AMATEUR RADIO STATION
NEW ZEALAND
A Bit of Geometry Behind the Real Distance
For simplicity, consider one hop with the following assumptions:
Using a spherical‐earth approach the actual path length for one hop (Lₕ) can be estimated by finding the two “legs” (from transmitter up to the reflection point and from the reflection point down to the receiver). Although the exact derivation requires applying the cosine law in a triangle with vertices at the Earth’s center, transmitter, and ionospheric reflection point, the key result is that the radio path ends up being a few percent longer than the ground distance.
For example, detailed spherical‐geometry calculations (with R ≃ 6,400 km and h ≃ 300 km) typically yield an actual hop path on the order of about 4,120 km when the ground distance is 4,000 km. That is, each hop adds roughly 120 km extra—that’s about a 3% overhead per hop.
Extrapolating to the Antipodal Path
The antipodal distance along the Earth’s surface is roughly 20,000 km (half of the Earth’s circumference). If conditions allow using the longest “optimum” hops (say, around 4,000 km per hop), then you would use about 5 hops to bridge that 20,000 km separation. In each hop the radio wave travels roughly 4,120 km instead of the 4,000 km measured along the surface. Multiplying,
Alternatively, if ionospheric conditions only permit shorter hops—for example, 3,000 km per hop—then you’d need roughly 7 hops for the antipodal path. In such cases the individual hop geometry tends to add around 5–6% extra per hop, so the cumulative extra distance could approach 1,100–1,200 km over 20,000 km—roughly a 5–6% overhead overall.
Summing It Up
In summary:
When propagating under optimum conditions (with long hops on the order of 4,000 km), the extra distance an HF signal travels compared to the straight great‐circle path is only about 3–4%. For an antipodal path (≈20,000 km along the Earth’s surface), this means the total radio path might be around 20,600–21,000 km—roughly 600 to 1,000 km longer than the geodesic distance. If circumstances force the use of shorter hops (around 3,000 km), the extra distance can rise to about 5–6% of the surface path.
These numbers are, of course, approximations. Real‐world factors—the variability of the ionosphere’s effective height, the precise angles of incidence, and even frequency‐dependent refractive effects—can slightly adjust these percentages. Nonetheless, the overall take‐home message is that even though the physical “skip” adds an extra loop to the journey, the additional path length is modest compared to the total distance.
For further reading, one might explore HF propagation models (such as those discussed in ITU recommendations and amateur technical articles on propagation paths) that examine how angles, heights, and environmental factors influence these extra distances in more detail. 1 2