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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] |
Gamma
Gamma (; uppercase , lowercase ; ) is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter normally represents a voiced velar fricative , except before either of the two front vowels (/e/, /i/), where it represents a Voiced palatal fricative#Palatal, voiced palatal fricative ; while /g/ in foreign words is instead commonly transcribed as γκ). In the International Phonetic Alphabet and other modern Latin-alphabet based phonetic transcription#Alphabetic, phonetic notations, it represents the voiced velar fricative. History The Greek letter Gamma Γ is a grapheme derived from the Phoenician alphabet, Phoenician letter (''gīml'') which was rotated from the right-to-left script of Canaanite to accommodate the Greek language's writing system of left-to-right. The Canaanite grapheme represented the /g/ phoneme in the Canaanite language, and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Precession
Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In other words, if the axis of rotation of a body is itself rotating about a second axis, that body is said to be precessing about the second axis. A motion in which the second Euler angle changes is called '' nutation''. In physics, there are two types of precession: torque-free and torque-induced. In astronomy, ''precession'' refers to any of several slow changes in an astronomical body's rotational or orbital parameters. An important example is the steady change in the orientation of the axis of rotation of the Earth, known as the precession of the equinoxes. Torque-free or torque neglected Torque-free precession implies that no external moment (torque) is applied to the body. In torque-free precession, the angular momentum is a constant, ... [...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] |
Radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at the centre of a circle by an Circular arc, arc that is equal in length to the radius. The unit was formerly an SI supplementary unit and is currently a dimensionless unit, dimensionless SI derived unit,: "The CGPM decided to interpret the supplementary units in the SI, namely the radian and the steradian, as dimensionless derived units." defined in the SI as 1 rad = 1 and expressed in terms of the SI base unit metre (m) as . Angles without explicitly specified units are generally assumed to be measured in radians, especially in mathematical writing. Definition One radian is defined as the angle at the center of a circle in a plane that wikt:subtend, subtends an arc whose length equals the radius of the circle. More generally, the magnit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar example of modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in , but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is ''congruent'' to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12). Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval (mathematics)
In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real interval can contain neither endpoint, either endpoint, or both endpoints, excluding any endpoint which is infinite. For example, the set of real numbers consisting of , , and all numbers in between is an interval, denoted and called the unit interval; the set of all positive real numbers is an interval, denoted ; the set of all real numbers is an interval, denoted ; and any single real number is an interval, denoted . Intervals are ubiquitous in mathematical analysis. For example, they occur implicitly in the epsilon-delta definition of continuity; the intermediate value theorem asserts that the image of an interval by a continuous function is an interval; integrals of real functions are defined over an interval; etc. Interval ... [...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] |
Orientation (mathematics)
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is orientable if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an orientation of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loops in it, and coming back to the starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed into its own mirror image . A Möbius strip is an example of a non-orientable space. Various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds oft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gimbal
A gimbal is a pivoted support that permits rotation of an object about an axis. A set of three gimbals, one mounted on the other with orthogonal pivot axes, may be used to allow an object mounted on the innermost gimbal to remain independent of the rotation of its support (e.g. vertical in the first animation). For example, on a ship, the gyroscopes, shipboard compasses, stoves, and even drink holders typically use gimbals to keep them upright with respect to the horizon despite the ship's pitching and rolling. The gimbal suspension used for mounting compasses and the like is sometimes called a Cardan suspension after Italian mathematician and physicist Gerolamo Cardano (1501–1576) who described it in detail. However, Cardano did not invent the gimbal, nor did he claim to. The device has been known since antiquity, first described in the 3rd c. BC by Philo of Byzantium, although some modern authors support the view that it may not have a single identifiable inventor. His ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
