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:
where is the distance you move in π₯Μ direction, is the distance you move in π¦Μ direction and 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 which changes the angle . is perpendicular to (and is thus, orthogonal) and by conventions changes the angle counter-clockwise. is called the azimuthal vector.
Intuition for Polar Coordinate
I get confused easily because there is an extra in for the differential area element, compared to cartesian coordinates which only has .
This is because the area, one of them is , and the arc is given by , NOT , 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
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.