Quaternions and spatial rotationUnit quaternions, known as versors, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation about an arbitrary axis. Rotation and orientation quaternions have applications in computer graphics, computer vision, robotics, navigation, molecular dynamics, flight dynamics, orbital mechanics of satellites, and crystallographic texture analysis.
Orientation (geometry)In geometry, the orientation, attitude, bearing, direction, or angular position of an object – such as a line, plane or rigid body – is part of the description of how it is placed in the space it occupies. More specifically, it refers to the imaginary rotation that is needed to move the object from a reference placement to its current placement. A rotation may not be enough to reach the current placement, in which case it may be necessary to add an imaginary translation to change the object's position (or linear position).
Relativistic angular momentumIn physics, relativistic angular momentum refers to the mathematical formalisms and physical concepts that define angular momentum in special relativity (SR) and general relativity (GR). The relativistic quantity is subtly different from the three-dimensional quantity in classical mechanics. Angular momentum is an important dynamical quantity derived from position and momentum. It is a measure of an object's rotational motion and resistance to changes in its rotation.
Kepler's laws of planetary motionIn astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbits and epicycles with elliptical trajectories, and explaining how planetary velocities vary. The three laws state that: The orbit of a planet is an ellipse with the Sun at one of the two foci. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
Rotation formalisms in three dimensionsIn geometry, various formalisms exist 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.
Rotations in 4-dimensional Euclidean spaceIn mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4. In this article rotation means rotational displacement. For the sake of uniqueness, rotation angles are assumed to be in the segment except where mentioned or clearly implied by the context otherwise. A "fixed plane" is a plane for which every vector in the plane is unchanged after the rotation.
StaticsStatics is the branch of classical mechanics that is concerned with the analysis of force and torque acting on a physical system that does not experience an acceleration, but rather, is in static equilibrium with its environment. If is the total of the forces acting on the system, is the mass of the system and is the acceleration of the system, Newton's second law states that (the bold font indicates a vector quantity, i.e. one with both magnitude and direction). If , then .
InertiaInertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law of motion. After some other definitions, Newton states in his first law of motion: LAW I. Every object perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.
Euler's equations (rigid body dynamics)In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with angular velocity ω whose axes are fixed to the body. Their general vector form is where M is the applied torques and I is the inertia matrix. The vector is the angular acceleration. Again, note that all quantities are defined in the rotating reference frame.
Euclidean vectorIn mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a directed line segment, or graphically as an arrow connecting an initial point A with a terminal point B, and denoted by . A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier".
Plane of rotationIn geometry, a plane of rotation is an abstract object used to describe or visualize rotations in space. The main use for planes of rotation is in describing more complex rotations in four-dimensional space and higher dimensions, where they can be used to break down the rotations into simpler parts. This can be done using geometric algebra, with the planes of rotations associated with simple bivectors in the algebra.
3D rotation groupIn mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry), and orientation (i.e., handedness of space). Composing two rotations results in another rotation, every rotation has a unique inverse rotation, and the identity map satisfies the definition of a rotation.
