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The position of all three axes An aircraft in flight is free to rotate in three dimensions: pitch , nose up or down about an axis running from wing to wing, yaw , nose left or right about an axis running up and down; and roll , rotation about an axis running from nose to tail. The axes are alternatively designated as lateral , vertical , and longitudinal . These axes move with the vehicle , and rotate relative to the Earth along with the craft.
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body . [ 1 ] To describe such an orientation in 3-dimensional Euclidean space three parameters are required. They can be given in several ways, Euler angles being one of them; see charts on SO(3) for others. Euler angles also represent three composed rotations that move a reference frame to a given referred frame. This is equivalent to saying that any orientation can be achieved by composing three elemental rotations (rotations around a single axis), and also equivalent to saying that any rotation matrix can be decomposed as a product of three elemental rotation matrices. Without considering the possibilities of different signs for the angles or moving the reference frame, there are twelve different conventions divided in two groups. One of them is called "proper" Euler angles and the other Tait–Bryan angles .
In mathematics , a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space . [ edit ] Introduction In lay terms, a frame of reference is a system of measuring rods used by an observer to measure the surrounding space by providing coordinates . A moving frame is then a frame of reference which moves with the observer along a trajectory (a curve ).