but I can't seem to find anything. environment. What is the Sandbox? This "Sandbox" is a place where Code Golf users can get feedback on prospective challenges they wish to post to the main page. My actual code/maths in short : 1) Rotate the position & orientation of lasers by roll & pitch. ]) orientation = Quaternion() translation = Vector3() scale = Vector3([1. Camera infos. A quaternion q*-1 corresponds to the same rotation as q; thus the sign of the reconstructed quaternion is arbitrary, and we return quaternions with positive w. # Rot is a 4 element tuple of the form [x,y,z,w] (which represents a quaternion) # If you plan to do matrix operations with these, make sure that you # convert them to numpy arrays with the correct dimensions!. There are some in tf. Homogeneous Transformation Matrices and Quaternions. moble / packages / quaternion 2018. Now, we rotate each point using the quaternion through the method rotatePoint as follows: #. quaternion) -> numpy. I am attempting to rotate a vector3 by a given quaternion. url bin/ bin/gcc/ bin/mingw/ code/M3Importer. stackoverflow. So the quaternion advantages would be:. Lindegren, U. To browse Academia. I am attempting to solve a problem involving Quaternions and 3D rotation. fftpack improvements. {"categories":[{"categoryid":387,"name":"app-accessibility","summary":"The app-accessibility category contains packages which help with accessibility (for example. Return type: float, numpy. @JaredSmith - the second I started reading this my first thought was "sounds like she may have been molested. I can use the following code to rotate object using accelerometer. I am attempting to rotate a vector3 by a given quaternion. There is a standard method to obtain a rotation matrix for this kind of angle-axis transformation, but please forget quaternions, they are difficult and in this very simple case, rather an overkill in my opinion. In experimenting with using video input I was very happy to find out how easy openCV makes accessing a computer’s webcam. It means that you have to first express the rotation about the axis as a quaternion (where $\phi$ is a randomly generated angle between 0 and $2\pi$):. Rotate(Vector3. import numpy as np from pyrr import Quaternion, Matrix33, Matrix44, Vector4 m = Matrix44 () * Quaternion # rotate a vector 4 by a matrix v = Matrix44. They are extracted from open source Python projects. x (and to fix a few bugs), and greatly expands the applications of quaternions. Rotate the new tooth pick on it's own axis, notice it isn't constrained to the x, y or z axis but is rotating on it's own 'new' and unique axis. To resolve this discrepancy, we swapped y and z components for the position and the quaternion for orientation. Welcome! This is the documentation for Numpy and Scipy. rotate_quaternion, does not result in properly rotated cylinder with ep1 = (0,0,0) and ep1 = (0,100,0) def FindRotation(ep1, ep2): First, create quaternion per. rotation_matrix_to_quaternion (rot_matrix) [source] ¶ Convert a rotation matrix-based representation of an orientation to a quaternion. InstAldebrn has sunset today, 2015 Sep 25! Enjoy this static snapshot! —@fasihsignal. Some Notes on Unit Quaternions and Rotation between quaternions more operations to rotate a vector using unit quaternions (although the details. # -*- coding: utf-8 -*-# transformations. Create a 3x3 rotation matrix from a unit 3-d vector and an angle in. url Website. Vector, point, quaternion, and matrix function arguments are expected to be "array like", i. 9+ Quaternion functions will work with vanilla Numpy, but they can also make use of this quaternion library. For debugging purposes, I have an object with which I've tested with all 3-axis rotation and movement and it's working as expected; the object in question as well as the camera class superseed a positions class for. ndarray] Converts a quaternion to angle axis format def quat_to_coeffs(quat: quaternion. I am working on a C# project where I need to rotate some vectors. Or, it can be used for input (e. 这是一篇翻译。原文：Python Numpy Tutorial翻译：我如果有什么问题欢迎留言私信指正探讨。正文：这篇教程是由Justin Johnson. 2 using the Boost 1. Calculate the rotation matrix required to rotate from one vector. It less useful for changing the point of view than the other rotation matrices. Helper function to rotate a vector by a quaternion def quat_to_angle_axis(quat: quaternion. Until now I was using a rather complicated solution using rotation matrices but now I also need support for quaternion rotation (and. ndarray – atomic coordinates for the ia atoms (including supercell offsets). OK, I Understand. By continuing to use Pastebin, you agree to our use of cookies as described in the Cookies Policy. Now imagine how significantly harder it would be to adjust 9 channels or curves for each individual limb (rotate, transform and bend x XYZ), instead of simply posing the character correctly. Robot control part 