Cassette teeth design
I hear that there are books and papers that discuss the shape of the teeth and they also have equations used to determine the shape using the pitch and roller diameter dimensions from the chain...  if anyone has any information on this or has a website/book/technical report where I can find more info about this topic please let me know.


This is a huge topic I'm going to bucket the conversation into two groups.
1. Information on products currently being manufactured.
2. General roller sprocket and roller chain design - this is the best source of detailed information. Google and/or search Amazon books "roller chain design" or "roller sprocket design".

1. Information on currently manufactured products.
Bicycle drive train makers (Shimano and others) have spent a great deal of money trying to make chains and cogs that shift better in hopes of giving them a competitive advantage in the market place. The details of their design are proprietary and patented. They will not publish enough details for you to re-create their designs.

There are people who have spent the time reverse engineering products.

For example:

Peter Verdone has a page that details a Shimano hyperglide 9 spd cassette interface.


Konstantin Shemyak spent some time examining the teeth of hyperglide sprocket teeth and has posted his findings.


GRABCAD has drawings of cassettes and chains.



Here is a deep dive into some Shimano chainring designs using information from the patent office from BIKERUMOR!.


Chasing the patent trail - Google has a nice patent search engine. Here is an example of a "bicycle sprocket assembly"


2. General roller sprocket and roller chain design
The roller chain / roller sprocket drive mechanism is over 100 years old. If you look for information that is not specific to current bicycle drive trains and generalize to sprocket and chain design you can find a world of technical information on the web an machinist handbooks like this document on Gears Educational Systems.  

Or an article on designing a single speed cog as just two examples of the mountain of information available.