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Rotations:

x' = xcosθ + ysinθ

y' = -xsinθ + ycosθ

Boosts:

x' = x - v

_{x}t

y' = y - v

_{y}t

Translations:

x' = x - d

_{x}

y' = y - d

_{x}

I wanted to combine these into a single pair of equations, so my first thought was to combine boosts and translations and plug into rotations. Doing that, you get:

x' = (x - d

_{x}- v

_{x}t)cosθ + (y - d

_{y}- v

_{y}t)sinθ

y' = -(x - d

_{x}- v

_{x}t)sinθ + (y - d

_{y}- v

_{y}t)cosθ

However, I realized that if you combined the equations differently, say by first plugging rotations into translations, and then plugging that into boosts, you get:

x' = xcosθ + ysinθ - v

_{x}t - d

_{x}

y' = -xsinθ + ycosθ- v

_{y}t - d

_{y}

Which is the correct order, if any, and why? It's also possible that I'm doing the composite wrong somehow, haven't really done that sort of thing since pre-calc.. I'm a junior now.