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## Main Question or Discussion Point

The Galilean transforms for rotations, boosts and translations in 2D are the follows:

Rotations:

x' = xcosθ + ysinθ

y' = -xsinθ + ycosθ

Boosts:

x' = x - v

y' = y - v

Translations:

x' = x - d

y' = y - d

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

y' = -(x - d

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

y' = -xsinθ + ycosθ- v

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.

Rotations:

x' = xcosθ + ysinθ

y' = -xsinθ + ycosθ

Boosts:

x' = x - v

_{x}ty' = y - v

_{y}tTranslations:

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)sinθy' = -(x - d

_{x}- v_{x}t)sinθ + (y - d_{y}- v_{y}t)cosθHowever, 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.