Projection / capability. Know what is being transformed, what is merely being re-described, what survives, and what information a representation discards.
Question this mathematics answers. When a view changes, what happened to the object—and what can you still recover?
Equation family
Rigid rotation: x′ = R x, with RᵀR = IProjection map: M = Π(ξ) (a representation may be many-to-one)Invariant example: ‖Rx‖² = xᵀRᵀRx = ‖x‖²
Symbols. x is a state/vector; R is an orthogonal rotation; Π is a representation/projection map; ξ is a detailed state; M is the retained representation.
Assumptions. A rigid rotation preserves Euclidean distance. A projection may intentionally discard coordinates.
Where the inputs come from. Coordinates come from the declared state description; R comes from the declared frame change; Π comes from the chosen representation or sensor/display pipeline.
Short derivation / justification
Proof sketch (standard linear algebra): Orthogonality means RᵀR=I. Therefore ‖Rx‖²=(Rx)ᵀ(Rx)=xᵀRᵀRx=xᵀx. The invariant is not “length always”; it is length under this stated class of transforms.
Worked example
Take x=(2,1). Rotate by 90°: x′=(-1,2). Coordinates changed, but both lengths are √5. If a separate projection keeps only the first coordinate, many different 2-D states can share the same displayed value.
Cadet or real-world application. Navigation and sensor fusion constantly move between vehicle, world, map, camera, and display coordinates.
What to look for. Ask: object or representation? transformation? survivor? lost information? invertible?
Common misinterpretation. “The picture changed, so the physical object changed.”
Break / domain boundary. A representation that preserves one quantity can discard another. An invariant claim is incomplete without the transformation.