Horizontal and vertical curves are important concepts in a highway design class.
However, many textbooks do not explain how to derive equations. When I
was preparing for Professional Engineering (PE) exam a few years ago, I spent
some time figuring out how to derive those equations. I will publish the
work on this web page. College students and engineers preparing for PE exam may
find them useful.

A simple horizontal curve is an arc (part of a circle). The curve looks
like this,

If you understand how to derive horizontal curve equation, the Horizontal Sightline Offset (HSO)
equation on page 3-115 of 7th edition of
Green Book
is very similar.