Let’s work through an example step-by-step. Suppose you’re calculating the “true” distance between two points on Earth, where you measure the two legs of a right triangle on the Earth’s surface.
Given:
- Leg a = 300 km
- Leg b = 400 km
- Earth's curvature radius R = 6,371 km
- For Earth’s spherical geometry, we set h = -1
Our Formula:
a² + b² + h (a²b² / R²) = c²
Step 1: Compute the Classical Sum
First, calculate the flat (Euclidean) part:
a² + b² = 300² + 400² = 90,000 + 160,000 = 250,000 km².
Step 2: Compute the Correction Term
Next, calculate the curvature correction:
(a²b² / R²) = (300² × 400²) / (6,371)²
- 300² = 90,000
- 400² = 160,000
- So, a²b² = 90,000 × 160,000 = 14,400,000,000 km⁴.
Now, compute R²:
R² = 6,371² = 40,600,000 km².
Thus,
(a²b² / R²) = 14,400,000,000 / 40,600,000 = 354.2 km².
Since h = -1, the correction term is:
h (a²b² / R²) = -354.2 km².
Step 3: Sum Up and Solve for c
Now, plug the values into the full equation:
c² = a² + b² + h (a²b² / R²) = 250,000 - 354.2 = 249,645.8 km².
Finally, take the square root to find c:
c = √249,645.8 = 499.65 km.
Interpretation:
- Classical Result: Without curvature, we would have c = √250,000 = 500 km.
- Corrected Result: The curvature correction reduces the geodesic distance slightly to about 499.65 km.
Even though the correction here is small (only about 0.35 km), in other contexts—especially over larger distances or in different curvature regimes (like cosmic scales)—this term can become much more significant.
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Summary of the Process:
1. Measure the sides a and b.
2. Identify the curvature radius R of the Earth (or the relevant space).
3. Set h = -1 for spherical (positively curved) geometry.
4. Plug in the values into the formula:
a² + b² + h (a²b² / R²) = c².
5. Solve for c to obtain the true geodesic distance.
Some of those results are approximations but I don't have the symbols for that on the DU.
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