Pitch is not flight path
With zero roll and sideslip, γ = θ − α. Change bank or sideslip and use the full matrix instead of this special-case shortcut.
05 / ORIENT · Flight mechanics
Reference Frames & Aircraft Angles
Reference framesBrowser calculation
Rotate the aircraft and transform the same air-relative velocity between body and North–East–Down axes.
ONE VECTOR. DIFFERENT COORDINATES.LIVE B
3D is unavailable. The calculated coordinate diagrams and curves below remain available.
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3D aircraft and body axes follow a 3–2–1 yaw–pitch–roll attitude. Earth axes are fixed; air velocity is white. The 3D arrows indicate direction only. The component bars below share one quantitative velocity scale. This is a kinematic state, not a force-balanced flight prediction. Drag to orbit; scroll to zoom.
Keep the vector length
Rotating a coordinate system changes components, not the magnitude of the physical velocity.
Separate heading and track
Yaw describes body orientation. Track comes from the Earth-horizontal components of velocity. Wind is zero in this experiment.
CALCULATED FROM THE SAME MODEL
Read the parameter study
B · solid turquoise A · dashed amber
Dots mark the selected settings; curves sweep one variable
Compare the evidence
| Quantity | A · saved | B · live | Unit |
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A remains fixed until you save again. Export includes settings, selected outputs and curve data.
Equations, assumptions and topic connections
Physical model
Rigid-body kinematics with orthonormal right-handed body axes x forward, y right, z down and Earth NED axes. Body-to-Earth matrix R = Rz(ψ) Ry(θ) Rx(φ). No translational/rotational dynamics or wind is assumed. Pitch is restricted away from Euler-angle singularities.
Vb = V[cosα cosβ, sinβ, sinα cosβ]ᵀ; Vned = R Vb. χ = atan2(VE,VN); γ = atan2(−VD, √(VN²+VE²)). RᵀR = I and det(R) = +1.
Topic connections
S44 · Week 2: slides 13–18, 21–31, 38–43. Body/velocity axes and matrix are original diagrams; source artwork not republished.
MIT · aircraft stability and control
Original schematic diagrams. No manufacturer performance, certification or maintenance-authorisation claim. Numerical verification checks the declared model; it is not experimental validation. Model version: course-foundations-v1.