An original variation of CTE: you sight from a point on the cue ball instead of its center, then pivot at a fixed bridge. For a left cut, sight the cue's left edge through that point to the object ball's right edge (mirrored for a right cut). Then pivot at the bridge to center ball and shoot. The page finds that sight point for any shot; the green band shows every point that still pockets.
Drag the balls or click a pocket.
setup cue (before pivot)cue after pivotshot lineghost ballpaths that pocket
The sight point
The cue ball, top view, object ball straight up. Orange: the setup cue, whose sighting edge (left for a left cut, right for a right cut) runs through the sight point S to the object-ball edge. Dashed: the cue after pivoting to center. Green: the band of sight points that still pocket the ball, drawn to true scale on the ball and on the millimetre ruler below it.
How it works
Cue: its centerline sits half a tip inside the sighting edge, and the pivot rotates it rigidly about the bridge.
Sight point:S =(b·e − b·d·tan θ − h·d) / (b + d). Here e = ±R is the object-ball edge on the ghost-ball side (right edge for a left cut, left edge for a right cut), θ the shot-line angle, h half the tip width (+ for a left cut, − for a right cut) and b the bridge's distance behind the cue ball's center (the slider's distance from the tip, plus R).
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.