Dated: 30-10-2024

Polar Coordinates

We were using \(x\) and \(y\) axes to find position of a point on a plane.¹
Now we will use \(r\) and \(\theta\) instead where \(r\) is the distance from the origin, also called pole and \(\theta\) is the measure of angle relative to \(x\) axis.

Conversion from Cartesian to Polar Coordinates

Using the Pythagorus theorem.

\[r^2 = x^2 + y^2\]

Using the slope,² we have

\[\tan (\theta) = \frac y x\]

Regular Coordinates for 3D Points

These coordinates are called rectangular coordinate system.
A point is represented as

\[P(x, y, z)\]

Cylindrical Coordinates for 3D Points

A point in this system is represented as

\[P(r, \theta, z)\]

Here \(r\) is the distance in \(xy\) plane, \(\theta\) is the \(\alpha\) and \(z\) is the height in the cylinder.

Spherical Coordinates for 3D Points

A point in this system is represented as

\[P(p, \theta, \phi)\]

Where \(p\) is the distance to the point from origin, \(\theta\) is the \(\alpha\) and \(\phi\) is \(\gamma\).

Conversion between Cylindrical and Rectangular

Cylindrical to Rectangular

\[(r, \theta, z) \rightarrow (x, y, z)\]

\[x = r \cdot \cos(\theta)\]

\[y = r \cdot \sin(\theta)\]

\[z = z\]

Rectangular to Cylindrical

\[(x, y, z) \rightarrow (r, \theta, z)\]

\[r = \sqrt{x^2 + y^2}\]

\[\theta = \tan^{-1} \frac y x\]

\[z = z\]

Conversion between Cylindrical and Spherical

Cylindrical to Spherical

\[(r, \theta, z) \rightarrow (p, \theta, \phi)\]

\[p = \sqrt{r^2 + z^2}\]

\[\theta = \theta\]

\[\phi = \tan^{-1} \frac r z\]

Spherical to Cylindrical

\[(p, \theta, \phi) \rightarrow (r, \theta, z)\]

\[r = p \cdot \sin(\phi)\]

\[\theta = \theta\]

\[z = p \cdot \cos (\phi)\]

Conversion between Spherical and Rectangular

Spherical to Rectangular

\[(p, \theta, \phi) \rightarrow (x, y, z)\]

\[\because r = p \cdot \sin(\phi)\]

\[\because x = r \cdot \cos (\theta)\]

\[x = p \cdot \sin(\phi) \cdot \cos(\theta)\]

\[\because y = r \cdot \sin(\theta)\]

\[y = p \cdot \sin(\theta) \cdot \sin(\phi)\]

\[z = p \cdot \cos(\phi)\]

Rectangular to Spherical

\[(x, y, z) \rightarrow (p, \theta, \phi)\]

\[x^2 + y^2 + z^2 = \left(p\sin \phi \cos \theta \right)^2 + \left( p \sin \phi \sin \theta \right)^2 + \left(p \cos \phi \right)^2\]

\[= p^2 \left( \sin^2 \phi \left( \cos^2 \theta + \sin^2 \theta \right) + \cos^2 \phi \right)\]

\[= p^2 \left( \sin^2 \phi + \cos^2 \phi \right) = p^2\]

\[\therefore p = \sqrt{x^2 + y^2 + z^2}\]

\[\theta = \tan^{-1} \frac y x\]

\[\phi = \cos^{-1} \frac{z}{\sqrt{x^2 + y^2 + z^2}}\]

Constant Surfaces in Rectangular Coordinate System

\[x = x_0\]

\[y = y_0\]

\[z = z_0\]

Constant Surfaces in Cylindrical Coordinate System

\(r = r_0\) defines a right cylinder, \(z = z_0\) defines a plane¹ and \(\theta = \theta_0\) defines a half plane¹ attached to the z axis.

Constant Surfaces in Spherical Coordinate System

\(p = p_0\) defines a sphere, \(\theta = \theta_0\) defines a half plane¹ and \(\phi = \phi_0\) defines a right cone.

Imagine z axis going through the north pole of the earth, x axis going through the prime meridian.
Then, the longitutde is specified in east and west degrees, represented by \(\theta\) and latitude is specified in north and south degrees, represented by \(\phi\).
Where \(p\) would be the radius of the earth.