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The curve that is used is a Clothoid:

https://en.wikipedia.org/wiki/Euler_spiral

Usually for any curve you go straight-clothoid-arc-clothoid-straight

For trajectory AND for pitch and roll

Back when I studied highway engineering they called it spiral-circle-spiral, and the vertical curves were more parabolic. The methods for laying them out were very low tech aimed at drafters and on site contractors rather than mathematicians. I can't remember the layout methods though.

Neat!

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