2: Jacobians, velocity, and force Jacobian matrices are a super useful tool, and heavily used throughout robotics and control theory. ndarray] Converts a quaternion to angle axis format def quat_to_coeffs(quat: quaternion. Direction Cosine Matrices. import numpy m=lambda s,p,t,q:[s*t-sum(p*q),s*q+t*p+numpy. org for current resources. Similarly, we will be able to express spin-weighted spherical harmonics directly in terms of quaternions, though with a simple translation to and from standard spherical coordinates. Choose a web site to get translated content where available and see local events and offers. Vector, point, quaternion, and matrix function arguments are expected to be "array like", i. Quaternions w+ix+jy+kz are represented as [w, x, y. Vector, point, quaternion, and matrix function arguments are expected to be "array like", i. To resolve this discrepancy, we swapped y and z components for the position and the quaternion for orientation. Quaternions ix+jy+kz+w are represented as [x, y, z, w]. Rotate a point about an arbitrary axis (3 dimensions) Written by Paul Bourke December 1992, Updated August 2002 Illustrative C code that implements the following algorithm is given here. It seens there is still some operations with quaternions that still blow my mind. Multiplying by a constant quaternion should rotate your data from one frame to the other, but I don't have a good enough intuitive feel for quaternions to give you the exact multiplier. a is a 3-by-3 matrix, with a plain float[9] array of uninitialized coefficients, b is a dynamic-size matrix whose size is currently 0-by-0, and whose array of coefficients hasn't yet been allocated at all. ndarray – The normalized vectors for qr and qd. I am attempting to solve a problem involving Quaternions and 3D rotation. I had fun finding out how to convert Eulers to Quaternions, you would think someone might have written a standard routine to do that, but apparently not, or not one I could find like numpy. quaternion_from_euler(). In next week’s blog post, we’ll be coding our very own RNN from the ground up in numpy and apply it to a language modeling task. The issue is I am not sure how to rotate/change the orientation of the model around it self and not the world axes. Multiplying. NumPy 利用ノート: Quaternion は NumPy を利用して実装されている。. html lammps-14Apr16/doc/Manual. Camera infos. Quaternions w+ix+jy+kz are represented as [w, x, y. An illustration of the problem is shown below for the simplest case of 3 corresponding points (the minimum required points to solve). None that I know, but it might be not too difficult writing one, given you do not ask for too heavy computations. In this example, we will start with the deformation gradient in Equation (4. Python, ROS geometry library, numpy. float64 precision. Tuple[float, numpy. Another way to express this is: q = Quat ( numpy. expand_dims taken from open source projects. The quaternion operation for determining v’ is by means of the formula. if it is connected to one incoming layer. pinv; assumes a is Hessian & returns no. 3D Geometric Transformation (Chapt. Conjugate()), and then retrieving the imaginary (vector) part. Here are the examples of the python api tf. The scalar componenet of is zero. The problem is that,the rotation around the X and Z axis are inverse (i rotate the device left,i get a right rotation etc) For what i know that quaternion rapresent orientation referring to an arbitrary ax. # define a known quaternion q = Quaternion (a = np. html lammps-14Apr16/doc/Manual. Gmsh is built around four modules: geometry, mesh, solver and post-processing. How do I create a view matrix directly from a quaternion and a position vector? because the quaternions rotate vectors Find symmetric pairs quickly in numpy. Releases into Groovy have been winding down and in the near future the build infrastructure for Groovy will be taken offline. The other thing is that the "rotation_mode" may change how information is updated. Return types are numpy arrays unless specified otherwise. rand ( 10 , 3 ) rotation = rowan. {"categories":[{"categoryid":387,"name":"app-accessibility","summary":"The app-accessibility category contains packages which help with accessibility (for example. Rotation Inversion. is like an axis to rotate around, quaternion components can be selected (as long as it's not zero) Interpolates between two 2x2 unitary NumPy matrices. The module also includes functions for the closely related angle, axis pair as a specification for rotation. Rotation Vectors. class Trackball: Trackball mechanism for interactive rotation Use the trackball utility to rotate a viewpoint around a fixed world-space coordinate (center). Considering that Quaternions are of little use without a set of functions to work with them, my question is: Where are the proper conversion functions for Quaternions?. Although a typical function call with rowan is, on