When can you no longer cross the tangent line??

[Bob Jewett]

... For a fairly full hit (less than a 1/4-ball hit), the CB will deflect about 2.5 times the cut angle (e.g., for a shot with a cut angle of 10 degrees, the CB will deflect about 25 degrees).

For a fairly thin hit, with a ball-hit-fraction greater than 3/4, the CB will deflect about 70% of the angle between the aiming line and the tangent line.
...

While this is true for perfect elasticity and no ball-ball friction, I find that using 3 times in the first case (for a nearly full-ball hit) and 75% in the second case seems to work pretty well. In the case of 75%, it's easy to bisect and then bisect again.

But rather than taking part of an angle, I think it is better to take part of a tangent, since that eliminates the tan(theta)=theta approximation.

 

[Patrick Johnson]

But the tangent line isn't like the one in your drawing. It passes through the center of the CB, not the OB - like the blue line in your drawing, not the yellow one. Maybe that's what has confused you?

That's not correct either. The tangent line does not pass through the center of either ball, but is tangent to both balls. It's halfway between the yellow line and the blue one.

That's the definition of tangent as it applies to geometry, but when we say the CB "travels along the tangent line" or "curves off the tangent line", I believe we commonly mean the line its center follows, not its edge.

pj
chgo

 

[dr_dave]

CB path shift down tangent line

As with the 30 degree rule and trisect system, the full-hit and thin-hit rules apply to the final direction of the CB. The actual final path of the CB is shifted down the tangent line with higher speed.

I think there is a fine point here that is being missed (or at least underemphasized). For a rolling ball, the initial path is a portion of a parabola. With higher speed the scale of the parabola increases, but it is still a parabola. I often see people here say in posts things like "the CB stays on the tangent line longer" for shots hit at higher speed. As I understand it, this is false. The path of the CB leaves the tangent line immediately. The better wording would be "the CB stays NEAR the tangent line longer". Please correct me if I am wrong.

You bring up a good point. Good illustrations of this can be found in my March '05 article. I still think my statement is valid about the final CB path being shifted down the tangent line. Diagram 2 in my June '05 article illustrates clearly what I mean.

Regards,
Dave

 

[dr_dave]

standard definition of "tangent line"

That's not correct either. The tangent line does not pass through the center of either ball, but is tangent to both balls. It's halfway between the yellow line and the blue one.

Some people define it this way, but this isn't how the term is most commonly used (e.g., in most books, articles, and forum postings). Diagram 1 in my July '04 article illustrates the "standard" definition.

Regards,
Dave

 

[dr_dave]

thin and full hit approximations

While this is true for perfect elasticity and no ball-ball friction, I find that using 3 times in the first case (for a nearly full-ball hit) and 75% in the second case seems to work pretty well. In the case of 75%, it's easy to bisect and then bisect again.

For a rolling ball, the effects of ball friction and inelasticity tend to cancel each other out (see my April '05 article). When I get some time, I'll try to look at this closer for thin and full hits.

But rather than taking part of an angle, I think it is better to take part of a tangent, since that eliminates the tan(theta)=theta approximation.

Agreed. If somebody is not good at visualizing 70% of an angle, your recommended line and distance trick is a good alternative.

Regards,
Dave