Taper Missing Diameter (Lathe Setup)
The inverse of the taper tile, for cutting a taper to a spec: given the taper per foot, one known end diameter, and the length, the missing end diameter is known -/+ (TPF/12) x L. A 1.000 in large end at 0.600 in/ft over 3 in makes a 0.850 in small end (drops 0.050 in per inch of length); the compound angle per side = atan(TPF/24) = 1.432 deg, which depends only on the TPF so the same setting cuts the taper at any length. First-principles trigonometry; the tool nose radius and setup govern the finished part.
Formula and source
change = (TPF/12) x L; small end = known - change (large end = known + change); angle per side = atan(TPF/24); included = 2 x (angle per side).
Taper setup inverse - the missing end diameter and the compound angle from the taper per foot - first-principles trigonometry as in Machinery's Handbook (Industrial Press), by name; public domain.
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