**representation of rotations**- A representation of rotations is a particular formalism, either algebraic or geometric, used to parametrize a rotation map. (wikipedia.org)
- The physics of the rotation around a fixed axis is mathematically described with the axis-angle representation of rotations. (wikipedia.org)

**orthogonal**- The axis (where present) and the plane of a rotation are orthogonal . (wikipedia.org)
- In some literature, the term rotation is generalized to include improper rotations, characterized by orthogonal matrices with determinant −1 (instead of +1). (wikipedia.org)
- The set of all orthogonal matrices of size n with determinant +1 forms a group known as the special orthogonal group SO(n), one example of which is the rotation group SO(3). (wikipedia.org)
- Suggested by Henry Felix Kaiser in 1958, it is a popular scheme for orthogonal rotation (where all factors remain uncorrelated with one another). (wikipedia.org)
- Factor analysis Empirical orthogonal functions Q methodology Rotation matrix Henry F. Kaiser (September 1958). (wikipedia.org)

**Cyclonic Rotation**- Cyclonic rotation or circulation is movement in the same direction as the Earth's rotation, as opposed to anticyclonic rotation. (wikipedia.org)
- The Coriolis effect causes cyclonic rotation to be in a counterclockwise direction in the northern hemisphere, and clockwise in the southern hemisphere. (wikipedia.org)

**rotate**- 2. Rotate so that the rotation axis is aligned with one of the principle coordinate axes. (siggraph.org)
- In SAS varimax rotation is available in PROC FACTOR using ROTATE = VARIMAX. (wikipedia.org)

**Euclidean**- Any direct Euclidean motion can be represented as a composition of a rotation about the fixed point and a translation. (wikipedia.org)
- In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. (wikipedia.org)

**algebraic**- Rotation matrices provide an algebraic description of such rotations, and are used extensively for computations in geometry, physics, and computer graphics. (wikipedia.org)

**rigid body**- Mathematically, a rotation is a rigid body movement which, unlike a translation, keeps a point fixed. (wikipedia.org)
- All rigid body movements are rotations, translations, or combinations of the two. (wikipedia.org)

**consequence of the rotation**- Another consequence of the rotation of a planet is the phenomenon of precession. (wikipedia.org)
- A consequence of the rotation method would be the aesthete's lack of commitment to any one thing. (wikipedia.org)

**clockwise rotation**- A clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude. (wikipedia.org)

**Thus**- Thus, the rotations around a point/axis form a group. (wikipedia.org)
- The choice of what opioid drug to use in which patient thus tends to be a balance between many different factors that must be considered, and the need for opioid rotation in chronic pain patients makes it advantageous for a wide range of different opioid drugs to be available, even though they may be broadly equivalent in action when used in shorter term treatment. (wikipedia.org)
- citation needed] Thus, tire rotation needs to occur more frequently for front-wheel drive vehicles. (wikipedia.org)

**vector**- Rotations of (affine) spaces of points and of respective vector spaces are not always clearly distinguished. (wikipedia.org)
- The former are sometimes referred to as affine rotations (although the term is misleading), whereas the latter are vector rotations . (wikipedia.org)
- The angular velocity vector (an axial vector) also describes the direction of the axis of rotation. (wikipedia.org)
- To perform the rotation using a rotation matrix R, the position of each point must be represented by a column vector v, containing the coordinates of the point. (wikipedia.org)
- Since matrix multiplication has no effect on the zero vector (the coordinates of the origin), rotation matrices describe rotations about the origin. (wikipedia.org)

**improper**- The " improper rotation " term refers to isometries that reverse (flip) the orientation. (wikipedia.org)

**Mathematically**- Mathematically, a rotation is a map . (wikipedia.org)

**differ**- Rotations in three-dimensional space differ from those in two dimensions in a number of important ways. (wikipedia.org)
- Is there some way to say how much their rotations differ? (opengl.org)
- Tire rotation is often recommended every 5,000 mi (≈8,000 km) to 8,000 mi(≈13,000 km), however this will differ between tire manufacturers and car makes. (wikipedia.org)

**inverse**- 4. Perform inverse rotation of 2. (siggraph.org)
- The reverse (inverse) of a rotation is also a rotation. (wikipedia.org)

