Find the scallop height from the stepover, or the stepover for a wanted height: ball end mills on flat, convex and concave surfaces, bull nose mills, and the feed marks of turning and boring.

This is the shape the tool's geometry leaves, nothing more. A real surface is rougher, from vibration, tool deflection and the marks of the cutting edge. Use it to choose the stepover, then measure the real surface to confirm.

What scallop height is

A ball end mill machines a surface one pass at a time, each pass a stepover ae from the last. The curved tips of neighbouring passes do not overlap completely, so a small ridge of material is left between them. Its height above the bottom of the passes is thescallop height h (cusp height). A smaller stepover gives a smoother surface, but more passes and a longer machining time; halving ae cuts h to about a quarter.

Ball end mill on a flat or inclined surface

The ball has a radius $R = D/2$ , so every pass has the same circular section. The scallop height is

h = R - \sqrt{R^2 - \left(\frac{a_e}{2}\right)^2} \qquad a_e = 2\sqrt{2Rh - h^2}

An inclined surface uses the same formula if ae is measured along the slope (the distance between passes at right angles to the cut, on the surface). CAM that sets the stepover horizontally (2D stepover) gives a longer distance on a slope, $a_e / \cos\alpha$ , where $\alpha$ is the slope angle, so scallops on steep walls are taller than on flat areas.

Ball end mill on a curved surface

On a surface of radius $R_w$ , the ball's centre runs on a circle of radius $R_w + R$ (convex) or $R_w - R$ (concave). Neighbouring passes are an angle $\theta = a_e / R_w$ apart when ae is measured along the surface, and the top of the scallop is where the two balls' circles cross.

Convex surface

h = (R_w + R)\cos\tfrac{\theta}{2} - \sqrt{R^2 - (R_w + R)^2 \sin^2\tfrac{\theta}{2}} - R_w

Concave surface

h = R_w - (R_w - R)\cos\tfrac{\theta}{2} - \sqrt{R^2 - (R_w - R)^2 \sin^2\tfrac{\theta}{2}}

At the same stepover, a convex surface has taller scallops than a flat one and a concave surface lower ones. A concave surface must have a larger radius than the ball, or the tool cannot fit.

Bull nose end mill on a flat surface

A bull nose mill with corner radius r has a flat bottom $D - 2r$ wide. If the stepover is no more than this, the flat bottoms of neighbouring passes overlap and no scallop is left; past it, the extra leaves a scallop from the two corner radii.

h = r - \sqrt{r^2 - \left(\frac{a_e - (D - 2r)}{2}\right)^2}

On inclined or curved surfaces the part of this tool that touches the surface changes with the surface angle, so this calculator handles bull nose mills on flat surfaces only.

Turning and boring

A turning tool or boring bar cuts a helix round the work, moving along the axis by the feed f (mm/rev) each revolution. The tip has a nose radius rε, so the marks of neighbouring revolutions leave ridges just as a ball end mill does. These are thefeed marks, given by the flat-surface formula with rε in place of R and f in place of ae.

h = r_\varepsilon - \sqrt{r_\varepsilon^2 - \left(\frac{f}{2}\right)^2} \approx \frac{f^2}{8\,r_\varepsilon}

For example, an rε 0.8 insert at 0.2 mm/rev gives h of about 6.3 µm and Ra of about 1.6 µm. Cylinders, faces, tapers and bored walls are straight lines in an axial section, so they use this formula; turning a curved profile uses the convex or concave formula above.

Bores are usually rougher than this. A boring bar sticks out a long way for its size, so it vibrates and deflects more easily than an external turning tool.
A larger nose radius gives a smoother finish but more radial cutting force. Boring and thin-walled work therefore often use a small radius to reduce chatter.
A very low feed usually does not improve the finish as the formula says. When the chip gets too thin, the edge rubs instead of cutting.
Wiper inserts have a trailing edge that wipes the ridges and give a smoother finish than this formula at the same feed. The formula is for single-radius inserts only.

At the same feed, the wiper edge trailing the tip wipes the top off the previous ridge, leaving about half the height. That is why insert makers say a wiper runs at about twice the feed for the same finish, or gives about twice as smooth a finish at the same feed. Wipers work best for straight turning and facing. The drawing shows the idea; real edge shapes differ between makers.

Surface roughness

The section of the scallops is an arc repeated every ae, so the height from peak to valley is h (Rz ≈ h), and when ae is small against the radius, the average roughness Ra is about $0.257\,h$ , as for turning feed marks. This is the geometric minimum; a real surface is usually rougher. The option from the Ra on the drawing runs this backwards: it takes h = Ra ÷ 0.257 and finds the stepover or feed that leaves it, so Ra 1.6 µm becomes h of about 6.2 µm.

Along the cutting pass of a mill there are small ridges from the feed per tooth fz in the same way. The flat-surface formula gives them with fz in place of ae; they are usually much lower than the scallops from the stepover.