- Forward


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


Prof. David Bernstein
James Madison University

Computer Science Department
bernstdh@jmu.edu

Print

Rectangular/Cartesian Coordinates
Back SMYC Forward
  • Reference Frames:
    • An origin
    • A basis (i.e., three linearly independent vectors)
    • Directional conventions
  • Quantities:
    • Three linear (i.e., length) coordinates
  • Visualization:
    • images/cartesian-coordinates-3d.gif
Cylindrical Coordinates
Back SMYC Forward
  • Quantities:
    • Polar coordinates in one plane (i.e., one angular and one linear coordinate)
    • Linear third coordinate
  • Visualization:
    • images/cylindrical-coordinates.gif
Converting between Cylindrical and Cartesian
Back SMYC Forward

Notation

images/cartesian-to-cylindrical.gif
Converting from Cylindrical to Cartesian
Back SMYC Forward
  • 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
Back SMYC Forward
  • 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
Back SMYC Forward
  • Quantities:
    • Two angular coordinates
    • One linear coordinate
  • Visualization:
    • images/spherical-coordinates.gif
Converting between Spherical and Cartesian Coordinates
Back SMYC Forward

Notation

images/cartesian-to-spherical.gif
Converting from Spherical to Cartesian
Back SMYC Forward
  • 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
Back SMYC Forward
  • 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
Back -