One of the reference lines is for the angle that the cue ball comes off of the object ball for a 45-degree cut. The result is that the cue ball's path has a slope of 1:4 forward of the tangent line (kiss line, line perpendicular to the object ball's path).
This reference line is a special case of a more general rule. If you know the rule, you can figure out the path of the cue ball for any angle as long as the cue ball is rolling smoothly on the cloth when it hits the object ball.
Here is an article from December 1998 that describes the system. Diagram 2 is the important one in that article, and it was slightly munged in publication. Here is the original:
![CropperCapture[11].png](https://forums.azbilliards.com/data/attachments/134/134677-cf1a7e6c6e114442eda3910364ff5667.jpg?hash=zxp-bG4RRE "CropperCapture[11].png")
In Ralph's reference line, the cue ball starts out with a slope of the cue ball's path relative to the kiss line of 1:1 (45 degrees) and comes off with a slope of 1:4 (14.04 degrees, roughly). In the shot shown above, the cue ball starts with a slope of 4:1 roughly and comes off the ball with a slope of 1:1, or 4 times less slope.
This reference line is a special case of a more general rule. If you know the rule, you can figure out the path of the cue ball for any angle as long as the cue ball is rolling smoothly on the cloth when it hits the object ball.
Here is an article from December 1998 that describes the system. Diagram 2 is the important one in that article, and it was slightly munged in publication. Here is the original:
In Ralph's reference line, the cue ball starts out with a slope of the cue ball's path relative to the kiss line of 1:1 (45 degrees) and comes off with a slope of 1:4 (14.04 degrees, roughly). In the shot shown above, the cue ball starts with a slope of 4:1 roughly and comes off the ball with a slope of 1:1, or 4 times less slope.
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