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,
Equations are shown on this PDF page:
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.
A vertical curve is a parabola which is more complicated than an arc.
The curve looks like this:
The equation derivation is on this PDF:
It can be proved that X = 1/2 * L on the below figure.
The proof is on this PDF page:
The PDF page below shows how to derive a minimum crest vertical curve equation on Page 3-166 of 7th edition of Green Book.