Description
radiusDerivative = centralAngleRadians*weight, the radius derivative of weighted spherical arc length with the angle fixed.
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.
radiusDerivative = centralAngleRadians*weight, the radius derivative of weighted spherical arc length with the angle fixed.
When to use this tool
Use Weighted Geodesic Radius Derivative when checking a stated spherical arc, chord or radius relationship.
How to use the tool and how it works
- Enter centralAngleRadians, weight. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read radiusDerivative. Use the output convention below.
Require centralAngleRadians in [0,pi] and nonnegative weight. radiusDerivative = centralAngleRadians*weight, the radius derivative of weighted spherical arc length with the angle fixed.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| centralAngleRadians | Central Angle Radians Unit: rad. | Required. Finite number; minimum 0. |
| weight | Weight | Required. Finite number; minimum 0. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| radiusDerivative | Radius Derivative | Required. Finite number. |
{
"input": {
"centralAngleRadians": 0.5,
"weight": 2
},
"output": {
"radiusDerivative": 1
}
}Limitations and assumptions
- This is a sphere-radius sensitivity, not an ellipsoidal geodesic calculation or a derivative allowing endpoints and central angle to change.
- 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
- Spherical Surface Same Latitude Chord
chord = 2*radius*cos(latitude)*abs(sin(longitudeDifference/2)).
- Distance: Spherical Arc Chord Ratio
Return arcToChordRatio = angle/(2*sin(angle/2)), using the continuous limit one at zero.