Definitions

A pinnacle point is a point from which no higher point can be seen. Across the globe, 601 have been found. These are all pinnacle points with more than 300 m of prominence. The algorithm used for finding pinnacle points is outlined below, along with sources of error. The curvature of the Earth, and atmospheric refraction are taken into account.

The prominence of a summit is how far down you have to go in order to get to higher elevation. It can be thought of as how much a summit rises above its surroundings.

The direct line of sight is the straight line connecting 2 points.

The light path is the bent path light takes in the atmosphere between 2 points as a result of refraction.

Sources

Mountains by Topographic Prominence

Thanks to Andrew Kirmse and the Prominence Group for finding 11,866,713 summits with a prominence greater than 100 feet (~30 m).

On-Top-Of-The-World Mountains

An on-top-of-the-world mountain (OTOTW) is a summit where no land rises above the horizontal plane from the summit. Since any land that rises above the horizontal plane would have higher elevation than the summit itself, if a summit is not an OTOTW then it can't be a pinnacle point either. In other words, pinnacle points are a subset of OTOTWs. Thanks to Kai Xu for finding 6,464 OTOTWs around the world. Andreas Geyer-Schulz deserves mention as well for his extremal peaks.

Atmospheric Refraction

Different layers of the atmosphere have different refractive indices due to varying temperatures and pressures. Although light's exact path in the air is difficult to calculate and depends on many factors, a ray's path can be approximated as the arc of a circle with radius 7 times greater than Earth's.

Error

1. When the OTOTWs were found, all summits with less than 300 m of prominence were omitted. This cutoff is carried forward to pinnacle points.
2. The earth is approximated as a sphere instead of an ellipsoid. This is done for simpler math.
3. Only 100 elevations are sampled when determining if two points have an unobstructed light path. This can lead to points that can block the light path being missed. By increasing the number of elevations sampled, more pinnacle points would be found.
4. To take atmospheric refraction into account, light rays are approximated as arcs of circles. The path light takes in the atmosphere is infact much more complex and depends on many factors. Note, since the distance you can see from a given point depends on temperature and pressure, the distance you can see technically changes with the season and even the time of day. Additionally, some assumptions used to take atmospheric refraction into account only hold true for heights small compared to the 8 km height of the homogeneous atmosphere. This project is slightly outside of this scope.
5. There is some inherent error in the data. The datasets have a resolution of 30 m (1 arcsecond), and the elevation API uses 90 m (3 arcseconds). All data sources are surface elevation models, so trees and buildings are included.

Algorithms

Finding Pinnacle Points

1. Categorize the 11,866,713 summits into patches based on longitude and latitude. This is done for faster processing later. The patch size is 10 deg x 10 deg with some extra to take into account the fact that summits may be able to see beyond the patch they are in.
2. Find the maximum horizon distance (MHD) defined as √(2*(7/6)*R_earth*Elevation) for each summit and OTOTW. The factor of 7/6 is to account for atmospheric refraction increasing the distance that can be seen.
3. Define a list of remaining OTOTWs as the full list of OTOTWs.
4. Find the highest elevation remaining OTOTW (HERO).
5. Find all other remaining OTOTWs where the sum of the OTOTW's MHD and the HERO's MHD is greater than the distance between them. In other words, find all OTOTWs that have a chance of being seen by the HERO. For each, do light path analysis (described below) to find which OTOTWs can actually be seen by the HERO. Each will have a lower elevation than the HERO, so they are removed from the list of remaining OTOTWs.
6. Find the HERO's patch.
7. Find all summits in the patch where both the summit's elevation is greater than the HERO's elevation and the sum of the summit's MHD and the HERO's MHD is greater than the distance between them. In other words, find all summits that have a chance of disqualifying the HERO from being a pinnacle point. For each, do light path analysis to determine if any summits can actually see the HERO. If any can, the HERO is not a pinnacle point and is removed from the list of remaining OTOTWs. Otherwise, the HERO is a pinnacle point and is added to a list.
8. Repeat 4-8 until there are no remaining OTOTWs.

Light Path Analysis

1. Generate a list of 100 equidistant latitude-longitude points between the two input points along the geodesic.
2. Find the elevation of each point using an API (Open-Meteo), and find the distance of each point to the first point.
3. Translate each distance-elevation point such that the first point is at (0, 0).
4. Apply a translation to each distance-elevation point based on how far away the point is to take the curvature of the earth into account.
5. Rotate the points to make the last point at the 0 elevation line. In other words, make direct line of sight along the x-axis.
6. Find the light path's distance above direct line of sight for each point.
7. If any of the points are of greater elevation than the light path, the light path is blocked. Otherwise, there is an unobstructed light path.

Contact

Contact me at jamiegbreault@gmail.com. Check out the latest on my github. There you will find a txt file of the pinnacle points, the algorithm used, and derivations of some of the math. Pinnacle points I have visited myself are marked with an asterisk [Count: 1].

Here's an article on OTOTWs and pinnacle points if you are interested.