Coordinates (elementary mathematics) This article describes some of the common coordinate systems that appear in elementary mathematics. For advanced topics, please refer to coordinate system. For more background, see Cartesian coordinate system. The coordinates of a point are the components of a tuple of numbers used to represent the location of the point in the plane or space. A coordinate system is a plane or space where the origin and axes are defined so that coordinates can be measured. Cartesian coordinates http://upload.wikimedia.org/wikipedia/en/d/d4/Cartesiancoordinates2D.JPG In the two-dimensional Cartesian coordinate system, a point P in the xy-plane is represent by a tuple of two components $(x,\; y)$. $x$ is the signed distance from the y-axis to the point P, and $y$ is the signed distance from the x-axis to the point P. In the three-dimensional Cartesian coordinate system, a point P in the xyz-space is represent by a tuple of three components $(x,\; y,\; z)$. $x$ is the signed distance from the yz-plane to the point P, $y$ is the signed distance from the xz-plane to the point P, and $z$ is the signed distance from the xy-plane to the point P. For advanced topics, please refer to Cartesian coordinate system. Polar coordinates The polar coordinate systems are coordinate systems in which a point is identified by a distance from some fixed feature in space and one or more subtended angles. The term polar coordinates often refers to circular coordinates (two-dimensional). Other commonly used polar coordinates are cylindrical coordinates and spherical coordinates (both three-dimensional). Circular coordinates The circular coordinate system, often referred to simply as the polar coordinate system, is a two-dimensional polar coordinate system, defined by an origin, O, and a semi-infinite line L leading from this point. L is also called the polar axis. In terms of the Cartesian coordinate system, one usually picks O to be the origin (0,0) and L to be the positive x-axis (the right half of the x-axis). http://upload.wikimedia.org/wikipedia/en/8/85/CircularCoordinates.png In the circular coordinate system, a point P is represented by a tuple of two components $(r,\; \backslash theta)$. Using terms of the Cartesian coordinate system, $0\backslash leq$ (radius) is the distance from the origin to the point P, and (azimuth) is the angle between the positive x-axis and the line from the origin to the point P. Cylindrical coordinates The cylindrical coordinate system is a three-dimensional polar coordinate system. http://upload.wikimedia.org/wikipedia/en/0/02/CylindricalCoordinates.png In the cylindrical coordinate system, a point P is represented by a tuple of three components $(r,\; \backslash theta,\; h)$. Using terms of the Cartesian coordinate system, $0\backslash leq$ (radius) is the distance between the z-axis and the point P, (azimuth or longitude) is the angle between the positive x-axis and the line from the origin to the point P projected onto the xy-plane, and $h$ (height) is the signed distance from xy-plane to the point P. Note: some sources use $z$ for $h$; there is no "right" or "wrong" convention, but it is necessary to be aware of the convention being used. Cylindrical coordinates involve some redundancy; $\backslash theta$ loses its significance if $r=0$. Cylindrical coordinates are useful in analyzing systems that are symmetrical about an axis. For example the infinitely long cylinder that has the Cartesian equation $x^2+y^2=c^2$ has the very simple equation $r=c$ in cylindrical coordinates. Spherical coordinates The spherical coordinate system is a three-dimensional polar coordinate system. http://upload.wikimedia.org/wikipedia/en/f/f7/Spherical_Coordinates.png In the spherical coordinate system, a point P is represented by a tuple of three components $(\backslash rho,\; \backslash phi,\; \backslash theta)$. Using terms of the Cartesian coordinate system, $0\backslash leq\backslash rho$ (radius) is the distance between the point P and the origin, $0\backslash leq\backslash phi\backslash leq\; 180^\backslash circ$ (colatitude or polar angle) is the angle between the z-axis and the line from the origin to the point P, and (azimuth or longitude) is the angle between the positive x-axis and the line from the origin to the point P projected onto the xy-plane. Note: some sources interchange the symbols $\backslash phi$ and $\backslash theta$ relative to this article, or use $r$ for ρ; there is