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Derivation of the 2D Rotation Matrix Changing View from Global to Local

Y. Y’. X Cos q. Y Sin q. X’. P. X Sin q. Y Cos q. q. X. Derivation of the 2D Rotation Matrix Changing View from Global to Local. X’ = X Cos q + Y Sin q. Y’ = Y Cos q - X Sin q. Y. Y’. X’. Y’ Cos q. P. X’ Sin q. X’ Cos q. q. Y’ Sin q. X.

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Derivation of the 2D Rotation Matrix Changing View from Global to Local

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  1. Y Y’ X Cos q Y Sin q X’ P X Sin q Y Cos q q X Derivation of the 2D Rotation Matrix Changing View from Global to Local X’ = X Cos q + Y Sin q Y’ = Y Cos q - X Sin q

  2. Y Y’ X’ Y’ Cos q P X’ Sin q X’ Cos q q Y’ Sin q X Derivation of the 2D Rotation Matrix Changing View from Local to Global X = X’ Cos q - Y’ Sin q Y = X’ Sin q + Y’ Cos q

  3. X’ Cos Sin -Sin Cos X = Y’ Y X Cos -Sin Sin Cos X’ = Y Y’ Conversion of Equations to Matrix Form From Global to Local: X’ = X Cos q + Y Sin q Y’ = Y Cos q - X Sin q From Local to Global: X = X’ Cos q - Y’ Sin q Y = X’ Sin q + Y’ Cos q

  4. Derivation of the 3D Rotation Matrix Y Y’ X’ P qz X X’ = X Cos q + Y Sin q Z’ Z Y’ = Y Cos q - X Sin q Z’ = Z

  5. Z Rotation from Global to Local

  6. Z axis rotations From Global to Local: X’ = X Cos qz + Y Sin qz Y’ = Y Cos qz- X Sin qz Z’ = Z From Local to Global: X = X’ Cos qz - Y’ Sin qz Y = X’ Sin qz + Y’ Cos qz Z = Z’

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