Regardless of the cut angle, you can always aim the cue ball's contact point at the object ball's contact point: the line between them runs parallel to the shot line. Or aim with the cue ball's center: up to a 30° cut, at twice the contact point's distance from center. For thinner cuts, aim its edge at a point stepped inward from the contact point.
Drag the balls or click a pocket.
shot lineghost ballcontact pointcuepaths that pocket
What you look at
Looking along the shot line. Blue: the cue ball's path. K: the object ball's contact point, x from center. K′: the cue ball's contact point, x on the other side. Green: your aim.
Top view
The geometry behind the method, seen from above.
shot lineghost ballcontact point to contact pointaim point
Looking down on the shot, lettered as in "Why doubling works" below: O the object ball's center, K the contact point, K′ the cue ball's contact point, G the ghost ball, P the aim point. The white line through O is the plane through the ball's center, square to the shot line, seen edge-on. Carried along the shot line, the contact point lands x along it; P is 2x along it, exactly where it meets the shot line. P is the aim point up to a 30° cut. Past that, aim the cue ball's edge 2x − R from O.
Why doubling works
Put the object ball's center at O. Let u be its direction toward the pocket, and φ the cut angle between the cue ball's path and u. Measure sideways distances at right angles to the shot line.
The contact point is K = O − R·u. Its sideways distance from O is x = R sin φ.
The ghost ball is G = O − 2R·u, and the shot line runs through it. Its sideways distance from O is 2R sin φ = 2x.
Result
Drop a perpendicular from O to the shot line. Its foot P is on the shot line, 2x from O. So aiming the cue-ball center at the point twice as far out as the contact point is aiming exactly along the shot line.
Thin cuts. Once 2x > R (cuts over 30°), P lies off the object ball. The cue ball's edge travels along a line parallel to the shot line, R to the side. That line passes 2x − R from O, which is always on the ball. Since 2x − R = x − (R − x), you find it by stepping inward from the contact point by the gap between the contact point and the edge.
Pure geometry. These pages draw the ideal lines. Two things shift real shots: throw, friction between the balls that pushes the object ball slightly toward the thick side, and perspective, your eye misjudging distances on the balls depending on where you look from. Players learn to adjust for both with practice.