Description
Return the relative arc-to-chord shortfall.
Spherical geometry distinguishes central angle, surface arc and straight chord. They coincide only in a small-angle limit, and a sphere-radius sensitivity differs from a geodesic calculation on an ellipsoid.
Return the relative arc-to-chord shortfall.
When to use this tool
Use Angular Chord Relative Error From Angle when checking a stated spherical arc, chord or radius relationship.
How to use the tool and how it works
- Enter angle. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read relativeError. Use the output convention below.
Return the relative arc-to-chord shortfall. Require angle in (0,pi] radians; relativeError = 1-2*sin(angle/2)/angle.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| angle | Central angle in radians, strictly positive and at most pi. | Required. Finite number; strictly above 0. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| relativeError | (arc length-chord length)/arc length; radius cancels. | Required. Finite number. |
json
{
"input": {
"angle": 3.141592653589793
},
"output": {
"relativeError": 0.3633802276324186
}
}Limitations and assumptions
- The denominator is arc length and radius cancels; zero angle is rejected rather than assigned its limit.
- Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Use endpoint coordinates and an appropriate spherical or ellipsoidal geodesic model for real geographic distance.
References
Similar or alternative tools
- Geodesic Chord Small Angle Error Term
Return the leading cubic arc-minus-chord error term.
- Fixed Radius Angular Gap Halfwidth
Return a fixed-radius arc-length envelope.