Origin (mathematics) in the context of Azimuth angle

In geometry, a Cartesian coordinate system (UK: /kɑːrˈtiːzjən/, US: /kɑːrˈtiːʒən/) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, called coordinate lines, coordinate axes or just axes (plural of axis) of the system. The point where the axes meet is called the origin and has (0, 0) as coordinates. The axes directions represent an orthogonal basis. The combination of origin and basis forms a coordinate frame called the Cartesian frame.

Similarly, the position of any point in three-dimensional space can be specified by three Cartesian coordinates, which are the signed distances from the point to three mutually perpendicular planes. More generally, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are the signed distances from the point to n mutually perpendicular fixed hyperplanes.

In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point P in space. Its length represents the distance in relation to an arbitrary reference origin O, and its direction represents the angular orientation with respect to given reference axes. Usually denoted x, r, or s, it corresponds to the straight line segment from O to P.In other words, it is the displacement or translation that maps the origin to P:

The term position vector is used mostly in the fields of differential geometry, mechanics and occasionally vector calculus.Frequently this is used in two-dimensional or three-dimensional space, but can be easily generalized to Euclidean spaces and affine spaces of any dimension.

In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature. It is a special case of a curve and an idealization of such physical objects as a straightedge, a taut string, or a ray of light. Lines are spaces of dimension one, which may be embedded in spaces of dimension two, three, or higher. The word line may also refer, in everyday life, to a line segment, which is a part of a line delimited by two points (its endpoints).

Euclid's Elements defines a straight line as a "breadthless length" that "lies evenly with respect to the points on itself", and introduced several postulates as basic unprovable properties on which the rest of geometry was established. Euclidean line and Euclidean geometry are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such as non-Euclidean, projective, and affine geometry.

In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system, whose origin, orientation, and scale have been specified in physical space. It is based on a set of reference points, defined as geometric points whose position is identified both mathematically (with numerical coordinate values) and physically (signaled by conventional markers).An important special case is that of an inertial reference frame, a stationary or uniformly moving frame.

For n dimensions, n + 1 reference points are sufficient to fully define a reference frame. Using rectangular Cartesian coordinates, a reference frame may be defined with a reference point at the origin and a reference point at one unit distance from the origin along each of the n coordinate axes.

State Street is a large south-north street, also one of the main streets, in Chicago, Illinois, United States and its south suburbs. Its intersection with Madison Street (41°52′55″N 87°37′40″W / 41.8820°N 87.6278°W / 41.8820; -87.6278) has marked the origin for Chicago's address system since 1909. State begins in the north at North Avenue, the south end of Lincoln Park, runs south through the heart of the Chicago Loop, and ends at the southern city limits, intersecting 127th Street along the bank of the Little Calumet River. It resumes north of 137th Street in Riverdale and runs south intermittently through Chicago's south suburbs until terminating at New Monee Road in Crete, Illinois.

From north to south, State Street traverses the following community areas of Chicago: Near North Side to the Chicago River, Chicago Loop to Roosevelt Road, Near South Side to 26th Street, Douglas to 39th Street, Grand Boulevard to 51st Street, Washington Park to 63rd Street, Grand Crossing to 79th Street, Chatham to 91st Street, Roseland to 115th Street, and West Pullman to 127th Street, where it terminates across from Riverdale Bend Woods. The street runs parallel and adjacent to the Dan Ryan Expressway from 65th Street south to just beyond 95th Street, where State Street crosses the I-94 Bishop Ford Memorial Freeway to enter Roseland.

An azimuth (/ˈæzəməθ/ ; from Arabic: اَلسُّمُوت, romanized: as-sumūt, lit. 'the directions') is the horizontal angle from a cardinal direction, most commonly north, in a local or observer-centric spherical coordinate system.

Mathematically, the relative position vector from an observer (origin) to a point of interest is projected perpendicularly onto a reference plane (the horizontal plane); the angle between the projected vector and a reference vector on the reference plane is called the azimuth.

The equatorial coordinate system is a celestial coordinate system widely used to specify the positions of celestial objects. It may be implemented in spherical or rectangular coordinates, both defined by an origin at the centre of Earth, a fundamental plane consisting of the projection of Earth's equator onto the celestial sphere (forming the celestial equator), a primary direction towards the March equinox, and a right-handed convention.

The origin at the centre of Earth means the coordinates are geocentric, that is, as seen from the centre of Earth as if it were transparent. The fundamental plane and the primary direction mean that the coordinate system, while aligned with Earth's equator and pole, does not rotate with the Earth, but remains relatively fixed against the background stars. A right-handed convention means that coordinates increase northward from and eastward around the fundamental plane.

In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is an isometry.

In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are

• the radial distance r along the line connecting the point to a fixed point called the origin;
• the polar angle θ between this radial line and a given polar axis; and
• the azimuthal angle φ, which is the angle of rotation of the radial line around the polar axis.

(See graphic regarding the "physics convention".)

A chart datum is the water surface serving as origin (or coordinate surface) of depths displayed on a nautical chart and for reporting and predicting tide heights. A chart datum is generally derived from some tidal phase, in which case it is also known as a tidal datum. Common chart datums are lowest astronomical tide (LAT) and mean lower low water (MLLW). In non-tidal areas, e.g., the Baltic Sea, mean sea level (MSL) is used.

The United States Capitol features a dome situated above its rotunda. The dome is 288 feet (88 m) in height and 96 feet (29 m) in diameter. Designed by Thomas U. Walter, the fourth Architect of the Capitol, it was constructed between 1855 and 1866 at a cost of $1,047,291 (equivalent to $17.6 million in 2024). The Statue of Freedom tops the lantern on the dome, and the dome is centered over the origin on street maps of Washington, D.C.

The dome is not stone, but rather cast iron carefully painted to appear to be made of the same stone as the capitol building below. It is actually two domes, one inside the other, and the total weight is 9.1 million pounds (4,100 t). The dome's cast iron frame was supplied and constructed by the iron foundry Janes, Fowler, Kirtland & Co. in the Bronx, New York. The interior of the dome includes a detailed geometric and floral pattern of lightly colored plasterwork, with Constantino Brumidi's monumental ceiling fresco, The Apotheosis of Washington, appearing in the oculus at the top of the inner dome.

In physical chemistry and fluid mechanics, a non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity, that is, it has variable viscosity dependent on stress. In particular, the viscosity of non-Newtonian fluids can change when subjected to force. Ketchup, for example, becomes runnier when shaken and is thus a non-Newtonian fluid. Many salt solutions and molten polymers are non-Newtonian fluids, as are many commonly found substances such as custard, toothpaste, starch suspensions, paint, blood, melted butter and shampoo.

Most commonly, the viscosity (the gradual deformation by shear or tensile stresses) of non-Newtonian fluids is dependent on shear rate or shear rate history. Some non-Newtonian fluids with shear-independent viscosity, however, still exhibit normal stress-differences or other non-Newtonian behavior. In a Newtonian fluid, the relation between the shear stress and the shear rate is linear, passing through the origin, the constant of proportionality being the coefficient of viscosity. In a non-Newtonian fluid, the relation between the shear stress and the shear rate is different. The fluid can even exhibit time-dependent viscosity. Therefore, a constant coefficient of viscosity cannot be defined.