I programmed a virtual FMS for testing of new airways and STARs, and I completely agree with the comment left by @MikeBrass. Ultimately, there is no standard for how FMS systems complete this task as long as they come to the same result.

It is very simple trigonometry and other, high school level, math concepts. The FMS knows the 3D coordinates for the waypoint (We'll call these X2,Y2,Z2) as well as the current GPS coordinates, altitude, true heading, and a myriad of other sensors that it uses to calculate the aircraft's current position in space (X1,Y1,Z1). It also uses the differences in GPS GS and CAS to work out the forward (Or rearward) component of the wind, and the differences in True Heading and TMG to calculate the sideways component of wind. We now have all the information required to complete calculations for nearly any scenario.

Required Crossing Altitude Scenario / Speed Restraints:

In this instance, the first calculation would be one of Altitude to gain (OR loose). This is as simple as Y2-Y1. Now, assuming a fixed (Pilot set speed or speed restrictions based on procedure) speed we can easily calculate a climb rate in FPM. This is done by creating a 2D line that travels between points X2,Y2 and X1,Y1 and calculating distance between those 2 points, along the line. Dividing this by GPS ground speed, and computing heading and GS changes along this leg similar to how a student pilot might do with a "Whiz Wheel". If the pilot changes speed, or the forward component of the wind changes during the climb or decent, you just need to run this calculation again but with the new position of the aircraft in 3D space.

Curved approach segments complicate this slightly but interpreting these as a 2D curve will greatly simplify your calculations.

If this doesn't answer your question, you may want to reword it.

I programmed a virtual FMS for testing of new airways and STARs, and I completely agree with the comment left by @MikeBrass. Ultimately, there is no standard for how FMS systems complete this task as long as they come to the same result.

It is very simple trigonometry and other, high school level, math concepts. The FMS knows the 3D coordinates for the waypoint (We'll call these X2,Y2,Z2) as well as the current GPS coordinates, altitude, true heading, and a myriad of other sensors that it uses to calculate the aircraft's current position in space (X1,Y1,Z1). It also uses the differences in GPS GS and CAS to work out the forward (Or rearward) component of the wind, and the differences in True Heading and TMG to calculate the sideways component of wind. We now have all the information required to complete calculations for nearly any scenario.

Required Crossing Altitude Scenario / Speed Restraints:

In this instance, the first calculation would be one of Altitude to gain (OR loose). This is as simple as Z2-Z1. Now, assuming a fixed (Pilot set speed or speed restrictions based on procedure) speed we can easily calculate a climb rate in FPM. This is done by creating a 2D line that travels between points X2,Y2 and X1,Y1 and calculating distance between those 2 points, along the line. Dividing this by GPS ground speed, and computing heading and GS changes along this leg similar to how a student pilot might do with a "Whiz Wheel". If the pilot changes speed, or the forward component of the wind changes during the climb or decent, you just need to run this calculation again but with the new position of the aircraft in 3D space.

Curved approach segments complicate this slightly but interpreting these as a 2D curve will greatly simplify your calculations.

If this doesn't answer your question, you may want to reword it.