average,. quaternion. This is mainly useful for specifying selections from phil parameters. rotate_quaternion, does not result in properly rotated cylinder with ep1 = (0,0,0) and ep1 = (0,100,0) def FindRotation(ep1, ep2): First, create quaternion per. The Python code for this calculation is. The primary purpose of this module is to provide compact. Because it is clear we are talking about vectors, and vectors only, we will omit the arrow used with vector notation. I am rotating n 3D shape using Euler angles in the order of XYZ meaning that the object is first rotated along the X axis, then Y and then Z. The azimuth ψ lies in the x and y quaternion terms and can be calculated using. The code was originally based on code by Martin Ling (which he wrote with help from Mark Wiebe), but has been rewritten with ideas from rational to work with both python 2. Use ↓ / ↑ to navigate through the list, Enter to go. represented as an axis ov er which the vehicle will rotate, and a rotation angle. Lindegren, U. Quaternions may not be cast to real or complex types. Bond Order¶ class freud. Also with the beginning of development for Jade Turtle, Groovy Galapagos will reach its end-of-life. Each has its own uses and drawbacks. Three-Dimensional Rotation Matrices 1. Rotations in 3D applications are usually represented in one of two ways: Quaternions or Euler angles. This post from stackoverflow was very helpful in creating a solid starting point to use quaternions to rotate a 3 dimensional cube using Python and OpenGL. For example, many signals are functions of 2D space defined over an x-y plane. If not click the link. The required quaternion can be calculated by multiplying these individual quaternions From our definitions the order of applying these rotations is heading,attitude then bank (about y,z then x). quaternions¶. ndarray and provides a way to manage anisotropic data. Readers & Writers ¶ This is the API PosFileWriter (rotate=False) The euler angle is converted to a quaternion and stored in the orientation of the frame. /ros-indigo-abb/ 07-Jun-2019 20:25 - ros-indigo-abb-driver/ 07-Jun-2019 20:24 - ros-indigo-abb-irb2400-moveit-config/ 07-Jun-2019 20:26. The module also includes functions for the closely related angle, axis pair as a specification for rotation. I am currently trying to use quaternions to rotate a block about an axis passing through any one of its edges, but I just can't seem to get that to work, I can make it rotate about its center but not about its axis!. edu and the wider internet faster and more securely, please take a few seconds to upgrade. acceleration. A lot of problems in statistical computing can be described mathematically using linear algebra. You can vote up the examples you like or vote down the ones you don't like. context_cache : openmmtools. The Quaternion Inverse block calculates the inverse for a given quaternion. i learnt what T from numpy is specifically the quaternions and octonions. Rotate(Vector3. rotVecByAxisAng(u, ax, theta) [source] ¶ Rotate the 3-vector u around axis ax for angle theta (radians), counter-clockwisely when looking at inverse axis direction. A tensor field can be decomposed into complex-valued functions by taking contractions with tangent vectors on the sphere and the normal to the sphere. And quaternions are 4-dimensional vectors used to describe 3 dimensional orientation, which provide stability and require less memory and compute but are more complicated to understand. Feb 11, 2017 · The space of 3-d rotations is represented in full by a sub-space of the quaternions, the space of unit quaternions, so you want to make sure that your quaternions are normalized. Anatomist: a python framework for interactive 3D visualization of neuroimaging data D Rivière1,2, D Geffroy1, I Denghien1, N Souedet1,3, Y Cointepas1,2 1IFR 49, Gif-sur-Yvette, France/2CEA, I2BM, Neurospin, Gif-sur-Yvette, France/3CEA, I2BM, MIRCEN, Fontenay-aux-. Lindegren, U. Hi, I am doing optimization on a vector of rotation angles tx,ty and tz using scipy. quaternion_matrix creates a 4x4 transformation matrix from the quaternion. N > 10000) where performance limitations become signi cant, how ever, rowan. Quaternions may not be cast to real or complex types. */ /* Homogeneous Transformation Matrices and Quaternions. rpm for CentOS 7 from EPEL repository. Rotating by the product lhs * rhs is the same as applying the two rotations in sequence: lhs first and then rhs, relative to the reference frame resulting from lhs rotation. Rotation Vectors. normalized, Vector3. I'm currently having an issue with moving the camera around the z-axis (roll). ndarray and provides a way to manage anisotropic data. void setSVOPosition (int frame_number) ¶. Tuple[float, numpy. environment. quaternion) -> numpy. Blog demonstrates how to update Python-embedded IDL graphics with gesture interactivity in order to enable mouse button & drag events in the