Pinnacle points are a points from which no higher point can be seen. In other words, at a pinnacle point you would be at the highest elevation in sight. Two points are defined to have line of sight if light can theoretically travel from one to the other unobstructed in clear atmospheric conditions. The curvature of the Earth, atmospheric refraction, and local topography are all taken into account. It is possible for two pinnacle points of equal elevation to have line of sight since neither is tall enough to disqualify the other.

Links

• Explanation of method
• Github directory
• Relative Hills Society article that (briefly) mentions pinnacle points

Data Sources

1. On-Top-Of-The-World Mountains

On-top-of-the-world (OTOTW) mountains are mountains where no land rises above the horizontal plane from its summit. Since any land that rises above the horizontal plane would have a higher elevation than the mountain itself, if a mountain is not an OTOTW mountain then it can not be a pinnacle point either. In other words, pinnacle points are a subset of OTOTW mountains. Kai Xu found all 6,464 OTOTW mountains on Earth with over 300 m of prominence, and I have identified which qualify as pinnacle points. Andreas Geyer-Schulz deserves mention as well for his extremal peaks, a nearly identical concept to OTOTW mountains developed completely independently.

2. Mountains by Prominence

Andrew Kirmse and Jonathan de Ferranti found all 11,866,713 summits on Earth with over 100 ft (~30 m) of prominence. Prominence is the minimum vertical distance one must descend to reach a higher point. Kai Xu identified OTOTW mountains using this dataset, so I use this dataset to identify which OTOTW mountains are pinnacle points. This source primarily uses the Copernicus GLO-30 DEM.

3. Mountains by Isolation

Andrew Kirmse and Jonathan de Ferranti found all 24,749,518 summits on Earth with over 1 km of isolation. Isolation is the distance to the nearest higher point. Extreme isolation points are strong pinnacle point candidates, so I use this dataset to find all pinnacle points with an isolation of at least 100 km. This source uses the STRM 90m DEM.

4. Open-Meteo's Elevation API

Open-Meteo offers an elevation API that can be used to find the elevation of any point on Earth. I host this API locally to find the elevation of points between summits that could obstruct line of sight. I also use this API to correct a few faulty summit elevations from the other data sources. This source uses the Copernicus GLO-90 DEM.

5. Beyond Horizons

Beyond Horizons has catalogued many of the longest lines of sight ever captured by photograph. I use these confirmed lines of sight to determine how to model light bending from atmospheric refraction over great distances.

Sources of Error

• There is some inherent error in the elevation data.
• The Earth is approximated as a sphere instead of an ellipsoid for simpler math.
• To take atmospheric refraction into account, light rays are approximated as arcs of circles. Though this is a commonly used approximation, the path light takes in the atmosphere is in fact much more complex and depends on many local factors. This is probably the largest source of error.
• Only summits with more than 300 m of prominence or more than 100 km of isolation are considered. The prominence threshold is determined by the OTOTW mountain dataset. The isolation threshold was chosen arbitrarily to be a pretty number that my algorithm can handle in a reasonable amount of time.
• When doing line of sight analysis, a discrete number of points between the observer and target are sampled to see if the ground obstructs line of sight. The samples are at most 100 m apart. It is possible for points that would block line of sight to not be captured. By increasing the number of samples, more pinnacle points could be found.