no widely accepted convention. The spherical coordinate system also involves some redundancy; $\backslash phi$ loses its significance if $\backslash rho=0$, and $\backslash theta$ loses its significance if $\backslash rho=0$ or φ=0 or φ=180°. To construct a point from its spherical coordinates: from the origin, go $$ along the positive z-axis, rotate $$ about y-axis toward the direction of the positive x-axis, and rotate $$ about the z-axis toward the direction of the positive y-axis. Spherical coordinates are useful in analyzing systems that are symmetrical about a point; a sphere that has the Cartesian equation $x^2+y^2+z^2=c^2$ has the very simple equation $\backslash rho=c$ in spherical coordinates. Spherical coordinates are the natural coordinates for physical situations where there is spherical symmetry. In such a situation, one can describe waves using spherical harmonics. Another application is ergonomic design, where $$ is the arm length of a stationary person and the angles describe the direction of the arm as it reaches out. See also: Celestial coordinate system Conversion between coordinate systems Cartesian and circular $x=r\backslash ,\backslash cos\backslash theta\; \backslash quad$ $y=r\backslash ,\backslash sin\backslash theta\; \backslash quad$ $r=\backslash sqrt$ $\backslash theta$ = \arctan\frac + \pi u_0(-x) \, \operatorname y where u0 is the Heaviside step function with $u\_0(0)=0$ and sgn is the signum function. Here the u0 and sgn functions are being used as "logical" switches which are used as shorthand substitutes for several if ... then statements. Some computer languages include a bivariate arctangent function atan2(y,x) which finds the value for θ in the correct quadrant given x and y. Cartesian and cylindrical $x=r\backslash ,\backslash cos\backslash theta$ $y=r\backslash ,\backslash sin\backslash theta$ $z=h\; \backslash quad$ $r=\backslash sqrt$ $\backslash theta$ =\arctan\frac + \pi u_0(-x) \, \operatorname y $h=z\; \backslash quad$ $$ \begindx\\dy\\dz\end= \begin \cos\theta&-r\sin\theta&0\\ \sin\theta&r\cos\theta&0\\ 0&0&1 \end\cdot \begindr\\d\theta\\dh\end $$ \begindr\\d\theta\\dh\end= \begin \frac}&\frac}&0\\ \frac&\frac&0\\ 0&0&1 \end\cdot \begindx\\dy\\dz\end Cartesian and spherical $=\backslash rho\; \backslash ,\; \backslash sin\backslash phi\; \backslash ,\; \backslash cos\backslash theta\; \backslash quad$ $=\backslash rho\; \backslash ,\; \backslash sin\backslash phi\; \backslash ,\; \backslash sin\backslash theta\; \backslash quad$ $=\backslash rho\; \backslash ,\; \backslash cos\backslash phi\; \backslash quad$ $=\backslash sqrt$ $=\backslash arccos\backslash frac$ $=\backslash arctan\backslash frac\}$ $$ =\arctan\frac + \pi\, u_0(-x)\, \operatorname y $$ \begindx\\dy\\dz\end= \begin \sin\phi\cos\theta&\rho\cos\phi\cos\theta&-\rho\sin\phi\sin\theta\\ \sin\phi\sin\theta&\rho\cos\phi\sin\theta&\rho\sin\phi\cos\theta\\ \cos\phi&-\rho\sin\phi&0 \end\cdot \begind\rho\\d\phi\\d\theta\end $$ \begind\rho\\d\phi\\d\theta\end= \begin \frac&\frac&\frac\\ \frac}&\frac}&\frac}\\ \frac&\frac&0 \end\cdot \begindx\\dy\\dz\end Cylindrical and spherical $=\backslash rho\; \backslash ,\backslash sin\backslash phi$ $=\backslash theta\; \backslash quad$ $=\backslash rho\; \backslash ,\backslash cos\backslash phi$ $=\backslash sqrt$ $$ =\arctan\frac + \pi \, u_0(-r) \, \operatorname h $=\backslash theta\; \backslash quad$ $$ \begindr\\d\theta\\dh\end= \begin \sin\phi&\rho\cos\phi&0\\ 0&0&1\\ \cos\phi&-\rho\sin\phi&0 \end\cdot \begind\rho\\d\phi\\d\theta\end $$ \begind\rho\\d\phi\\d\theta\end= \begin \frac}&0&\frac}\\ \frac&0&\frac\\ 0&1&0 \end\cdot \begindr\\d\theta\\dh\end See also Coordinate system Cartesian coordinate system Parabolic coordinate system Curvilinear coordinates Coordinate rotation Vector (spatial) Vector fields in cylindrical and spherical coordinates Nabla in cylindrical and spherical coordinates For spherical coordinates: Gimbal lock Spherical harmonics Euler angles Yaw, pitch and roll Credit to original articles: Polar coordinates Cylindrical coordinate system Spherical coordinate system External links Frank Wattenberg has made some nice animations illustrating spherical and cylindrical coordinate systems. http://www.physics.oregonstate.edu/bridge/papers/spherical.pdf is a description of the different conventions in use for naming components of spherical coordinates, along with a proposal for standardizing this. 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