Inclined Fire

D.M., may his days be blessed, done the English translation of the "Inclined fire" series of articles. He is too modest to claim proper credit, but without his help this site would not be. My gratitude is as deep as thorough.

Further down:

Definitions (what are we talking about -- a must read to understand the rest)
When angles matter, and when they don't
Comparative analysis of existing approximation methods (NDS, Rifeman's rule, Improved Rifleman's Rule, Sierra's)
The best method -- 2I2R (invented it all by my little self)
And how to use it in real life

[See also: all notes on inclined fire as a single page.]

Brace for impact.

TL/DR summary for the lazy ones

(If all of a sudden Zombie Apocalypse strikes, and you just don't have time to read it all.)

1. NDS

Accuracy: bad
Domain: up to ≃400-500 m
Complexity: moderate -- 1 table lookup, 2 mathematical operations
Pros: none
Cons: inelegant and useless
Conclusion: Zombies would laugh their heads off

2. Rifleman's Rule

Accuracy: acceptable
Domain: 30º, < 500 m. Maybe a little more if holding a correction.
Complexity: very simple -- 1 table lookup, 1 mathematical operation, often automated
Pros: beautiful simplicity, independence from rifle and caliber, no need to amend or extend existing ballistic tables
Cons: surprisingly limited applicability
Conclusion: Not far away zombies suffer

3. Improved Rifleman's Rule

Accuracy: good
Domain: up to ±30º -- all distances. Up to ±40º -- within 800 m with additional hold
Complexity: easy -- 1 table lookup, 1 mathematical operation
Pros: reasonable accuracy, simplicity, can use existing tables.
Cons: depends on zero distance
Conclusion: Zombies start getting really, really scared.

4. Sierra's method

Accuracy: A+!
Domain: as far as the eye can see
Complexity: complex! 2 table lookups, 3 mathematical operations
Pros: accuracy is as good as it gets
Cons: too complex to be useful in the field, requires extra data (regular ballistic tables are not enough)
Conclusion: Zombies would have time to eat your ass off before you sort out the dope.

5. World's premiere -- 2I2R (also for MOA turrets)

Accuracy: A
Domain: all practical angles, distances, and calibers
Complexity: easy -- 1 table lookup, 1.5 mathematical operations
Pros: accuracy, simplicity, can use existing ballistic tables
Cons: only works with 100m / 100 yd zero; procedure depends on click value
Conclusion: Zombies stand no chance