# Coordinate System

This was focused on heavily for ECE106, and also learned in MATH119.

See [[notes/Differentials#Differential Length Element|Differentials#Differential Length Element]]

See Coordinate Frame if youβre referring for robotics.

### Cartesian Coordinate

Cartesian coordinates are defined by 3 vectors with directions given by π₯Μ, π¦Μ, π§Μ.

In math, we use πΜ, πΜ, πΜ. π΄β in terms of the coordinate system will be given as: $Aβ=Axx^+Ayy^β+Azz^$

where $A_{x}$ is the distance you move in π₯Μ direction, $A_{y}$ is the distance you move in π¦Μ direction and $A_{z}$ is the distance you move in π§Μ direction. [[notes/Differentials#Differential Surface Elements|Differentials#Differential Surface Elements]]

### Polar Coordinates

We have two vectors which define the polar coordinates, vector πΜ which changes the radius of the compass and vector $Οβ$ which changes the angle $Ο$. $Οβ$ is perpendicular to $r$ (and is thus, orthogonal) and by conventions changes the angle $Ο$ counter-clockwise. $Οβ$ is called the **azimuthal vector**.

Intuition for Polar Coordinate

I get confused easily because there is an extra $r$ in $rdrdΟ$ for the differential area element, compared to cartesian coordinates which only has $dxdy$.

This is because the area, one of them is $dr$, and the arc is given by $rdΟ$, NOT $dΟ$, $Ο$ is just an angle.

### Cylindrical Coordinate

Cylindrical coordinates are basically polar coordinates in 3D.

We follow the left-hand rule and thus, πΜ is the first vector, β Μ is the second vector and π§Μ is the third vector.

#### Conversion from Cartesian to Polar Coordinates

We simply let $x=rcosΟandy=rsinΟ$

### Spherical Coordinate

We will make the z-axis as a principle axis of symmetry. So the sphere will be symmetrically created around the z-axis. We introduce a new vector, πΜ which changes the angle of the line with respect to the z-axis. This vector is called the polar vector.

Understand the 3 differential elements, should be easy now

Danger

In spherical coordinates, r is the distance of the point from the origin. In cylindrical coordinates, it is the perpendicular distance of the point from the z-axis.