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Conversion Between Quaternions And Euler Angles
Spatial rotations in three dimensions can be parametrized using both Euler angles and unit quaternions. This article explains how to convert between the two representations. Actually this simple use of "quaternions" was first presented by Euler some seventy years earlier than Hamilton to solve the problem of magic squares. For this reason the dynamics community commonly refers to quaternions in this application as "Euler parameters". Definition There are two representations of quaternions. This article uses the more popular Hamilton. A quaternion has 4 real values: (the real part or the scalar part) and (the imaginary part). Defining the norm of the quaternion as follows: \lVert q \rVert = \sqrt A ''unit quaternion'' satisfies: \lVert q \rVert = 1 We can associate a quaternion with a rotation around an axis by the following expression :\mathbf_w = \cos(\alpha/2) :\mathbf_x = \sin(\alpha/2)\cos(\beta_x) :\mathbf_y = \sin(\alpha/2)\cos(\beta_y) :\mathbf_z = \sin(\alpha/ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation Formalisms In Three Dimensions
In geometry, there exist various rotation formalisms to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational (or angular) kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space. According to Euler's rotation theorem, the rotation of a rigid body (or three-dimensional coordinate system with a fixed origin) is described by a single rotation about some axis. Such a rotation may be uniquely described by a minimum of three real parameters. However, for various reasons, there are several ways to represent it. Many of these representations use more than the necessary minimum of three parameters, although each of them still ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right-hand Rule
In mathematics and physics, the right-hand rule is a Convention (norm), convention and a mnemonic, utilized to define the orientation (vector space), orientation of Cartesian coordinate system, axes in three-dimensional space and to determine the direction of the cross product of two Euclidean vector, vectors, as well as to establish the direction of the force on a Electric current, current-carrying conductor in a magnetic field. The various right- and left-hand rules arise from the fact that the three axes of three-dimensional space have two possible orientations. This can be seen by holding your hands together with palms up and fingers curled. If the curl of the fingers represents a movement from the first or x-axis to the second or y-axis, then the third or z-axis can point along either right thumb or left thumb. History The right-hand rule dates back to the 19th century when it was implemented as a way for identifying the positive direction of coordinate axes in three dime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation In Three Dimensions
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersecting anywhere inside or outside the figure at a ''center of rotation''. A solid figure has an infinite number of possible axes and angles of rotation, including chaotic rotation (between arbitrary orientations), in contrast to rotation around a axis. The special case of a rotation with an internal axis passing through the body's own center of mass is known as a spin (or ''autorotation''). In that case, the surface intersection of the internal ''spin axis'' can be called a ''pole''; for example, Earth's rotation defines the geographical poles. A rotation around an axis completely external to the moving body is called a revolution (or ''orbit''), e.g. Earth's orbit around the Sun. The ends of the external ''axis of revolution'' can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation Matrix
In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation (mathematics), rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end rotates points in the plane counterclockwise through an angle about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates , it should be written as a column vector, and matrix multiplication, multiplied by the matrix : : R\mathbf = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end \begin x \\ y \end = \begin x\cos\theta-y\sin\theta \\ x\sin\theta+y\cos\theta \end. If and are the coordinates of the endpoint of a vector with the length ''r'' and the angle \phi with respect to the -axis, so that x = r \cos \phi and y = r \sin \phi, then the above equations become the List of trigonometric identities#Angle sum and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Angles
The Euler angles are three angles introduced by Leonhard Euler to describe the Orientation (geometry), orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478PDF/ref> They can also represent the orientation of a mobile frame of reference in physics or the orientation of a general Basis (linear algebra), basis in three dimensional linear algebra. Classic Euler angles usually take the inclination angle in such a way that zero degrees represent the vertical orientation. Alternative forms were later introduced by Peter Guthrie Tait and George H. Bryan intended for use in aeronautics and engineering in which zero degrees represent the horizontal position. Chained rotations equivalence Euler angles can be defined by elemental geometry or by composition of rotations (i.e. chained rotations). The geometrical definition demonstrates that three consecutive ''elemental rotations'' (rotatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation Operator (vector Space)
