Ballnose Feed-Direction Cusp and the Governing Finish

It is the same geometry rotated ninety degrees, h = R - sqrt(R^2 - (fz/2)^2) with feed per tooth in place of stepover -- and the reason to see both at once is that the surface you get is the LARGER of the two. A 1/2 in ballnose at 0.030 in stepover and 0.006 in per tooth leaves 0.000450 in across the passes and only 0.000018 in along the feed, so the stepover governs by 25x and tightening the feed for finish is wasted cycle time. The two balance exactly when feed per tooth equals stepover. Both scale with the SQUARE of their spacing, so halving either cuts its cusp by four. Geometric headroom on the feed is not permission -- chip load, deflection, and the machine usually bind first; it only means finish is not the reason to hold back.

cusp h = R - sqrt(R^2 - (w/2)^2) evaluated twice: with w = stepover for the cusp across the passes (delegated to ballnose-scallop-height) and with w = feed per tooth for the cusp along the feed. Governing cusp = max of the two; the two are equal when feed per tooth = stepover. Near the bottom both reduce to w^2/(8R). Feedrate = feed per tooth x flutes x rpm.

Ballnose cusp geometry as used in CAM and mold machining. Both directions are the same circular-segment construction; nothing here is tabulated or proprietary.

Circular-segment geometry. No chart, table, or manufacturer data is used.

Estimate. AHJ and licensed professional govern.

Field names used by the API: r_in, stepover_in, feed_per_tooth_in, rpm, flutes, stepover_cusp_in, feed_cusp_in, governing_cusp_in, ratio, feedrate_ipm, balanced_feed_in

• Same geometry both ways h = R - sqrt(R^2 - (w/2)^2) with w = stepover or feed per toothcircular-segment geometry
• Governing cusp the larger of the two is what appears on the partsurface geometry
• Balance point the cusps are equal when feed per tooth = stepoverfollows from the shared formula
• Stepover cusp delegated to computeBallnoseScallopHeight, not reimplementedlanded sibling tile
• Feed headroom geometric only; chip load, deflection, and the machine usually bind firststated caution
• Not Ra peak-to-valley geometry on a flat surface at the tool tip; slope, deflection, and runout excludedstated scope limit