**Givens**- Givens rotation Golub, Gene H. (wikipedia.org)

**Earth's**- Unlike Earth's rotation, a chaotic rotation may not have a fixed axis or period. (wikipedia.org)

**anticyclonic**- Note that while the supercell itself is rotating in a cyclonic direction, the couplet is forced into an anticyclonic rotation due to the forward flank downdraft and inflow intersecting. (wikipedia.org)

**orientation**- The rotation group is a point stabilizer in a broader group of (orientation-preserving) motions . (wikipedia.org)
- But a (proper) rotation also has to preserve the orientation structure . (wikipedia.org)
- In 2D the axis of rotation is always perpendicular to the xy plane, i.e., the Z axis, but in 3D the axis of rotation can have any spatial orientation. (siggraph.org)
- A rotation is simply a progressive radial orientation to a common point. (wikipedia.org)
- These combine proper rotations with reflections (which invert orientation). (wikipedia.org)

**counterclockwise**- The examples in this article apply to active rotations of vectors counterclockwise in a right-handed coordinate system by pre-multiplication. (wikipedia.org)
- If a standard right-handed Cartesian coordinate system is used, with the x-axis to the right and the y-axis up, the rotation R(θ) is counterclockwise. (wikipedia.org)

**reflections**- A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (n − 1)-dimensional flat of fixed points in a n-dimensional space. (wikipedia.org)

**crops**- Crop rotation , the successive cultivation of different crops in a specified order on the same fields, in contrast to a one-crop system or to haphazard crop successions. (britannica.com)
- Throughout human history, wherever food crops have been produced, some kind of rotation cropping appears to have been practiced. (britannica.com)
- The first number in a rotation ratio refers to cultivated row crops, the second to close-growing grains, and the third to sod-forming, or rest, crops. (britannica.com)
- It is desirable to include legumes alone or in mixtures with nonlegume sod-forming crops as a regular crop in many field rotations. (britannica.com)
- In addition to the many beneficial effects on soils and crops, well-planned crop rotations also provide the business aspects of farming with advantages. (britannica.com)
- Kierkegaard likens the rotation method to the crop rotation used in agriculture, in which the soil is constantly rotated in order to maximize the use of nutrients for the crops. (wikipedia.org)

**angle**- For rotation about X axis we need to find cos A, sin A where A = angle between projection of U (in YZ plane) and Z axis. (siggraph.org)
- How about computing the angle between x and x', y and y', and z and z', where x, y, and z are the rows of the target matrix, and x', y', z' are the rows of the rotation matrix attempting to match the target matrix? (opengl.org)
- but to apply the rotation, the angle itself is not required. (wikipedia.org)
- As previously remarked, we need never explicitly compute the rotation angle ϑ. (wikipedia.org)
- They constitute a mixed axes of rotation system, where the first angle moves the line of nodes around the external axis z, the second rotates around the line of nodes and the third one is an intrinsic rotation around an axis fixed in the body that moves. (wikipedia.org)
- Axis-angle representation (pictured at the right) specifies an angle with the axis about which the rotation takes place. (wikipedia.org)
- An alternative convention uses rotating axes, and the above matrices also represent a rotation of the axes clockwise through an angle θ. (wikipedia.org)

**great rotation**- Stephen Davies, CEO, Javelin Wealth Management says talk of a great rotation out of bonds into equities has been overdone. (cnbc.com)
- A new Great Rotation? (reuters.com)
- Michael Hartnett and Kate Moore, equity strategists at Bank of America Merrill Lynch, call the coming move to stocks "The Great Rotation" and argue it will affect profoundly what does well and what poorly. (reuters.com)
- The Great Rotation argues for a complete shift into the assets that have underperformed in recent years, a move that would clearly compromise the leadership of growth, yield and quality," they write in a note to clients. (reuters.com)
- To be sure, the Great Rotation will be a powerful trend once it is established, and like the move out of stocks will be self-reinforcing. (reuters.com)

**axes**- We will first look at rotation around the three principle axes (X, Y, Z) and then about an arbitrary axis. (siggraph.org)
- A three-dimensional object can always be rotated around an infinite number of imaginary lines called rotation axes (/ˈæksiːz/ AK-seez). (wikipedia.org)
- Rotations around the x, y and z axes are called principal rotations. (wikipedia.org)
- A basic rotation (also called elemental rotation) is a rotation about one of the axes of a Coordinate system. (wikipedia.org)

