The Boston projectThree straight lines through 26 fenland waterways
Fit each waterway three ways, then rotate the map one degree at a time and see which answers move.
The learner sends one request to the Overpass service for every named drain, canal, river and ditch in a box around Boston, converts the coordinates to metres on the British National Grid, and places a point every 10 m along each waterway so that heavily mapped stretches do not count for more. Twenty-six named waterways have at least 1.5 km inside the box. For each one the program draws three straight lines. The first is ordinary least squares with northing predicted from easting, the line that makes the vertical gaps smallest. The second swaps the roles, making the horizontal gaps smallest. The third is total least squares, also called orthogonal regression, which makes the perpendicular distance from each point to the line smallest and treats north and east alike.
Five of the 26 waterways (our Python on OpenStreetMap geometry; bearings in degrees, folded to 0 to 180)| Waterway | Length in box | Total least squares bearing | Typical distance from that line | Northing-on-easting line differs by | Easting-on-northing line differs by | Worst northing-on-easting error as the map turns |
|---|
| New Hammond Beck | 5.8 km | 80.9 | 10.1 m | 0.0 | 0.0 | 0.8 |
| Maud Foster Drain | 4.2 km | 169.3 | 47.9 m | 0.5 | 0.0 | 16.3 |
| Hobhole Drain | 13.4 km | 3.5 | 179.8 m | 2.1 | 0.0 | 14.7 |
| Old Hammond Beck | 8.5 km | 91.5 | 291.2 m | 0.0 | 33.8 | 64.5 |
| South Forty Foot Drain | 26.9 km | 50.0 | 2,236.1 m | 5.6 | 8.0 | 89.9 |
Read the table from the top. New Hammond Beck is almost ruler straight, 10.1 m from its own line on average over 5.8 km, and all three methods agree. As a waterway wanders, they part company. Hobhole Drain runs close to north and south, and the method that measures vertical gaps tilts its answer by 2.1 degrees, because for a line that steep the vertical gap is a poor measure of how far a point really is from it. Old Hammond Beck runs close to east and west, and there it is the other ordinary method that goes wrong, by 33.8 degrees. Across all 26, the northing-on-easting line differed from total least squares by more than a degree for 11 waterways and the easting-on-northing line for 13.
The last column is the real test. The program turns the whole map one degree at a time, 180 times, refits, and turns the answer back. Total least squares gives the same direction every time, because perpendicular distance does not care which way is up. Ordinary least squares does care. For 22 of the 26 waterways its answer swung by more than 10 degrees at some angle, and for the long South Forty Foot Drain by 89.9 degrees, which means it ended up almost at right angles to the truth. Only New Hammond Beck and Clay Dike, the two straightest, held within a degree. A method whose answer depends on which way you hold the map is telling you about your choice of axes, not about the drain.
Ages 8 to 11
Draw a steep line of dots on squared paper, fit a line by eye, then turn the paper and try again.
Ages 11 to 15
Code ordinary least squares in Python for a handful of points and swap which variable is predicted.
Ages 15 and up
Fit all three lines to the real waterways, add the rotation test, and explain why only one stays still.
Sources and limits
Waterway geometry © OpenStreetMap contributors under the Open Database Licence. The resampling, the fits and the rotations are ours. A straight line is a crude summary of a winding river; the point of the exercise is the fitting method, not a survey of the channels, and a different box would clip them differently.
