Why were some tube station platforms built curved?

I believe there was no other choice. When most of the deep level network was built builders had to avoid digging under buildings because they'd have to pay compensation for the disruption, so for instance at Bank on the Central line, there are several curves all around the station, therefore to place a station at the location they wanted they had to build it on a curve. Remember in those days the tube was such a novelty that no-one cared if the station was curved, and there wasn't much H&S...

You're not an engineer are you? Try drawing a smooth curve on a piece of paper then see what you have to do to put a straight bit in. You'll find that the transition bits have to be far steeper curves - too steep in railway terms. And as for a complete re-route, well
1. The central area stations "are where they are". In other words you're joining up the dots rather than starting with an (underground) 'green field' site* and
2. As sydneynick says, the routes were constrained by needing to go under highways rather than propeties wherever possible.


* On the northern extensions, which mainly were greenfield sites, the route indeed is far straighter with very few curves and no curved stations. That's what happens when the designers do indeed have a choice.

Bank (central) curves to avoid the Bank of England's vault's at points, so a valid reason there.

The Victorians and Edwardians didn't have RVAR to comply with so didn't need to be so obsessive about platform/train gaps.
Generally the cars were shorter, which helps - less end throw and centre throw on curves. However, the early Tube cars do seem to have had lower floors, so the train floor was broadly level with the platforms. The gradual move of train equipment below the floors in successive designs pushed the floors upwards to what we see today on most deep Tubes.