**commutative**- Rotations in three dimensions are generally not commutative , so the order in which rotations are applied is important even about the same point. (wikipedia.org)
- Note that the two-dimensional case is the only non-trivial (i.e. not one-dimensional) case where the rotation matrices group is commutative, so that it does not matter in which order multiple rotations are performed. (wikipedia.org)

**Practice**- While there is good evidence for the efficacy of opioid rotation as a treatment approach in general, there is less evidence for what particular opioid analgesics are most suitable, and in practice the choice of opioid drugs used depends on many factors such as patient characteristics, prescriber preferences and safety. (wikipedia.org)
- Some customers are fully aware of the practice of rotation, and will reach towards the back of the shelf in order to get newer (and therefore slightly better) produce. (wikipedia.org)
- Tire rotation is the practice of moving the wheels and tires of an automobile from one position to another, to ensure even tire wear. (wikipedia.org)

**Coordinate**- Special case: The rotation axis is parallel to a principle coordinate axis. (siggraph.org)
- 3. Perform rotation of object about coordinate axis. (siggraph.org)

**factor**- Tom Simpson, from the National Billiard Academy, discusses why luck is a small factor in American rotation. (wikipedia.org)
- If these conditions hold, the factor loading matrix is said to have "simple structure," and varimax rotation brings the loading matrix closer to such simple structure (as much as the data allow). (wikipedia.org)
- A summary of the use of varimax rotation and of other types of factor rotation is presented in this article on factor analysis. (wikipedia.org)
- https://support.sas.com/documentation/cdl/en/statug/63347/HTML/default/viewer.htm#statug_factor_sect029.htm Factor rotations in Factor Analyses by Herve Abdi About Varimax Properties of Principal Components http://www2.imm.dtu.dk/pubdb/views/edoc_download.php/4041/pdf/imm4041.pdf This article incorporates public domain material from the National Institute of Standards and Technology website http://www.nist.gov. (wikipedia.org)

**displaystyle**- From the definition, a π {\displaystyle \pi } -consistent rotation map is consistently labeled. (wikipedia.org)

**refers**- Left rotation refers to the following In an array, moving all items to the next lower location. (wikipedia.org)

**compute**- Now compute Ry(B) for rotation to z-axis. (siggraph.org)

**argues**- The rotation method, Kierkegaard argues, would eventually lead aesthetes to a state of despair, because all activities, no matter how unique or new, will eventually become boring: I don't feel like doing anything. (wikipedia.org)

**geometry**- Rotation in mathematics is a concept originating in geometry . (wikipedia.org)

**translation**- A plane rotation around a point followed by another rotation around a different point results in a total motion which is either a rotation (as in this picture), or a translation . (wikipedia.org)
- Now we can perform the first translation (of the rotation axis to pass through the origin) by using the matrix T (-x1, -y1, -z 1), i.e., move the point P1 to the origin. (siggraph.org)
- Since you just said rotation difference, I'm assuming there is no translation difference, or you don't care about that. (opengl.org)
- However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than a rotation, e.g. a translation. (wikipedia.org)

**spatial**- That is to say, any spatial rotation can be decomposed into a combination of principal rotations. (wikipedia.org)

**invariant**- The plane of rotation is a plane that is invariant under the rotation. (wikipedia.org)
- Preserving orthogonality requires that it is a rotation that leaves the sub-space invariant. (wikipedia.org)

**transformations**- These two types of rotation are called active and passive transformations . (wikipedia.org)

**axis passes**- 1. Translate so that rotation axis passes through origin. (siggraph.org)

**principle**- So the first one who migrated the principle of Rotation Curation to Twitter was Maxim Loick, a.k.a @Pausanias. (wikipedia.org)

**pitch**- The overworked and unhealthy bullpen would suffer if bother Moyer and Martinez, neither of whom is likely to regularly pitch deep into games, were both in the rotation. (philly.com)
- In flight dynamics, the principal rotations are known as yaw, pitch, and roll (known as Tait-Bryan angles). (wikipedia.org)

**different**- Rotations about different points, in general, do not commute. (wikipedia.org)
- citation needed] Additionally, some vehicles are designed (or retrofitted) with front and rear wheels of different sizes and unidirectional rotation treads, making rotation impossible[citation needed]. (wikipedia.org)

