## Recap on coordinate systems Here we describe some most common coordinates systems in Euclidean space. ### Cartesian coordinates Cartesian coordinates can be characterized by an origin and orthogonal coordinate basis $\mathbf{i},\ \mathbf{j},\ \mathbf{k},\ \cdots$. These vectors are connected to the origin $O$ with coordinates $(1,0,0,\cdots)$, $(0,1,0,\cdots)$, $(0,0,1,\cdots)$, etc. A vector with coordinates $(x_{0}, y_{0}, z_{0}, \cdots)$ is of the linear combination form $\mathbf{V}=x_{0}\ \mathbf{i}+y_{0}\ \mathbf{j}+z_{0}\ \mathbf{k}+\cdots$ The system preserves Euclidean distances, meaning the coordinate unit matches the Euclidean unit of length. This is important because we may construct orthogonal affine coordinate system from orthogonal basis but not Cartesian coordinates by defining basis being non-unity. Depending on the arrangement of the basis vectors (clockwise or counter-clockwise from $\mathbf{i}$ to $\mathbf{j}$, along the short path), the system can be either right-handed or left-handed. For $2D$ Cartesian coordinates, it is characterized by three continues degrees of freedom (2 for location of origin and one for direction of $x$ axis). For $3D$ case, degrees of freedom is 6. ### Affine coordinates Affine coordinate can be characterized by an origin and linear independent coordinate basis $\mathbf{i},\ \mathbf{j},\ \mathbf{k},\ \cdots$. For vector $\mathbf{V}$, the decomposition would be $\mathbf{V}=x_{0}\ \mathbf{i}+y_{0}\ \mathbf{j}+z_{0}\ \mathbf{k}+\cdots$ Affine coordinate basis do not have to be orthogonal and unitary. Any two affine coordinate systems are related by an invertible liner transformations with a shift. Since Cartesian system is a specific type of affine coordinate system, any affine coordinate can be linearly transformed into a Cartesian system and vice versa. ### Polar coordinates Polar coordinates system is defined in $2D$. It is characterized by an arbitrary point $O$ as pole (or origin) and a ray from $O$ as polar axis. Each point $A$ in the plane is assigned two coordinates, $r$ and $\theta$, where $r$ is Euclidean distance from the pole and $\theta$ is the angle between $OA$ and the polar axis. The counter-clockwise direction is defined as positive (right-handed). A rescaled polar coordinate system would be called scaled polar coordinates, if Euclidean distance could not be satisfied. ### Cylindrical coordinates Cylindrical coordinate system is an extension of polar coordinate system in higher, typically three, dimensions. For three dimensional case, a plane called coordinate plane is first selected and a polar system is constructed in this plane, then a coordinate axis is constructed perpendicular to the coordinate plane from origin $O$. A point $A$ is characterized by its orthogonal projection onto the coordinate plane and the distance from coordinate axis, giving $(r, \theta, z)$. The sign conversion is usually right-handed. ### Spherical coordinates Spherical coordinate system is another extension of polar, typically in three-dimension. A coordinate plane with the origin $O$ and a polar axis is selected. Point $A$ could be fully determined with an Euclidean distance $r$ from $O$, an azimuthal angle $\phi$, and a longitudinal angle $\theta$ between $OA$ and longitudinal axis, a line passes through $O$ and orthogonal to the coordinate plane. ### Relations among common coordinate systems (2D and 3D) - Polar system and Cartesian system $\begin{aligned} x(r, \theta) &= r \cos \theta \\ y(r, \theta) &= r \sin \theta \end{aligned}$ $\begin{aligned} r(x, y) &= \sqrt{x^2 + y^2} \\ \theta(x, y) &= \arctan \frac{y}{x} \end{aligned}$ - Cylindrical system and Cartesian system $\begin{aligned} x(r, \theta, z) &= r \cos \theta \\ y(r, \theta, z) &= r \sin \theta \\ z(r, \theta, z) &= z \end{aligned}$ $\begin{aligned} r(x, y, z) &= \sqrt{x^2 + y^2} \\ \theta(x, y, z) &= \arctan \frac{y}{x} \\ z(x, y, z) &= z \end{aligned}$ - Spherical system and Cartesian system $\begin{aligned} x(r, \theta, \phi) &= r \sin \theta \cos \phi \\ y(r, \theta, \phi) &= r \sin \theta \sin \phi \\ z(r, \theta, \phi) &= r \cos \theta \end{aligned}$ $\begin{aligned} r(x, y, z) &= \sqrt{x^2 + y^2 + z^2} \\ \theta(x, y, z) &= \arcsin \frac{z}{\sqrt{x^2 + y^2 + z^2}} \\ \phi(x, y, z) &= \arctan \frac{y}{x} \end{aligned}$