L02_Coordinate_Transformation

L02_Coordinate_Transformation - CE507 Lecture 2 Coordinate...

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CE507 Lecture 2 Coordinate Transformation Start with a coordinate system 123 (, , ) x xx a right-hand system with origin O. Rotate it to a new coordinate system, x ′′ with the same origin O. Let , , ˆˆˆ eee be the unit vectors along the ,, x −− axes, and , , ′′′ the corresponding unit vectors along the x axes. Define: angle between the - (new) axis and - (old)axis. ij i j α = and a = cos ( ) = ˆˆ ee ′⋅ , called the direction cosines . We will study the matrix [] ( ) ij aa = in this lecture. Note that a given vector, x ± , say, can have two different representations w.r.t. the two coordinate systems.
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11 2 2 33 ˆˆˆ ˆ ˆ ii x xe xe xe x ex e == + + ′′ ′′ ′′ ′′ ++ ± Objective: To find the relationship between the two representations 123 (, , ) x xx and x ′′′ of a given point in the two coordinate systems. (1) From ˆˆ x ± we know ˆ ˆ jj x x ⋅= ± ± j x is the projection of x ± along ˆ j e , similarly for i x being that of x ± along ˆ i e .
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L02_Coordinate_Transformation - CE507 Lecture 2 Coordinate...

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