinen Namen, meine E-Mail-Adresse und meine Website in diesem Browser speichern, bis ich wieder kommentiere. axes(2,) array_like But even though the declarations of np.array objects from np.matrix look very similar, their behavior can be very different in many contexts. Parameters marray_like Array of two or more dimensions. The matrix relating to the overall rotation is calculated by multiplying the 3 matrices in the reverse order. output formats supported, consult the individual methods examples. Are Githyanki under Nondetection all the time? So for our example, we will start from a simple point on the X axis described by the following vector. Find centralized, trusted content and collaborate around the technologies you use most. Asking for help, clarification, or responding to other answers. What is the deepest Stockfish evaluation of the standard initial position that has ever been done? (Once you've mapped p to q you are still free to rotate around q). By executing we obtain the three angles, which are then the same ones we had inserted at the beginning. Deine E-Mail-Adresse wird nicht verffentlicht. Making statements based on opinion; back them up with references or personal experience. and A = I a be the skew-symmetric matrix associated to a, i.e. Are Githyanki under Nondetection all the time? OpenGL - find rotation vector for glRotate(), Julia AffineTransforms sign of rotation angle, Given a unit vector find the two angles of rotation that aligns that vector with an axis. SM1 - SM2 - SM3 - SM4 - SM5 - SM6 - SM7 - SM8 - SM9 - SM10 - SM11 - SM12 - SM13 - SM14 - SM15 - SM16 - SM17 - SM18 - SM19 - SM20 - SM21 - SM22 -, Leitfden und Kommentare zur Programmierung und zur Informatik im Allgemeinen. How can I best opt out of this? Now just discard the real element and you have your rotated vector! I know that for rotating w.r.t OX axis we can just compute new coords as: Shouldn't axis be x, y or z? Learn how your comment data is processed. array([[0. , 0. , 0.70710678, 0.70710678]. By running the code you will get the graphical representation. scipy.spatial.transform.Rotation.from_rotvec, http://vpython.org/contents/docs/visual/VisualIntro.html, http://git.automatics.dyndns.dk/?p=pymath3d.git, Making location easier for developers with new data primitives, Stop requiring only one assertion per unit test: Multiple assertions are fine, Mobile app infrastructure being decommissioned, 2022 Moderator Election Q&A Question Collection. A quaternion is a number with one real and 3 imaginary dimensions usually written as q = w + xi + yj + zk where 'i', 'j', 'k' are imaginary dimensions. Connect and share knowledge within a single location that is structured and easy to search. kinteger Number of times the array is rotated by 90 degrees. The argument is the rotation vector (a unit vector) multiplied by the rotation angle in rads. This is very neat, would you be so kind to add the equivalent for 2D? In an even simpler way, any rotation can be described by a sequence of three successive rotations, also called elementary rotations, which occur around one of the three coordinate axes X, Y and Z. LLPSI: "Marcus Quintum ad terram cadere uidet.". Hi, ljetbo, I think this sounds difficult, or just not very easy with math3d. Further, you seem to be talking about a scalar field over the plane (R(2)), whereas math3d deals with the Special Euclidean group (SE+(3)). Finding features that intersect QgsRectangle but are not equal to themselves using PyQGIS. , so it's time expensive, but readable and secure. Some coworkers are committing to work overtime for a 1% bonus. Deine E-Mail-Adresse wird nicht verffentlicht. Where the 3x3 components represent vector components in. But there is a limitation to the use of Euler angles, which is often referred to with the term Gimbal Lock. In this article we will see what Euler angles are, how they are calculated and how the rotational motion of a rigid body in three-dimensional Euclidean space can be calculated. Estimate a rotation to optimally align two sets of vectors. While special classes for rotations can be convenient, in some cases one needs rotation matrices (e.g. Stack Overflow for Teams is moving to its own domain! If we want to apply a rotation at this point it will be sufficient to multiply this vector precisely with the rotation matrix and thus obtain another vector. What value for LANG should I use for "sort -u correctly handle Chinese characters? For example in the case of rotation around the z axis, the rotation will be positive depending on the arrangement of the X and Y axes in the representation. In fact, it can be tempting to use the more common np.array. It provides a vector class which has a method A.rotate(theta,B). You may want to look up homogenous coordinates / transformations, the latter being the matrices used to transform a set of coordinates from one coordinate system to another. with: The following operations on rotations are supported: Indexing within a rotation is supported since multiple rotation transforms tcolorbox newtcblisting "! The 4 element vectors (w, x, y, z) are constructed as follows: First, a numpy array of 4 elements is constructed with the real component w=0 for both the vector to be rotated vector and the rotation axis rot_axis. 3D rotations can be represented using rotation matrices, which are 3 x 3 real orthogonal matrices with determinant equal to +1 [1]. Rotation.as_matrix() #. new_yaxis = -np.cross(new_xaxis, new_zaxis) # new axes: nnx, nny, nnz = new_xaxis, new_yaxis, new_zaxis . Not the answer you're looking for? Erforderliche Felder sind mit * markiert. numpy.rot90(m, k=1, axes=(0, 1)) [source] # Rotate an array by 90 degrees in the plane specified by axes. Copyright 2008-2022, The SciPy community. [ 1.00000000e+00, 2.22044605e-16, 0.00000000e+00], [ 0.00000000e+00, 0.00000000e+00, 1.00000000e+00]]), array([0. , 0. , 1.57079633]), array([0. , 0. , 0.70710678, 0.70710678]). To subscribe to this RSS feed, copy and paste this URL into your RSS reader. In a future article we will learn how rotations in space are implemented with Hamilton quaternions, what they are and how to use them in Python. This matrix represents the rotation from A to B in the base consisting of the following column vectors: normalized vector projection of B onto A: u = (A B)A (A B)A = A normalized vector rejection of B onto A: v = B (A B)A B (A B)A the cross product of B and A: w = B A Those vectors are all orthogonal, and form an orthogonal basis. All at the basis of the 3D engines with which many video games are developed. I used the following simple function: Thanks for contributing an answer to Stack Overflow! When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. notation) and projecting the normal dist on a plane that is also now rotated by theta in reference to xy? Short story about skydiving while on a time dilation drug. can be stored within a single Rotation instance. The list of rotation matrices U should satisfy: On a point-by-point basis, the problem reduces to being able to compute a rotation matrix for a rotation of some angle about some axis. For All with step-by-step practical tests developed in Python. The problem is that neither np.diag nor np.outer accept axis as an argument. . ? efficiently calculate list of 3d rotation matrices in numpy or scipy, Making location easier for developers with new data primitives, Stop requiring only one assertion per unit test: Multiple assertions are fine, Mobile app infrastructure being decommissioned, 2022 Moderator Election Q&A Question Collection. Which in Python is implemented as follows. Rotation of an angle around the y axis (current), Rotation of an angle around the z axis (current). matrixndarray, shape (3, 3) or (N, 3, 3) Shape depends on shape of inputs used for initialization. For example with rotations of 90 degrees or 180 degrees. To create Rotation objects use from_ methods (see examples below). In Python, the matrix object of the numPy library exists to express matrices. using any of the from_ functions. The virtual realities we often play with on our PCs are based on 3D engines, i.e. Therefore, by multiplying in the reverse order we obtain the matrix relating to the overall rotation: 158/5000We also see this case in Python. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA.
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