**order**- In philosopher Søren Kierkegaard's Either/Or, the rotation method is the mechanism used by higher level aesthetes in order to avoid boredom. (wikipedia.org)
- Rotation maps were first introduced by Reingold, Vadhan and Wigderson ("Entropy waves, the zig-zag graph product, and new constant-degree expanders", 2002) in order to conveniently define the zig-zag product and prove its properties. (wikipedia.org)

**typically**- A rotation about an external point, e.g. the Earth about the Sun, is called a revolution or orbital revolution, typically when it is produced by gravity. (wikipedia.org)
- In Linux log rotation is typically performed using the logrotate command. (wikipedia.org)
- Log rotation is typically performed according to a schedule (e.g., hourly, daily, weekly) or when a log file reaches a certain size. (wikipedia.org)
- The rotation pattern is typically moving the back wheels to the front, and the front to the back, but crossing them when moving to the back. (wikipedia.org)

**mathematics**- In mathematics, a rotation map is a function that represents an undirected edge-labeled graph, where each vertex enumerates its outgoing neighbors. (wikipedia.org)

**general**- Also, unlike the two-dimensional case, a three-dimensional direct motion, in general position , is not a rotation but a screw operation . (wikipedia.org)
- It was a low-risk, high reward move intended to replace Rodrigo Lopez in the rotation, and insure against the possibility that general manager Ruben Amaro Jr. would not be able to add a pitcher before yesterday's trade deadline. (philly.com)
- In the general case, rotation about an arbitrary axis is more complicated. (siggraph.org)
- In general, most rotations are confined to time limits of eight years or less. (britannica.com)

**particularly**- While revolution is often used as a synonym for rotation, in many fields, particularly astronomy and related fields, revolution, often referred to as orbital revolution for clarity, is used when one body moves around another while rotation is used to mean the movement around an axis. (wikipedia.org)

**Examples**- Examples of chaotic rotation include Hyperion, a moon of Saturn which rotates so unpredictably that the Cassini probe could not be reliably scheduled to pass by unexplored regions, and Pluto's Nix, Hydra, and possibly Styx and Kerberos and Neptune's Nereid. (wikipedia.org)
- AVL trees and red-black trees are two examples of binary search trees that use the left rotation. (wikipedia.org)

**commonly**- In astronomy, rotation is a commonly observed phenomenon. (wikipedia.org)

**patterns**- citation needed] More complex rotation patterns are required if the vehicle has a full-size spare tire that is part of the rotation, or if there are snow tires. (wikipedia.org)
- Sector rotation is a theory of stock market trading patterns. (wikipedia.org)

**symmetric**- In fact, we can reproduce the symmetric update determined by Qkℓ by retaining only the three values k, ℓ, and t, with t set to zero for a null rotation. (wikipedia.org)
- Because of the conservation of angular momentum, chaotic rotation is not seen in objects that are spherically symmetric or well isolated from gravitational interaction, but is the result of the interactions within a system of orbiting bodies, similar to those associated with orbital resonance. (wikipedia.org)

**time**- It has the ability to trigger rotation based on file size, time or interval (or any combination thereof). (wikipedia.org)
- Opioid rotation requires strict monitoring in patients with ongoing levels of high opioid doses for extended periods of time, since long term opioid use can lead to a patient developing tolerance to the analgesic effects of the drug. (wikipedia.org)
- New evidence suggests that our galaxy and other have settled into an orderly, disk-like rotation over the past 8 billion years[not in citation given] and that other galaxies are slowly following suit over time. (wikipedia.org)
- A single left rotation is done in O(1) time but is often integrated within the node insertion and deletion of binary search trees. (wikipedia.org)

**group**- All rotations about a fixed point form a group under composition called the rotation group (of a particular space). (wikipedia.org)
- The rotation group is a Lie group of rotations about a fixed point . (wikipedia.org)
- A simple rotation would be one crop from each group with a 1:1:1 ratio. (britannica.com)

**bonds**- Reuters) - One of these days, and it might start soon, investors are going to begin to reverse their multi-year rotation out of stocks and into bonds. (reuters.com)

**bodies**- from careful observation by the Keck and Hubble Telescopes of hundreds of galaxies, a trend was discovered that suggests galaxies such as our own Milky Way used to have a very chaotic rotation, with planetary bodies and stars rotating randomly. (wikipedia.org)

