Reduced Distance Calculator

Reduced Distance Formula:

\[ K = R \times \sqrt{\frac{(D - (H_2 - H_1)) \times (D + (H_2 - H_1))}{(R + H_1) \times (R + H_2)}} \]

Earth Radius (R):

Distance Traveled (D):

Elevation of a (H₁):

Elevation of b (H₂):

Reduced Distance (K):

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1. What is Reduced Distance?

Reduced distance is the distance which is reduced over the ellipsoid between the projections of the two points onto the ellipsoid. It accounts for the curvature of the Earth and elevation differences between points.

2. How Does the Calculator Work?

The calculator uses the reduced distance formula:

\[ K = R \times \sqrt{\frac{(D - (H_2 - H_1)) \times (D + (H_2 - H_1))}{(R + H_1) \times (R + H_2)}} \]

Where:

\( K \) — Reduced distance (m)
\( R \) — Earth radius (km)
\( D \) — Distance traveled (m)
\( H_1 \) — Elevation of point a (m)
\( H_2 \) — Elevation of point b (m)

Explanation: The formula accounts for the Earth's curvature and elevation differences to calculate the reduced distance over the ellipsoid.

3. Importance of Reduced Distance Calculation

Details: Calculating reduced distance is crucial for accurate geodetic measurements, surveying, and navigation applications where the Earth's curvature and elevation differences must be considered.

4. Using the Calculator

Tips: Enter Earth radius in km, distance traveled in meters, and elevations of both points in meters. All values must be valid (Earth radius > 0, distance ≥ 0).

5. Frequently Asked Questions (FAQ)

Q1: Why is Earth radius typically given in km but converted to meters?
A: The formula requires consistent units. Earth radius is commonly referenced in km, but the calculation converts it to meters to match the other distance measurements.

Q2: What is the typical value for Earth radius?
A: Earth radius ranges from approximately 6,357 km to 6,378 km, depending on the specific ellipsoid model used.

Q3: When should reduced distance be used instead of straight-line distance?
A: Reduced distance should be used when accounting for Earth's curvature and elevation differences is important, such as in geodetic surveying and long-distance navigation.

Q4: Are there limitations to this formula?
A: This formula assumes a spherical Earth model. For extremely precise calculations, more complex ellipsoidal models may be required.

Q5: How does elevation difference affect the reduced distance?
A: Greater elevation differences between points result in larger corrections to the measured distance when calculating the reduced distance over the ellipsoid.