Euler angles

The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body. To describe such an orientation in 3-dimensional Euclidean space three parameters are required. They can be given in several ways, Euler angles being one of them; see charts on SO(3) for others. Euler angles are also used to describe the orientation of a frame of reference (typically, a coordinate system or basis) relative to another. They are typically denoted as α, β, γ, or φ, θ, ψ.

Euler angles represent a sequence of three elemental rotations, i.e. rotations about the axes of a coordinate system. For instance, a first rotation about z by an angle α, a second rotation about x by an angle β, and a last rotation again about y, by an angle γ. These rotations start from a known standard orientation. In physics, this standard initial orientation is typically represented by a motionless (fixed, global, or world) coordinate system; in linear algebra, by a standard basis.

Any orientation can be achieved by composing three elemental rotations. The elemental rotations can either occur about the axes of the fixed coordinate system (extrinsic rotations) or about the axes of a rotating coordinate system, which is initially aligned with the fixed one, and modifies its orientation after each elemental rotation (intrinsic rotations). The rotating coordinate system may be imagined to be rigidly attached to a rigid body. In this case, it is sometimes called a local coordinate system. Without considering the possibility of using two different conventions for the definition of the rotation axes (intrinsic or extrinsic), there exist twelve possible sequences of rotation axes, divided in two groups:

Yaw

Yaw may refer to:

Flight dynamics

Flight dynamics is the study of the performance, stability, and control of vehicles flying through the air or in outer space. It is concerned with how forces acting on the vehicle influence its speed and attitude with respect to time.

In fixed-wing aircraft, the changing orientation of the vehicle with respect to the local air flow is represented by two critical parameters, angle of attack ("alpha") and angle of sideslip ("beta"). These angles describe the vector direction of airspeed, important because it is the principal source of modulations in the aerodynamic forces and moments applied to the aircraft.

Spacecraft flight dynamics involve three forces: propulsive (rocket engine), gravitational, and lift and drag (when traveling through the earths or any other celestial atmosphere). Because aerodynamic forces involved with spacecraft flight are very small, this leaves gravity as the dominant force.

Aircraft and spacecraft share a critical interest in their orientation with respect to the earth horizon and heading, and this is represented by another set of angles, "yaw," "pitch" and "roll" which angles match their colloquial meaning, but also have formal definition as an Euler sequence. These angles are the product of the rotational equations of motion, where orientation responds to torque, just as the velocity of a vehicle responds to forces. For all flight vehicles, these two sets of dynamics, rotational and translational, operate simultaneously and in a coupled fashion to evolve the vehicle's state (orientation and velocity) trajectory.

Yam (god)

Yam, Yamm, or Yammu was a Levantine sea and river god, popular in the Bronze and Early Iron Ages.

Yam, from the Canaanite word Yam (Hebrew: ים‎) meaning "Sea", also written Yaw, is one name of the Ugaritic god of Rivers and Sea. Also titled Judge Nahar ("Judge River"), he is also one of the 'ilhm (Elohim) or sons of El, the name given to the Levantine pantheon. Others dispute the existence of the alternative names, claiming it is a mistranslation of a damaged tablet. Despite linguistic overlap, theologically this god is not a part of the later subregional monotheistic theology, but rather is part of a broader and archaic Levantine polytheism.

Yam is the deity of the primordial chaos and represents the power of the sea, untamed and raging; he is seen as ruling storms and the disasters they wreak. The gods cast out Yam from the heavenly mountain Sappan (modern Jebel Aqra; Sappan is cognate to Tsephon). The seven-headed dragon Lotan is associated closely with him and he is often described as the serpent. He is the Canaanite equivalent of the Sumerian Tiamat, the primordial mother goddess.

Angle

In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles.

Angle is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation.

Angle (album)

Angle is an album by English jazz pianist Howard Riley, which was released on CBS in 1969 as part of their Realm Jazz Series, and reissued on CD by Columbia in 1999. It features his working trio of that period, with bassist Barry Guy and drummer Alan Jackson. The fully notated "Three Fragments" is a flute-piano duet with Barbara Thompson.

Reception

The Penguin Guide to Jazz notes that "Riley is the only credited composer on the album, which perhaps accounts for its thoughful and rather reserved character."

Track listing

Personnel

References

Glossary of professional wrestling terms

Professional wrestling has accrued a considerable nomenclature through its long existence. Much of it stems from the industry's origins in the days of carnivals and circuses, and the slang itself is often referred to as "carny talk." In the past, wrestlers used such terms in the presence of fans so as not to reveal the worked nature of the business. In recent years, widespread discussion on the Internet has popularized these terms. Many of the terms refer to the financial aspects of pro wrestling in addition to performance-related terms.

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References