Python GUI. 444 445 This function first generates a rotation matrix via euler_*_to_R() and then uses R_to_axis_angle() to convert t. I want to work towards writing some scripts that together take EBSD-based orientation data which are relative to a sample surface reference frame and rotate the data for visualization relative to a kinematic or geographic reference frame. So the quaternion advantages would be:. Not sure about Blender, but Maya has 1001 ways to fiddle with the tangents. Later we will see that x will correspond to the source parameters and the coefficient describing the attitude. Feb 11, 2017 · The space of 3-d rotations is represented in full by a sub-space of the quaternions, the space of unit quaternions, so you want to make sure that your quaternions are normalized. Holds a unit quaternion. quaternions ¶. Quaternion or set of quaternions, specified as an m-by-4 matrix containing m quaternions, or a single 1-by-4 quaternion. Similarly, we will be able to express spin-weighted spherical harmonics directly in terms of quaternions, though with a simple translation to and from standard spherical coordinates. The corresponding rotation matrix is obtained from quat. A little experimentation should suffice. When performing operations on images, you can now add the return_transform kwarg that will return both the new image and the transform that created the image. 19e-07 0 1 0. Tuple[float, numpy. import numpy as np from math import cos, sin, radians. Also with the beginning of development for Jade Turtle, Groovy Galapagos will reach its end-of-life. Below is a listing of all executables for which no corresponding man page is available yet. You're using an out-of-date version of Internet Explorer. Faster way to generate a rotation matrix?. You can do so in just the way you would normalize any 4-vector (i. The 3 Euler angles are. Class for handling dual quaternions and their interpolations. Euler angle <-> Rotation matrix <-> quaternion conversions. Note that the (4,4) element of a matrix can be used for uniform scaling, too. We will apply a rotation and a translation to a loaded point cloud and display then result. Because it is clear we are talking about vectors, and vectors only, we will omit the arrow used with vector notation. Index; About Manpages; FAQ; Service Information; stretch / Contents. Note: this is an entirely separate implementation from the PyOpenGL quaternion class. Another difference is that numpy matrices are strictly 2-dimensional, while numpy arrays can be of any dimension, i. To resolve this discrepancy, we swapped y and z components for the position and the quaternion for orientation. def quat_mult (q1, q2): """ Multiply two quaternions. rotate_vector function Rotates a 3D vectors based on the orientation of the quaternions. 19e-07 0 1 0. Quaternions and Key Bindings: Simple 3D Visualization in Matplotlib Sat 24 November 2012 Matplotlib is a powerful framework, but its 3D capabilities still have a lot of room to grow. Angles are in radians unless specified otherwise. There are some in tf. These functions are intended to provide basic support for common mathematical operations associated with displaying stimuli (e. By using the a and n keywords, we tell the function that we are representing the quaternion by defining the angle and axis of rotation. 3f1, I m trying to sync the rotation of a child gameobject, in local works perfectly fine but it doesnt show up in other clients. returns a quaternion to rotate about the x-axis by the input angle. I get the inverses to b & c. py # Copyright (c) 2006-2019, Christoph Gohlke # Copyright (c) 2006-2019, The Regents of the University of California. If you want, you can verify that rotating around a coordinate axis is a special case of this matrix. • direction of the vector is the axis to rotate about orientation matrix ( quaternion can be represented as matrix as well) quaternions or orientation matrix. Geometric Algebra (GA) is a universal algebra which subsumes complex algebra, quaternions, linear algebra and several other independent mathematical systems. html lammps-14Apr16/doc/Manual. y = linspace(a,b,n) generates a row vector y of n points linearly spaced between and. Tried with: PhysX 3. This sample shows how to use a RotateTransform3D and a ScaleTransform3D to rotate and change the scale of a 3-D model. This module implements Geometric Algebras (a. Since the norm of u is one, not all its entries can be zero. mplot3d import Axes3D import numpy as np # Rotate the u. First rotate around Z with alfa, then around the new Y with beta, then around new Z with gamma. Holds a unit quaternion. The linspace function generates linearly spaced vectors. Gmsh is built around four modules: geometry, mesh, solver and post-processing. Pictorial Presentation:. This paper describes a commonly used set of Tait-Bryan Euler angles, shows how to convert from Euler angles to a rotation matrix and back, how to rotate objects in both the forward and reverse direction, and how to concatenate multiple rotations into a single rotation matrix. ])) By using the a and n keywords, we tell the function that we are representing the quaternion by defining the angle and axis of rotation. fftpack improvements.  