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- Dictionaryquaternion/kwəˈtəːnɪən/
noun
- 1. a complex number of the form w + xi + yj + zk, where w, x, y, z are real numbers and i, j, k are imaginary units that satisfy certain conditions.
- 2. a set of four people or things. rare
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Quaternions are a four-dimensional number system that extends the complex numbers and can represent rotations in three-dimensional space. Learn about their discovery by Hamilton, their algebraic properties, and their uses in mathematics and physics.
The quaternion is implemented as Quaternion[a, b, c, d] in the Wolfram Language package Quaternions` where however , , , and must be explicit real numbers. Note also that NonCommutativeMultiply (i.e., **) must be used for multiplication of these objects rather than usual multiplication (i.e., *).
Learn how to use quaternions, a 4-tuple algebra, to represent and compose rotations in 3D. See the definition, properties and examples of quaternion operators, and how they relate to rotation matrices.
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Oct 10, 2021 · A quaternion of the form \(a=a+0i+0j+0k\leftrightarrow (a,0,0,0)\) is called a scalar quaternion or a real quaternion. A quaternion of the form \(xi+yj+zk\leftrightarrow (0,x,y,z)\) is called a pure quaternion or an imaginary quaternion.
- Overview
- Basics
- Rotation and Orientation
- Anchoring Quaternions to Coordinates and Directions
- Rotating A Vector Using A Quaternion
- Specifying A Quaternion That Describes The Rotation from One Vector to Another
- Specifying A Quaternion That Describes Rotation Around An Axis
- Rotating An Orientation, and Composing Rotations
- Finding The Quaternion That Rotates One Orientation to Another
- Combining These Operations
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This article demonstrates how to use quaternions for practical applications. In it, we build intuition about how to use them as building blocks to solve engineering and geometry problems. Applications include computer graphics, attitude systems for air and spacecraft, biomechanics, and structural chemistry models. You may have read that quaternions...
A quaternion is described by a set of 4 numbers, labeled w, x, y, and z. In code, this may be represented by a struct (or class) with 4 floating-point fields, and functions (eg methods) that perform various operations, and overload arithmetic operators. We can format it like this: Q=[w,x,y,z] All quaternions in this article are unit quaternions; th...
Orientation and rotation quaternions are equivalent: The distinction is that ones used for orientation describe the operation that rotates the identity quaternion to the orientation. The identity quaternion is defined as having a wcomponent equal to 1, and all others equal to 0. The distinction is how you use them; it's important to keep track of w...
Let's examine a 3d model of a boat. Its orientation is stored as a quaternion; one that represents the result of its rotation on the unit quaternion. How do we orient it so that its hull is pointed down? The key in how the model is defined, and how the graphics engine is set up. If you'd like the unit quaternion to represent an upright and level or...
Rotating a vector is one of the most common applications of quaternions, and is a building block for other operations. Here's how it's done, assuming a unit quaternion and unit vector. ∘means compose operations; ie multiplying quaternions by other quaternions, and vectors. v^′=Q∘v^∘Q−1 Where v^ is the original vector, v^′ is the rotated vector, Q i...
There are multiple rotations that can describe this; the equation above describes the shortest rotation. R is the quaternion that rotates unit vector v1 to unit vector v2. w=1+v1⋅v2[x,y,z]=v1×v2R=[w,x,y,z]||[w,x,y,z]|| Note: The above approach will fail if the vectors are parallel; to correct this in code, check that the dot product is equal to 1 o...
To create a quaternion R that describes the rotation of amount θ around axis v, use this equation: w=cosθ2x=sinθ2vxy=sinθ2vyz=sinθ2vz R=[w,x,y,z] One example of this use is changing the orientation of computer-graphics objects in response to user input. For example, modifying camera orientation in response to mouse movements. You could create a...
To rotate a quaternion representing an orientation, perform quaternion multiplication with the rotation quaternion on the left: O′=R∘O To compose multiple rotations, perform quaternion multiplication with the first rotation to apply on the right, and the last on the left. When multiple rotations are composed in this way, they are applied right to l...
This is a re-arrangement of the quaternion multiplication procedure above that rotates an orientation, using this division definition: Q1Q2=Q1∘Q2−1 R=O2∘O1−1
The operations above are powerful when combined - they can solve many orientation and orientation problems. For example, to align a 3D model with a certain direction, construct a quaternion that rotates the model's desired axis (for example, the up vector) with a vector pointing in the desired direction: This quaternion can be used as the orientati...
Learn how to use quaternions for practical applications in engineering and geometry. This article explains quaternions as building blocks for orientation and rotation transformations, and provides examples and references.
Learn about the history, definition and properties of quaternions, a 4-dimensional algebra that can do rotations in 3D. See how quaternions are related to complex numbers, vectors and Euler's formula.
The quaternions were invented by Sir William Rowan Hamilton about 1850. Hamilton was perhaps the first to note that complex numbers could be thought of as a way to multiply points in the plane. He then had the idea of trying to find a way to multiply points in R3 so that the field axioms would be satisfied.
