Geometry: why yaw, pitch, roll in that order?

  1. It is basically a question in computer graphics but I guess math sub-forum will suit this question.

    While rotating a point, why we first apply yaw, then pitch and then roll in that order? Of course if we change the order, final rotation changes but why this specific order works? Some proof or logical explanation will help.

  2. jcsd
  3. SteamKing

    SteamKing 10,999
    Staff Emeritus
    Science Advisor
    Homework Helper

    Who or what applies yaw, pitch, and roll in that order?

    As long as you specify a certain order of rotations, derive the transformations for that order, and stick to it, the order doesn't matter.
  4. D H

    Staff: Mentor

    Any permutation of yaw/pitch/roll will "work". There are six such permutations. All six "work" in the sense that any one these rotation sequences can represent any possible orientation of a body in three dimensional space.

    In fact, there are six more sequences that "work" in the sense of being able to represent any possible orientation in three dimensional space. There's a yaw/roll/yaw sequence (that's the canonical Euler sequence), but also yaw/pitch/yaw, etc. Even better (even worse?), there are twelve more such Euler-like sequences that "work" in this sense. Typically a yaw/pitch/roll sequence involves a rotation about the z axis, then about the y' axis, then about the x'' axis. Well, what if you rotate about the original z axis, then the original y axis, and then the original x axis? That "works", too, in the sense of being able to represent any possible orientation in three dimensional space. This makes for a total of 24 such Euler-like rotation sequences.

    I intentionally put "works" in quotes. Not one of these 24 Euler-like sequences is very useful mathematically. They are useful only because we humans have a hard time grasping the mathematics of the group SO(3).
  5. Stephen Tashi

    Stephen Tashi 4,474
    Science Advisor
    2014 Award

    Given that "apply" means to perform a linear transformation relative to the current coordinate frame, I find it easier to visualize applying yaw first. It gives the general "heading". Likewise, I think its easier to visualize the effect of applying pitch first and then roll. Thinking of an airplane, suppose you apply roll first. If you then apply yaw, the direction of the plane is turned about the current vertical axis, not the original vertical axis.
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