The rcount and ccount kwargs supersedes rstride and. quaternion (4x1 array of float) – Quaternion describing rotation after translation. Detecting rotation and line spacing of image of page of text using Radon transform - output. Quaternion components are stored as doubles. quaternions¶. The bond orientational order diagram (BOOD) is a way of studying the average local environments experienced by particles. "the breakthrough finally came on Monday 16 October 1843 in Dublin, when Hamilton was on his way to the Royal Irish Academy where he was going to preside at a council meeting. For large ar-rays (e. In this section we explain how to implement them. This lecture is meant to serve as a review of concepts you have covered in linear algebra courses. rotate_angles ( angles , amount ) [source] ¶. This is usually done by defining the zero-point of some coordinate with respect to the coordinates of the other frame as well as specifying the relative orientation. The 3 Euler angles are. understanding is that quaternions are just one way to solve the import numpy as np #Define data. import numpy as np # import quaternion EPSILON = 1e-10 # make sure quaternions are normalized (to be unit quaternions) # unit quaternions are a way to compactly represent 3D rotations # while avoiding singularities or discontinuities (e. This segment covers the clockwise rotation. The identity quaternion has real part 1 and vector part 0. matplotlib. Output is a list containing scalar part and vector part of the resulting quaternion, the latter being represented as. This implementation assumes that Numeric python will be available, and provides only those methods and helpers. Return types are numpy arrays unless specified otherwise. "the breakthrough finally came on Monday 16 October 1843 in Dublin, when Hamilton was on his way to the Royal Irish Academy where he was going to preside at a council meeting. 75i+-4j+-5k") The program will calculate the sum or product of the two quaternions. How to Find the Angle Between Two Vectors. array) - The first quaternion(s). transformations matrix to rotate about axis defined by point cs = ci*sk 1137 sc = si*ck 1138 ss = si*sk 1139 1140 quaternion = numpy. Re: composing Euler rotation matrices Instead of trying to decipher what someone wrote on a Wikipedia, why don't you look at a working piece of source code? e. The Quaternion Inverse block calculates the inverse for a given quaternion. For large ar-rays (e. rotate_quaternion, does not result in properly rotated cylinder with ep1 = (0,0,0) and ep1 = (0,100,0) def FindRotation(ep1, ep2): First, create quaternion per. A linear transformation of the plane $$\mathbb R^2$$ is a geometric transformation of the form. cstride for default sampling method for wireframe plotting. The code is fairly optimized as it is, bu. If, for some reason you are using NumPy's matrix subclass, then it is also possible to initialize with a string:. (Note that quaternion multiplication is not commutative. Angles are in radians unless specified otherwise. Indexing within a rotation is supported since multiple rotation transforms can be stored within a single Rotation instance. Although a typical function call with rowan is, on average,. rotation = Quaternion. Only applicable if the layer has exactly one input, i. polyfit() to fit polynomials to the points. Various distance metrics for quaternions. While python lists can contain values corresponding to different data types, arrays in python can only contain values corresponding to same data type. ]) orientation = Quaternion() translation = Vector3() scale = Vector3([1. Now, we rotate each point using the quaternion through the method rotatePoint as follows:. I have models written in SDF that I want to rotate at specific extrinsic angles before starting a Gazebo simulation with them. How to rotate a Quaternion Reply Quote. We use cookies for various purposes including analytics. You have to pass it the UAVO_ObjectName which is the class to cast the appropriate elements too. For large ar-rays (e. Linear Algebra and Linear Systems¶. You can omit any prefix from the symbol path; adding a. The code was originally based on code by Martin Ling (which he wrote with help from Mark Wiebe), but has been rewritten with ideas from rational to work with both python 2. translation ( 3x1 array of float, optional ) – Translation to be performed before rotation. x : pip install pyopengl pygame numpy Notable Mentions. quat1 (numpy. With the code above, there is a high probability that you’ll trigger an assertion like the following (meaning Eigen detects that the