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersecting anywhere inside or outside the figure at a ''center of rotation''. A solid figure has an infinite number of possible axes and angles of rotation, including chaotic rotation (between arbitrary orientations), in contrast to rotation around a axis. The special case of a rotation with an internal axis passing through the body's own center of mass is known as a spin (or ''autorotation''). In that case, the surface intersection of the internal ''spin axis'' can be called a ''pole''; for example, Earth's rotation defines the geographical poles. A rotation around an axis completely external to the moving body is called a revolution (or ''orbit''), e.g. Earth's orbit around the Sun. The ends of the external ''axis of revolution'' can b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Definition
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitions (which try to list the objects that a term describes).Lyons, John. "Semantics, vol. I." Cambridge: Cambridge (1977). p.158 and on. Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions. In mathematics, a definition is used to give a precise meaning to a new term, by describing a condition which unambiguously qualifies what the mathematical term is and is not. Definitions and axioms form the basis on which all of modern mathematics is to be constructed. Basic terminology In modern usage, a definition is something, typically expressed in words, that at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conversion
Conversion or convert may refer to: Arts, entertainment, and media * ''The Convert'', a 2023 film produced by Jump Film & Television and Brouhaha Entertainment * "Conversion" (''Doctor Who'' audio), an episode of the audio drama ''Cyberman'' * "Conversion" (''Stargate Atlantis''), an episode of the television series ''Stargate Atlantis'' * "The Conversion" (''The Outer Limits''), a 1995 episode of the television series ''The Outer Limits'' * " Chapter 19: The Convert", an episode of the television series ''The Mandalorian'' Business and marketing * Conversion funnel, the path a consumer takes through the web toward or near a desired action or conversion * Conversion marketing, when a website's visitors take a desired action * Converting timber to commercial lumber Computing, science, and technology * Conversion of units, conversion between different units of measurement Computing and telecommunication * CHS conversion of data storage, mapping cylinder/head/sector tuples ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gimbal Lock
Gimbal lock is the loss of one degree of freedom (mechanics), degree of freedom in a multi-dimensional mechanism at certain alignments of the axes. In a three-dimensional three-gimbal mechanism, gimbal lock occurs when the axes of two of the gimbals are driven into a parallel configuration, "locking" the system into rotation in a degenerate two-dimensional space. The term can be misleading in the sense that none of the individual gimbals is actually restrained. All three gimbals can still rotate freely about their respective axes of suspension. Nevertheless, because of the parallel orientation of two of the gimbals' axes, there is no gimbal available to accommodate rotation about one axis, leaving the suspended object effectively locked (i.e. unable to rotate) around that axis. The problem can be generalized to other contexts, where a coordinate system loses definition of one of its variables at certain values of the other variables. Gimbals A gimbal is a ring that is suspen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atan2
In computing and mathematics, the function (mathematics), function atan2 is the 2-Argument of a function, argument arctangent. By definition, \theta = \operatorname(y, x) is the angle measure (in radians, with -\pi 0, \\[5mu] \arctan\left(\frac y x\right) + \pi &\text x < 0 \text y \ge 0, \\[5mu] \arctan\left(\frac y x\right) - \pi &\text x < 0 \text y < 0, \\[5mu] +\frac &\text x = 0 \text y > 0, \\[5mu] -\frac &\text x = 0 \text y < 0, \\[5mu] \text &\text x = 0 \text y = 0. \end Instead of the tangent, it can be convenient to use the half-tangent as a representation of an angle, partly because the angle has a unique half-tangent, (See tangent half-angle formula.) The expression with in the denominator should be used when and to avoid possible loss of significance in computing . When an function is unavailable, it can be computed as twice the arctangent of the half-tangent . That is, |
Right Angle
In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. The term is a calque of Latin ''angulus rectus''; here ''rectus'' means "upright", referring to the vertical perpendicular to a horizontal base line. Closely related and important geometrical concepts are perpendicular lines, meaning lines that form right angles at their point of intersection, and orthogonality, which is the property of forming right angles, usually applied to Euclidean vector, vectors. The presence of a right angle in a triangle is the defining factor for right triangles, making the right angle basic to trigonometry. Etymology The meaning of ''right'' in ''right angle'' possibly refers to the Classical Latin, Latin adjective ''rectus'' 'erect, straight, upright, perp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arctan
In mathematics, the inverse trigonometric functions (occasionally also called ''antitrigonometric'', ''cyclometric'', or ''arcus'' functions) are the inverse functions of the trigonometric functions, under suitably restricted domains. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry. Notation Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: , , , etc. (This convention is used throughout this article.) This notation arises from the following geometric relationships: when measuring in radians, an angle of radians will correspond to an arc whose length is , where is the radius of the circle. Thus in the unit circle, the cosine of x f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
