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

Coordinate Systems for

$\mathbb{R}^{3}$
An Introduction

Prof. David Bernstein
James Madison University

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 $\mu = \cos^{-1} \frac{p_{3}}{||\bs{p}||}$