Conventional Turning Scallop and Ra Geometry
Calculate an ideal circular-nose cusp height and the common geometric Ρα approximation from feed per revolution and nose radius. The worksheet deliberately separates ideal tool-path geometry from measured surface texture.
Controlled inputs
Ideal-geometry results
Method, units and model boundary
Circle geometry: (r − h)² + (f/2)² = r²
Exact ideal cusp: h = r − √(r² − (f/2)²)
Stable equivalent: h = (f²/4) / [r + √(r² − (f/2)²)]
Small-feed cusp: h ≈ f²/(8r)
Common geometric Ra approximation: Ra,geom ≈ f²/(32r)
- f and r are entered in millimetres. The three height outputs are converted from millimetres to micrometres by multiplying by 1000.
- The exact cusp equation is the direct circle construction for adjacent ideal paths and requires 0 < f < 2r. This mathematical domain is not a manufacturer-approved operating range.
- The f²/(8r) equation is the small-feed expansion of the exact cusp geometry. The displayed relative difference shows its departure from the exact circular construction for the entered pair.
- The f²/(32r) equation is a common basic theoretical Ρα model for conventional turning. It is an approximation, not the exact mean deviation of the circular arc and not a formula issued by ISO 21920.
- Within this isolated model, decreasing feed or increasing radius lowers the geometric values. In a real process, insert limits, forces, vibration, chip formation, minimum chip thickness, edge wear and wiper geometry can invalidate that simple trend.
What the two-input model does not include
| Excluded factor | Why it matters | Required control |
|---|---|---|
| Wiper, profile or form insert | The active edge is not represented by one circular nose radius, so the equations do not describe its generated profile. | Use the insert manufacturer’s geometry-specific guidance or a validated profile model. |
| Machine and setup dynamics | Runout, chatter, compliance and vibration can dominate the measured surface. | Verify setup stability and measure the produced surface. |
| Material and cutting condition | Adhesion, ploughing, built-up edge, cutting speed, depth of cut and wear affect actual texture. | Qualify the process for the actual material, tool and condition. |
| Surface-texture specification | Measured parameters depend on the specified profile, operator, nesting/index and evaluation conditions. | Apply the current drawing specification and the applicable ISO 21920 measurement chain. |
| Conformity decision | A calculated ideal value has no measurement uncertainty and cannot prove acceptance. | Use measured results, uncertainty and the contractually applicable decision rule. |
Evidence and status
Identifies Ra = f²/(32rₑ) as a basic theoretical model, then explicitly explains that it omits process imperfections such as vibration and chip adhesion and can disagree with experiments at low feed.
University-hosted peer-reviewed paperGives f²/(8R) for the ideal peak-to-valley feed-groove geometry under an explicitly ideal tool path. This worksheet calls it a cusp-height approximation, not a measured ISO Rt value.
Boston University author-hosted paperExplains that a larger nose radius can permit higher feeds while a smaller radius may be needed when vibration occurs, and shows that wiper inserts generate a different feed/finish relationship.
Official manufacturer handbookThe official ISO card states that this published document specifies terms, definitions and profile surface-texture parameters. It replaced withdrawn ISO 4287:1997 and is currently at stage 90.92, to be revised. No closed clause is used to claim that the cutting formula is an ISO formula.
Official ISO recordSources accessed 16 July 2026. The full ISO parameter and operator requirements require the applicable licensed standards and drawing specification.
Interpretation questions
Is f²/(32r) an ISO formula?
No. It is a common two-input geometric approximation used in machining literature. ISO 21920 defines the surface-texture parameter framework; it does not turn this tool-path model into a standardized measurement or acceptance calculation.
Is f²/(8r) the same as measured Rt?
No. Here it is the small-feed approximation to one ideal circular-nose cusp. A measured profile parameter depends on the specified measurement and evaluation procedure and includes the actual generated surface.
Can actual Ra be estimated by multiplying by two?
Not generally. The ratio is process-dependent and may change with tool geometry, feed regime, material behaviour, vibration, wear and measurement conditions. This worksheet therefore supplies no universal actual-Ra multiplier.
Does a larger nose radius always improve the part?
No. It lowers these ideal geometric values at fixed feed, but it can also change cutting forces and vibration tendency. Use the tool manufacturer’s operating data and qualify the real process.
Revision: 16 July 2026. Result classification: approximate conventional-turning geometry only.
