coordinate system

(noun)

a method of representing points in a space of given dimensions by coordinates from an origin

Related Terms

Examples of coordinate system in the following topics:

  • Three-Dimensional Coordinate Systems

    • The three-dimensional coordinate system expresses a point in space with three parameters, often length, width and depth ($x$, $y$, and $z$).
    • Each parameter is perpendicular to the other two, and cannot lie in the same plane. shows a Cartesian coordinate system that uses the parameters $x$, $y$, and $z$.
    • This is a three dimensional space represented by a Cartesian coordinate system.
    • The cylindrical coordinate system is like a mix between the spherical and Cartesian system, incorporating linear and radial parameters.
    • Identify the number of parameters necessary to express a point in the three-dimensional coordinate system

  • Cylindrical and Spherical Coordinates

    • While Cartesian coordinates have many applications, cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
    • A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis.
    • Then the $z$ coordinate is the same in both systems, and the correspondence between cylindrical $(\rho,\varphi)$ and Cartesian $(x,y)$ are the same as for polar coordinates, namely $x = \rho \cos \varphi; \, y = \rho \sin \varphi$.
    • A spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.
    • A cylindrical coordinate system with origin $O$, polar axis $A$, and longitudinal axis $L$.

  • Polar Coordinates

    • We use coordinate systems every day, even if we don't realize it.
    • For example, if you walk 20 meters to the right of the parking lot to find the car, you are using a coordinate system.
    • The coordinate system you are most likely familiar with is the $xy$-coordinate system, where locations are described as horizontal ($x$) and vertical ($y$) distances from an arbitrary point.
    • This is called the Cartesian coordinate system.
    • The $xy$ or Cartesian coordinate system is not always the easiest system to use for every problem.

  • Vectors in Three Dimensions

    • The mathematical representation of a physical vector depends on the coordinate system used to describe it.
    • Other vector-like objects that describe physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors.
    • In the Cartesian coordinate system, a vector can be represented by identifying the coordinates of its initial and terminal point.
    • Typically in Cartesian coordinates, one considers primarily bound vectors.
    • A bound vector is determined by the coordinates of the terminal point, its initial point always having the coordinates of the origin $O = (0,0,0)$.

  • Surfaces in Space

    • To say that a surface is "two-dimensional" means that, about each point, there is a coordinate patch on which a two-dimensional coordinate system is defined.
    • For example, the surface of the Earth is (ideally) a two-dimensional surface, and latitude and longitude provide two-dimensional coordinates on it (except at the poles and along the 180th meridian).
    • In spherical coordinates, the surface can be expressed simply by $r=R$.

  • Physics and Engeineering: Center of Mass

    • In this case, the distribution of mass is balanced around the center of mass and the average of the weighted position coordinates of the distributed mass defines its coordinates.
    • In the case of a system of particles $P_i, i = 1, \cdots, n$, each with a mass, $m_i$, which are located in space with coordinates $r_i, i = 1, \cdots, n$, the coordinates $\mathbf{R}$ of the center of mass satisfy the following condition:
    • If the mass distribution is continuous with respect to the density, $\rho (r)$, within a volume, $V$, then the integral of the weighted position coordinates of the points in this volume relative to the center of mass, $\mathbf{R}$, is zero, that is:
    • COM can be defined for both discrete and continuous systems.

  • Center of Mass and Inertia

    • In the case of a distribution of separate bodies, such as the planets of the Solar System, the center of mass may not correspond to the position of any individual member of the system.
    • In the case of a system of particles $P_i, i = 1, \cdots , n$, each with mass $m_i$ that are located in space with coordinates $\mathbf{r}_i, i = 1, \cdots , n$, the coordinates $\mathbf{R}$ of the center of mass satisfy the condition:
    • If the mass distribution is continuous with the density $\rho (r)$ within a volume $V$, then the integral of the weighted position coordinates of the points in this volume relative to the center of mass $\mathbf{R}$ is zero; that is:

  • Double Integrals in Polar Coordinates

    • In $R^2$, if the domain has a cylindrical symmetry and the function has several particular characteristics, you can apply the transformation to polar coordinates, which means that the generic points $P(x, y)$ in Cartesian coordinates switch to their respective points in polar coordinates.
    • The polar coordinates $r$ and $\varphi$ can be converted to the Cartesian coordinates $x$ and $y$ by using the trigonometric functions sine and cosine:
    • The Cartesian coordinates $x$ and $y$ can be converted to polar coordinates $r$ and $\varphi$ with $r \geq 0$ and $\varphi$ in the interval $(−\pi, \pi]$:
    • In general, the best practice is to use the coordinates that match the built-in symmetry of the function.
    • This figure illustrates graphically a transformation from cartesian to polar coordinates

  • Triple Integrals in Spherical Coordinates

    • When the function to be integrated has a spherical symmetry, change the variables into spherical coordinates and then perform integration.
    • When the function to be integrated has a spherical symmetry, it is sensible to change the variables into spherical coordinates and then perform integration.
    • It's possible to use therefore the passage in spherical coordinates; the function is transformed by this relation:
    • Points on $z$-axis do not have a precise characterization in spherical coordinates, so $\theta$ can vary from $0$ to $2 \pi$.
    • Spherical coordinates are useful when domains in $R^3$ have spherical symmetry.

  • Triple Integrals in Cylindrical Coordinates

    • When the function to be integrated has a cylindrical symmetry, it is sensible to integrate using cylindrical coordinates.
    • When the function to be integrated has a cylindrical symmetry, it is sensible to change the variables into cylindrical coordinates and then perform integration.
    • Also in switching to cylindrical coordinates, the $dx\, dy\, dz$ differentials in the integral become $\rho \, d\rho \,d\varphi \,dz$.
    • Finally, it is possible to apply the final formula to cylindrical coordinates:
    • Cylindrical coordinates are often used for integrations on domains with a circular base.