Machining · toolpath geometry
Cusp Height & Stepover Calculator
Calculate theoretical cusp height or stepover for adjacent planar passes of a spherical ball-nose profile.
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Calculate theoretical cusp height or stepover
Uses planar circle geometry for adjacent passes of a spherical ball-nose profile.
Example planar geometry values are preloaded to demonstrate the equation. They are not a finishing-process recommendation.
Show calculation steps
| Step | Expression | Raw value |
|---|---|---|
| Theoretical cusp height | R − √(R² − (s/2)²) | 0.025063 mm |
How to use the cusp-height calculator
Choose whether to solve theoretical cusp height or stepover, select units, and enter the ball diameter plus the known value. The tool uses the radius internally and blocks geometry outside the circle model.
For radius R and stepover s:
h = R − √(R² − (s/2)²)
The inverse is:
s = 2√(2Rh − h²)
Both equations are independently derived from a right triangle between the ball center, pass midpoint and intersection height. No manufacturer table is copied.
Worked example
A 10 mm diameter ball has radius 5 mm. With 1 mm planar stepover, theoretical cusp height is 5 − √(25 − 0.25) = 0.025063 mm. Feeding that height back into the inverse returns 1 mm.
Model boundary
The result is a two-dimensional theoretical planning value for adjacent planar passes of a spherical profile. It does not predict measured roughness or complete three-dimensional surface finish. Tool deflection, runout, tilt, surface slope, projection of the toolpath, machine motion, material response, tool wear and non-spherical geometry are excluded.
Common mistakes
- Entering radius in a diameter field.
- Treating theoretical cusp height as Ra or another measured roughness parameter.
- Applying planar stepover geometry unchanged to a steep or changing surface.
- Using the formula for a bull-nose, barrel, lens or other non-spherical tool.
- Assuming a small theoretical cusp guarantees the machine and process can reproduce it.
FAQ
Is cusp height the same as surface roughness?
No. It is a theoretical geometric remainder between adjacent paths. Measured finish depends on the complete process.
Why can stepover not exceed ball diameter?
Beyond the diameter, the two adjacent circle profiles do not overlap within this construction.
What does zero stepover mean?
The paths coincide and theoretical cusp height is zero. It is mathematically valid but not a productive toolpath recommendation.
References and further reading
These sources define or explain the calculation. Source editions and model limits are part of the result—not decorative citations.
- Ball Nose Surface Finish Calculator, Kennametal; accessed 2026-08-22. CUSP-C01
- Scallop Height Data Calculator, Kennametal; accessed 2026-08-22. CUSP-C02
- NIST Guide to the SI, National Institute of Standards and Technology; accessed 2026-08-22. CHIP-C04