Euler Angles for Space Shuttle

The Euler (or Eulerian) angles, usually designated , , and , uniquely determine the orientation of a rigid body in three-dimensional space. There are several conflicting conventions for defining the Euler angles, depending on the choice of axes and the order in which rotations about these axes are performed. This Demonstration uses the convention described in MathWorld, hyperlinked below in Related Links. This is most commonly used in physics and is known as the " convention." For topical relevance, NASA's space shuttle is chosen as the rigid body. The angles and are analogous to the spherical polar coordinates orienting the main axis of the shuttle, while describes rotation about this axis. The ranges of the three Euler angles are given by: , , and . The corresponding motions of a rigid body are termed nutation, precession and intrinsic rotation.
Any rotation of a rigid body can be represented as the product of three successive rotations , with matrix representations
, , .
A unit vector with Euler angles , , and can also be represented by a quaternion , where
, , , ,
and successive rotations obey the algebra of quaternion multiplication. The same combination rule pertains to a linear combination of Pauli spin matrices: , where is the 2×2 unit matrix and means .
In aeronautic or astronautic applications, the " convention" is most often used. The angles , , and are then known as Tait-Bryan rotations, called pitch, yaw, and roll, respectively.
  • Contributed by: S. M. Blinder
  • With a correction by Joeri Gerlo, University of Ghent


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Related Curriculum Standards

US Common Core State Standards, Mathematics

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+