Original Post
I'm just starting out and am doing a quick refresher on linear algebra / trigonometry to set the stage for delving into 3D programming. Math has never been my strong-point, and while I have an intuitive understanding of the literal analogy of a gimbal and its 3 rings and why that would physically happen, I'm afraid I don't understand the mathematical connection here as to WHY this happens.
I feel like I'm missing something obvious in the following explanation (quoted from "3D Math Primer for Graphics and Game Development 2nd Ed." by Fletcher Dunn):
"if we head right 45 deg and then pitch down 90 deg, this is the same as pitching down 90 deg and then banking 45 deg. In fact, once we chose ±90 deg as the pitch angle, we are restricted to rotating about the vertical axis. This phenomenon, in which an angle of ±90 deg for the second rotation can cause the first and third rotations to rotate about the same axis, is known as Gimbal lock. To remove this aliasing from the canonical set of Euler angle triples, we assign all rotation about the vertical axis to heading in the Gimbal lock case. In other words, in the canonical set, if pitch is ±90 deg, then bank is zero."
He lost me early here - "once we chose ±90 deg as the pitch angle, we are restricted to rotating about the vertical axis" - first of all what is the vertical axis referring to here, in this case, the world space canonical Y-axis, the model space Y-axis or does vertical axis refer to the x-axis and the "verticality" of the pitch rotation? Why the pitch axis and not the other axis? Why does ±90 deg restrict the rotation? With a gimbal there is a physical reason because of the rings, but mathematically why does this happen?
I can't seem to find a solid mathematical example with numbers, like a worked example, that explains how and why this happens / how to identify it mathematically etc.
I feel like I'm missing something obvious in the following explanation (quoted from "3D Math Primer for Graphics and Game Development 2nd Ed." by Fletcher Dunn):
"if we head right 45 deg and then pitch down 90 deg, this is the same as pitching down 90 deg and then banking 45 deg. In fact, once we chose ±90 deg as the pitch angle, we are restricted to rotating about the vertical axis. This phenomenon, in which an angle of ±90 deg for the second rotation can cause the first and third rotations to rotate about the same axis, is known as Gimbal lock. To remove this aliasing from the canonical set of Euler angle triples, we assign all rotation about the vertical axis to heading in the Gimbal lock case. In other words, in the canonical set, if pitch is ±90 deg, then bank is zero."
He lost me early here - "once we chose ±90 deg as the pitch angle, we are restricted to rotating about the vertical axis" - first of all what is the vertical axis referring to here, in this case, the world space canonical Y-axis, the model space Y-axis or does vertical axis refer to the x-axis and the "verticality" of the pitch rotation? Why the pitch axis and not the other axis? Why does ±90 deg restrict the rotation? With a gimbal there is a physical reason because of the rings, but mathematically why does this happen?
I can't seem to find a solid mathematical example with numbers, like a worked example, that explains how and why this happens / how to identify it mathematically etc.
Take a look at the diagrams in the "Mechanical Engineering" section.