How Accurate Is the Spherical Law of Cosines for Calculating Distances?

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SUMMARY

The spherical law of cosines is a mathematical formula used to calculate distances between two points on a sphere. This discussion highlights its accuracy and the method for calculating error in approximations, specifically when ignoring ellipsoidal effects. To determine the uncertainty in distance due to positional errors, recalculating the distance for adjusted positions is recommended over complex differentiation. This approach simplifies the process while maintaining accuracy.

PREREQUISITES
  • Understanding of spherical geometry
  • Familiarity with the spherical law of cosines
  • Basic calculus for error analysis
  • Knowledge of positional uncertainty concepts
NEXT STEPS
  • Research the derivation and applications of the spherical law of cosines
  • Learn about error propagation techniques in geometric calculations
  • Explore methods for calculating distances on ellipsoidal models
  • Investigate numerical methods for approximating distances with positional uncertainties
USEFUL FOR

Mathematicians, geographers, and engineers involved in spatial analysis and distance calculations will benefit from this discussion.

moonman239
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Let's say I want to use the spherical law of cosines to determine the distance between two points. How do I calculate the error in the approximation (ignoring ellipsoidal effects)?
 
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Is the law of cosines an approximation for a sphere?

If you have an uncertainty in the position of each point and you want the uncertainty in the distance - you could do some horrible differentiation of the formula, OR you could just re-calcuate the distance for one position +/- the error and use that
 

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