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Storing geodata in neural networks

Chapter 4 · The Earth is a sphere

Deutsche Fassung · Translated from the German original.

Chester with his blue cap and a tape measure around his neck curves a flat world map over a sphere and sews the edge together. Aiko, a glowing neural network, stands on the table on the left and holds the thread.

Contents:

  • Why are longitude and latitude a poor input for a neural network?
  • How do you describe a place as a point on a sphere instead?
  • What does this new approach change at the date line and at the poles?

Starting point

In chapters 2 and 3 the neural network got longitude and latitude as input – the way places are usually given on a world map. For people that is practical. For the neural network it is not so easy.

Two catches with longitude and latitude

On the left a sphere with two neighbouring points A and B at the date line, 22 km apart, and the North Pole as a single point. On the right a rectangular world map on which A lies at the far right at +179.9° and B at the far left at −179.9°; the top edge of the map is the North Pole.

The seam. Longitude runs from −180° to +180°. A point at 179.9° east and one at 179.9° west are only 22 km apart on Earth – both in the Pacific, very close to the date line, which roughly follows the 180th meridian.

For the neural network, however, they are the numbers +179.9 and −179.9: as different as two longitudes can be. The network would have to learn that two completely different inputs have almost the same elevation. With the method from chapter 3, which builds a surface from many straight ramps, that is hard.

The poles. At the North Pole longitude has no meaning: every longitude between −180° and +180° describes the same point. In the grid, the top row is nevertheless a whole row of 2 160 cells with different longitudes.

For the neural network these are 2 160 different inputs for a single place. At the South Pole it is the same.

Both come from the way we describe places: the rectangle of the world map has edges, the globe has none.

The point on the sphere

The solution: you describe a place not by two angles (longitude, latitude) but by its position in space.

To do so, imagine the Earth as a sphere with radius 1 and the centre of the Earth as the origin. Every place is then a point on this sphere and has three coordinates x, y and z.

A sphere with three axes: x points towards 0° east, 0° north, y towards 90° east, 0° north, z towards the North Pole. Regensburg is marked as a point on the sphere. On the right the conversion formulas and the values for Regensburg: x = 0.64, y = 0.14, z = 0.75.

From longitude and latitude they are computed like this:

\[ x = \cos(\text{latitude}) \cdot \cos(\text{longitude}) \]
\[ y = \cos(\text{latitude}) \cdot \sin(\text{longitude}) \]
\[ z = \sin(\text{latitude}) \]

What the three formulas mean is shown step by step with the example of Regensburg:

Three sections. 1: In the side view, Regensburg lies 49° above the equator; this gives the height z = sin(49°) = 0.75 and the distance to the Earth's axis r = cos(49°) = 0.66. 2: Seen from above onto the North Pole, Regensburg lies on a circle with radius 0.66, rotated by 12.1° from the direction 0° east; the share in the direction 0° east is x = 0.66 · cos(12.1°) = 0.64. 3: The share in the direction 90° east is y = 0.66 · sin(12.1°) = 0.14.

  1. Latitude determines the height. The further north a place lies, the higher it lies above the plane of the equator: z = sin(latitude). At the same time it moves closer to the Earth's axis; the distance to the axis is r = cos(latitude).
  2. Longitude distributes this distance between two directions. Seen from above onto the North Pole, all places of the same latitude lie on a circle with radius r. Longitude is the angle on this circle. The share in the direction 0° east is x = r · cos(longitude).
  3. The rest points towards 90° east: y = r · sin(longitude).

Substituting r = cos(latitude) gives exactly the three formulas above.

A few examples:

Place Longitude Latitude x y z
Gulf of Guinea 0° 0° 1 0 0
Indian Ocean 90° E 0° 0 1 0
North Pole any 90° N 0 0 1
Regensburg 12.1° E 49.0° N 0.64 0.14 0.75
Pacific, point A 179.9° E 0° −1.000 +0.002 0
Pacific, point B 179.9° W 0° −1.000 −0.002 0

The two points at the date line now have almost the same numbers, just as their position on Earth suggests. And at the North Pole it does not matter which longitude you use: because cos(90°) = 0, the result is always (0, 0, 1) – one place, one description.

In short: places that are close together on Earth now also have similar coordinates. There are no edges and no seam any more for the network to stumble over.

The price is small. Instead of two inputs the network has three, and every neuron of the first layer needs one more weight – with 128 neurons that is 128 additional weights. The 50 049 weights of the simple network from chapter 2 thus become 50 177.

What difference it makes

How large the difference is shown by a comparison: a neural network with the same architecture, a simple MLP (preset Plain MLP) with 4 hidden layers, the same grid, the same training – once with longitude and latitude, once with spherical coordinates as input.

The picture shows the Pacific on both sides of the 180th meridian.

Three maps of the Pacific between 160° east and 160° west, from 50° south to 10° north, with the 180th meridian as a dashed line. On the left the original with ocean trenches and New Zealand. In the middle the network learned with longitude and latitude: along the dashed line runs a vertical edge, to its left it is lighter than to its right. On the right the network learned with spherical coordinates: no edge at the line.

With longitude and latitude, the network has a visible seam at the 180° line: the elevation jumps there, although the terrain continues on both sides. Along the whole line from the South to the North Pole the jump is 690 m on average. With spherical coordinates it almost completely disappears: 0.2 m on average.

Both networks are almost the same size (50 049 and 50 177 weights) and trained for the same time.

Try it out

In the sidebar under 2 · Network, choose the preset Plain MLP (fails on purpose) and open Architecture details.

At the very top is Coordinate input. The default is xyz – the spherical coordinates from this chapter. The alternative lonlat passes longitude and latitude directly into the network.

Sidebar with Architecture details expanded: the Coordinate input selection is open and shows the two options xyz and lonlat.

For comparison, first train with lonlat – that is the network from chapter 2 – and look at the error map at the right and left edges in the Maps & Error tab.

Then train again with xyz and compare. The weighted RMSE drops from around 700 m to around 600 m (see the appendix).

In the Globe 3D tab the Earth can be turned. Set the View direction (longitude) slider to 180: then you look straight at the seam.

Ideas to explore further

  • On the error map, look not only at the seam but also at the polar regions: what does Antarctica look like with lonlat and with xyz?
  • Under 1 · Data, set the Grid resolution low (180 × 360) so that training is quick, and switch several times between lonlat and xyz.
  • Convert a well-known place into spherical coordinates, for example with a calculator and the formulas above. For which places is x negative, where y, where z?

Further reading

Overview · Back: Chapter 3 · Architecture of a neural network · Next: Chapter 5 · Elevations as waves

This tutorial was created by Gerhard Völkl using Claude Code (Anthropic). It is meant to help you learn more about storing geodata in neural networks. Suggestions for improvement are welcome at gkvoelkl@nelson-games.de.