Quaternion coefficients are not aligned) :. Parameters-----move_set : list of tuples (MCMCMove, float_ Each tuple associate an MCMCMoves to its probability of being selected on apply(). ` The rcount and ccount kwargs supersedes rstride and. Here are the examples of the python api numpy. Return type: float, numpy. It relies on the quaternion extension to numpy found here. rotate_vectors (R, v, axis=-1) [source] ¶ Rotate vectors by given quaternions. linspace(start, stop, num=50, endpoint=True, retstep=False, dtype=None) # "Calculate the exponential of all elements in the input array. N > 10000) where performance limitations become significant, however,rowan outstrips pyquaternion by roughly two orders of magnitude and approaches the perfor-mance of numpy-quaternion. The converter can therefore also be used to normalize a rotation matrix or a quaternion. 2 Astronomical Coordinate Systems The coordinate systems of astronomical importance are nearly all. A lot of problems in statistical computing can be described mathematically using linear algebra. If you want, you can verify that rotating around a coordinate axis is a special case of this matrix. Hernandez The reference is usually made in the following way: Ref. Static archive of the old Ogre Addon forums. 19e-07 0 1 0. ndarray Converts a quaternion into the coeffs format the backend expects. How do I rotate the camera around the world with the touchpad ? Please note — don’t actually rotate the camera via the touchpad — it makes people sick fast. How to Find the Angle Between Two Vectors. save (bool, optional) – If this flag is True, each iteration image is saved in a sequential number. If one quaternion represents a rotation and the second quaternion represents another rotation, what does their multiplication represent? Many web sites talk about formulas and matrices, but I want to know what exactly are we doing multiplication for?. the vector to rotate: Returns. {"categories":[{"categoryid":387,"name":"app-accessibility","summary":"The app-accessibility category contains packages which help with accessibility (for example. Helper function to rotate a vector by a quaternion def quat_to_angle_axis(quat: quaternion. acceleration. Similarly, we will be able to express spin-weighted spherical harmonics directly in terms of quaternions, though with a simple translation to and from standard spherical coordinates. 3D Geometric Transformation (Chapt. quaternion. We also have the determinant. Stay tuned until then… References. Quaternion components are stored as doubles. Hi guys I would have though there would be a function within the Vector3 class or similar to convert a position vector into the equivalent rotation vector in eulerangles. C# Sharp Basic: Exercise-50 with Solution. I can use the following code to rotate object using accelerometer. Rotations in Space: Euler Angles, Matrices, and Quaternions¶ This notebook demonstrates how to use clifford to implement rotations in three dimensions using euler angles, rotation matices and quaternions. Retrieves the input mask tensor(s) of a layer. And quaternions are 4-dimensional vectors used to describe 3 dimensional orientation, which provide stability and require less memory and compute but are more complicated to understand. This should show the age of the page Package. The class se3. quaternion_matrix creates a 4x4 transformation matrix from the quaternion. Two rotation quaternions can be combined into one equivalent quaternion by the relation: ′ = in which q′ corresponds to the rotation q 1 followed by the rotation q 2. 2 Creating Data Types. Another difference is that numpy matrices are strictly 2-dimensional, while numpy arrays can be of any dimension, i. The following are code examples for showing how to use tf. It might be convenient to use the formulation given here which uses quaternions to simplify the calculation of the rotation matrix. Complex types may also be cast to quaternions, with their single imaginary component becoming the first imaginary component of the quaternion. If I understand correctly multiplying two quaternions together is equivalent to applying the rotation of the first quaternion then the rotation of the second. All input is normalized to unit quaternions and may therefore mapped to different ranges. animation, posing, rendering, etc. Gets the position of each bone in a skeletal mesh as a quaternion. $$x \in \{s, a\}$$, s being the the set of 5 source parameter and a being the coeffeicients of the splines for each of the four quaternions parameters describing the attitude of the satellite at each time). Note: all angles are assumed to be specified in radians. Linear Algebra and Linear Systems¶. where indicates the quaternion multiplcation and is a pure quaternion representation of the vector. A 3-vector can also be represented by a Quaternion object who's scalar part is 0 and vector part is the required 3-vector.