What is the cusp height formula?

Cusp (scallop) height h = R - sqrt(R^2 - (s / 2)^2), with R the ball radius and s the step-over in the same unit. A 0.25 in radius at 0.010 in step-over gives 0.25 - 0.249950 = 0.000050 in, or 50 microinches.

How do I calculate step-over from a target scallop height?

Rearrange the formula: step-over = 2 x sqrt(R^2 - (R - h)^2). For a 63 microinch limit on a 0.25 in radius ball nose, that is 2 x sqrt(0.0625 - 0.249937^2) = about 0.0112 in. A close approximation is 2 x sqrt(2 x R x h).

Is scallop height the same as surface roughness Ra?

No. Scallop height is the peak-to-valley height of the theoretical ridge; Ra averages the deviation over the surface and is much smaller for the same profile. Use the scallop height against a peak-to-valley limit, and measure Ra if the print calls for it.

Does tool diameter or step-over matter more?

Step-over. Cusp height grows with the square of the step-over but only falls in proportion to the radius: doubling the step-over roughly quadruples the cusp, while doubling the tool diameter roughly halves it.

Does the formula work for metric tools?

Yes, use any unit as long as R and step-over share it. A 10 mm ball nose (R = 5 mm) at 0.3 mm step-over leaves 5 - sqrt(25 - 0.0225) = 0.00225 mm, or 2.25 micrometers.

Engineering and Process

Scallop Height Formula (Cusp Height)

Scallop height, also called cusp height, is the ridge a ball-nose cutter leaves between adjacent passes. Use the formula to pick a step-over that meets a surface-finish limit, or to check the finish a CAM step-over will leave.

Formula

Scallop Height = R - sqrt(R^2 - (Step-Over / 2)^2)

Variables

R: Ball-nose tool radius (half the tool diameter), in inches or mm
Step-Over: Distance between adjacent passes, measured across the surface, in the same unit
Scallop Height: Height of the cusp left between passes on a flat surface, in the same unit

Understanding the Scallop Height Formula (Cusp Height)

A ball-nose cutter leaves a circular groove on each pass; where two grooves meet, a ridge remains. Its height is the radius minus the distance from the tool center to the surface at half the step-over, which is the square-root term. In the example, a 0.25 in radius at 0.010 in step-over leaves 0.000050 in, or 50 microinches.

For small step-overs the formula is close to step-over squared / (8 x R), which shows why finishing passes are expensive: halving the cusp height needs only about 29% less step-over, but quartering it doubles the number of passes. Solve for step-over when you start from a finish limit: step-over = 2 x sqrt(R^2 - (R - h)^2), which gives 0.0112 in for 63 microinches on the 1/2 in tool.

The formula assumes a flat surface and a true ball. On a slope, the effective step-over across the surface grows, and on a convex or concave surface the cusp changes again; CAM systems that hold a constant scallop adjust the step-over for that. Runout and tool deflection add to the theoretical cusp, so leave margin to the print limit.

Worked Example

A 1/2 in ball nose (R = 0.25 in) finishes a flat area at 0.010 in step-over. The print allows 63 microinches of cusp.

Half step-over = 0.010 / 2 = 0.005 in
sqrt(0.25^2 - 0.005^2) = sqrt(0.0625 - 0.000025) = 0.249950 in
Scallop height = 0.25 - 0.249950 = 0.000050 in = 50 microinches
Margin to the 63 microinch limit = 63 - 50 = 13 microinches

Result: 50 microinches (0.000050 in) of cusp height, 13 under the limit

Common Mistake

Treating cusp height as surface roughness Ra. The formula gives the peak-to-valley height of the theoretical cusp on a flat surface; Ra is an average and comes out at a fraction of that, and slopes and tool runout change the real figure.

Frequently Asked Questions

What is the cusp height formula?: Cusp (scallop) height h = R - sqrt(R^2 - (s / 2)^2), with R the ball radius and s the step-over in the same unit. A 0.25 in radius at 0.010 in step-over gives 0.25 - 0.249950 = 0.000050 in, or 50 microinches.
How do I calculate step-over from a target scallop height?: Rearrange the formula: step-over = 2 x sqrt(R^2 - (R - h)^2). For a 63 microinch limit on a 0.25 in radius ball nose, that is 2 x sqrt(0.0625 - 0.249937^2) = about 0.0112 in. A close approximation is 2 x sqrt(2 x R x h).
Is scallop height the same as surface roughness Ra?: No. Scallop height is the peak-to-valley height of the theoretical ridge; Ra averages the deviation over the surface and is much smaller for the same profile. Use the scallop height against a peak-to-valley limit, and measure Ra if the print calls for it.
Does tool diameter or step-over matter more?: Step-over. Cusp height grows with the square of the step-over but only falls in proportion to the radius: doubling the step-over roughly quadruples the cusp, while doubling the tool diameter roughly halves it.
Does the formula work for metric tools?: Yes, use any unit as long as R and step-over share it. A 10 mm ball nose (R = 5 mm) at 0.3 mm step-over leaves 5 - sqrt(25 - 0.0225) = 0.00225 mm, or 2.25 micrometers.