**citation**- citation needed] If the tires are unidirectional, the rotation can only be rotated front to back on the same side of the vehicle to preserve the rotational direction of the tires. (wikipedia.org)
- citation needed] In some cases (for example, BMW), automobile manufacturers may recommend performing no tire rotation at all. (wikipedia.org)

**implementation**- There are no clinical guidelines outlining the use and implementation of opioid rotation. (wikipedia.org)

**proper**- How important is proper tire rotation? (wikipedia.org)

**Guidelines**- The official rule book, Guidelines for American Rotation, is available from the American Billiard Club website. (wikipedia.org)

**broader**- There are now also several science-themed rotation curation accounts used for science outreach to a broader community, including@RealScientists, @Biotweeps, and @Astrotweeps. (wikipedia.org)

**active**- Stellar rotation is measured through Doppler shift or by tracking active surface features. (wikipedia.org)

**origin**- This (common) fixed point is called the center of rotation and is usually identified with the origin . (wikipedia.org)
- Rotations about the origin have three degrees of freedom (see rotation formalisms in three dimensions for details), the same as the number of dimensions. (wikipedia.org)
- Any rotation about the origin can be represented as the composition of three rotations defined as the motion obtained by changing one of the Euler angles while leaving the other two constant. (wikipedia.org)

**dimensions**- This definition applies to rotations within both two and three dimensions (in a plane and in space, respectively. (wikipedia.org)

**passive**- A sector rotation investment strategy is not a passive investment strategy like indexing, and requires periodic review and adjustment of sector holdings. (wikipedia.org)
- Tactical asset allocation and sector rotation strategies require patience and discipline, but have the potential to outperform passive indexing investment strategies. (wikipedia.org)

**point**- Any rotation is a motion of a certain space that preserves at least one point . (wikipedia.org)
- Composition of rotations sums their angles modulo 1 turn , which implies that all two-dimensional rotations about the same point commute . (wikipedia.org)
- This is similar to 2D rotation about an arbitrary point. (siggraph.org)
- This is directly analogous to the 2D case of rotation about a point. (siggraph.org)
- A rotation is a circular movement of an object around a center (or point) of rotation. (wikipedia.org)
- If a rotation around a point or axis is followed by a second rotation around the same point/axis, a third rotation results. (wikipedia.org)
- This presentation is convenient only for rotations about a fixed point. (wikipedia.org)

**usually**- The motion of the components of galaxies is complex, but it usually includes a rotation component. (wikipedia.org)
- Reduced stock is usually included in the rotation of stock, and is therefore moved to the front of the shelf ahead of any unreduced stock. (wikipedia.org)

**matrix**- Because my rotation in 3D is either described by a quaternion or by a 3x3 matrix. (opengl.org)
- that is, a square matrix R is a rotation matrix if and only if RT = R−1 and det R = 1. (wikipedia.org)
- The rotation matrix can be derived from the Euler Formula, by considering how a complex number is transformed under multiplication by another complex number of unit modulus. (wikipedia.org)
- See below for other alternative conventions which may change the sense of the rotation produced by a rotation matrix. (wikipedia.org)

**provide**- Short rotations are not likely to provide the best crop balances, and long rotations on a larger number of fields may introduce complications. (britannica.com)

**basic**- An investor or trader may describe the current market movements as favoring basic material stocks over semiconductor stocks by calling the environment a sector rotation from semiconductors to basic materials. (wikipedia.org)

**ways**- A three-dimensional rotation can be specified in a number of ways. (wikipedia.org)

**Short**- The American Rotation Championship Series ("the ARCS" for short), is a three-staged cycle of player qualifications that culminates in a national championship. (wikipedia.org)

**various**- In the major food-producing regions of the world, various rotations of much shorter length are widely used. (britannica.com)

**represents**- Lee, the 2008 American League Cy Young Award winner, represents an enormous upgrade the top of a rotation that spent much of the first half of this season pitching inconsistently. (philly.com)

**particular**- For a particular rotation: The axis of rotation is a line of its fixed points. (wikipedia.org)
- In statistics, a varimax rotation is used to simplify the expression of a particular sub-space in terms of just a few major items each. (wikipedia.org)