BIZOMAT MAKEOS · Manufacturing
CNC Arc Linearisation Chord Error
What chord error do these line segments leave on an arc, or what segment length meets a tolerance?
Chordal deviation of straight line segments approximating an arc or contour, the longest segment for a given tolerance, and the number of segments for the arc.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Calculate | choice | Find the segment length for a tolerance, or the error for a segment length. |
| Arc radius | mm | Radius of the arc or local radius of curvature of the contour. |
| Chord tolerance | mm | Allowed deviation between chord and arc. |
| Segment length | mm | Length of each straight segment (error mode). |
| Arc angle | ° | Angle of the arc, for the number of segments. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Calculate
- What it is
- Whether the tool finds the longest straight segment allowed for a chord tolerance, or the chord error produced by a segment length you already use.
- Units and format
- Choose one of 2 options from the list: Segment length for a tolerance; Error for a segment length.
- Example
- The form opens with “Segment length for a tolerance” selected as a worked example. Choose the option that matches your case.
What this means: Arc radius
- What it is
- The radius of the arc being approximated by straight moves, or the local radius of curvature at the tightest part of a freeform contour.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger radius allows longer segments for the same tolerance.
- Get it right
- For a contour, use the smallest radius of curvature, since that is where chord error is largest.
What this means: Chord tolerance
- What it is
- The largest gap you accept between a straight segment and the true arc, measured at the middle of the segment. Used when finding the segment length.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.005 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Allowed segment length rises roughly with the square root of the tolerance.
- Get it right
- This is a one-sided deviation; it should be a small share of the part tolerance, since other errors add to it.
What this means: Segment length
- What it is
- The length of each straight move the CAM system or control uses along the arc. Used only when calculating the chord error.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Chord error rises roughly with the square of the segment length.
- Get it right
- It must be shorter than the arc diameter.
What this means: Arc angle
- What it is
- The angle the arc sweeps around its centre, used to count how many segments are needed to cover it.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.001° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 90° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The number of segments rises in proportion to the arc angle.
- Get it right
- A full circle is 360°; a quarter arc between two quadrant points is 90°.
Method and formulas
- Error e = R (1 − cos(θ/2)), where θ is the angle subtended by one segment.
- Segment length L = 2√(2Re − e²).
- Segments = arc angle ÷ θ, rounded up.
- Example: R 50 mm with a 0.005 mm tolerance allows 1.4142 mm segments, 56 for 90°.
Limitations
- Error on one side of the profile only (chords lie inside a convex arc); CAM systems may split the tolerance inside and outside.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Set CAM output tolerance
- Check whether a posted program meets the profile tolerance
- Estimate block count for a contour
Who it is for
- Machinists
- CNC programmers
- Manufacturing engineers
What you get
- Segment length and angle
- Chord error
- Segments for the arc
Interpreting the result
- Very small tolerances create huge programs; use arc output (G02/G03) or spline interpolation where the control allows.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 1279–1300, 77–80. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Profiles that cut cleanly
In Drafting Geometry by the Numbers
What it covers: Prepare 2D geometry for laser, plasma and waterjet cutting: arcs versus segments, chord tolerance, layers, units, lead-ins, cut length and pierce count, with a cutting estimate.
Method explained here · BIZOMAT original book
Draft, splits and releasing the model
What it covers: Split a moulded part into half-shells, check draft against the pull direction, account for what draft does to flow area, and export and release a surface-built model a toolmaker can trust.
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