Represent and describe rotations in three dimensions
Course:- Computer Networking
Reference No.:- EM131944173

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Computer Graphics Euler Angle Object Rotation Assignment -

There are many, many ways to represent and describe rotations in three dimensions. Different references have different orientations, descriptions, and notations and it is easy to get quite confused. Be especially careful about terminology and notation when you cross-reference this document with any other references.

Euler Angle Rotation of an Object

In three-dimensional space, to orient a vector rooted at the origin in any direction requires only two angles. One way to do this orientation is to turn the vector about the axis and then raise or lower the vector's and point until the vector is pointing in the desired direction. Even though this is happening in a three-dimensional space, two angles suffice since we are identifying only a direction for the vector and not a specific point ∈ R3.

However to orient a rigid body with respect to a fixed coordinate system in three-dimensional space requires three angles, one for each axis. Leonhard Euler created a systematic way to do this (in 1776) and thus we generally call these angles the Euler Angles. Peter Tait and George Bryan later extended Euler's work (in 1911) and therefore some refer to certain classes of these angles and their application as Tait-Bryan Angles. We will stick with the generic name Euler angles to avoid having to switch back and forth between two designations.

For the purposes of this class we will denote the three Euler angles as φ, θ, and ψ.

  • φ represents the rotation about the axis and is know as Roll.
  • θ represents the rotation about the axis and is known as Pitch.
  • ψ represents the rotation about the axis and is known as Yaw.

The names Roll, Pitch, and Yaw come from the aeronautical world, where they represent the aircraft's bearing, elevation, and bank angle.

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