Coordinate Systems for \mathbb{R}^{3}

Coordinate Systems for \(\mathbb{R}^{3}\)
An Introduction

Computer Science Department

bernstdh@jmu.edu

Rectangular/Cartesian Coordinates

Reference Frames:
- An origin
- A basis (i.e., three linearly independent vectors)
- Directional conventions
Quantities:
- Three linear (i.e., length) coordinates
Visualization:

Cylindrical Coordinates

Quantities:
- Polar coordinates in one plane (i.e., one angular and one linear coordinate)
- Linear third coordinate
Visualization:

Converting between Cylindrical and Cartesian

Notation

Converting from Cylindrical to Cartesian

What We Know:
- \(\theta\), \(d\) and \(h\)
Using the Right Triangle:
- \(\cos \theta = p_{1} / d\) so \(p_{1} = d \cos \theta\)
- \(\sin \theta = p_{2} / d\) so \(p_{2} = d \sin \theta\)
- \(p_{3} = h\)

Converting from Cartesian to Cylindrical

What We Know:
- \(\bs{p}\)
Using the Right Triangle:
- \(\tan \theta = p_{2} / p_{1}\) for \(p_{1} \gt 0\) so \(\theta = \tan^{-1} p_{2} / p_{1}\)
Using the Norm:
- \(d = \sqrt{p_{1}^{2} + p_{2}^{2}}\)
By Definition:
- \(h = p_{3}\)

Spherical Coordinates

Converting between Spherical and Cartesian Coordinates

Notation

Converting from Spherical to Cartesian

What We Know:
- \(\theta\), \(\mu\) and \(r\)
Finding \(d\):
- \(\sin \mu = d / r\) so \(d = r \sin \mu\)
Finding \(p\):
- \(\cos \theta = p_{1} / d \) so \(p_{1} = d \cos \theta = r \sin \mu \cos \theta\)
- \(\sin \theta = p_{2} / d \) so \(p_{2} = d \sin \theta = r \sin \mu \sin \theta\)
- \(\cos \mu = p_{3} / r \) so \(p_{3} = r \cos \mu\)

Converting from Cartesian to Spherical

What We Know:
- \(\bs{p}\)
Finding \(d\):
- \(d = \sqrt{p_{1}^{2} + p_{2}^{2}}\)
Finding \(r\):
- \(r = ||\bs{p}|| = \sqrt{p_1^2 + p_2^2 + p_3^2}\)
Using the Right Triangles:
- \(\tan \theta = p_{2} / p_{1}\) so \(\theta = \tan^{-1} p_{2}/p_{1}\)
- \(\cos \mu = p_{3} / r\) so \(\mu = \cos^{-1} \frac{p_{3}}{||\bs{p}||}\)

There's Always More to Learn