The Three-Shadowpoint method for finding an east-west line, based on an old Roman surveying method. This article was written by Thomas Stein, based on work by J.A.F. de Rijk, Th. de Vries, H. Janssen, and Ton vd Beld

1.      In Figure 1, a gnomon “g” with foot “F” and top “T” is erected.

2.      In Fig. 2 at three different times, the shadow of g is recorded on the ground.

Let the tip of top T be recorded in shadowpoints A, B0 and C – the three shadowpoints.

Point A would be the first point, Point B0 the second, and Point C latest of the three points.

3.      In Fig. 3, assuming B0 to be closest to the foot of the gnomon F, then the distance from top T to
shadowpoint B0 will be the shortest of the distances. We measure the distance TB
from T towards B0, so obtaining the distance “ß”.

4.      In Fig. 3, we measure this distance ß from T to shadowpoint A, and reach sub-point B’A.

Dropping a vertical from point B’A to the ground, we reach sub-point BA.

5.      Likewise in Fig. 3, we measure this distance ß from T to shadowpoint C, and reach sub-point B’C.

Dropping a vertical from point B’C to the ground, we reach sub-point BC.

In Fig. 4 all of these sub-points are visible on the 3D representation.

6.      In Fig. 5, we then extend a line through sub-points BA and BC. This line generally runs along the ground. We also extend a second line through sub-points B’A and B’C. Note that this second line is elevated because both sub-points B’A and B’C are off the ground.

7.      In Fig. 5, we can see that these two lines will intersect in sub-point BAC.

8.      In Fig. 6, we then connect sub-point BAC to shadowpoint B0.

This line is the local East-West direction!

The whole procedure can be carried out with pegs and strings, and no calculations are necessary.

Why it works

Suppose we have a plane H and a point B0 in it. Above the plane two arbitrary points B’A and B’C "float".

Call the plane through B0, B’A and B’C a plane E. Find the intersection of planes H and E.

Solution

Drop perpendiculars from B’A and B’C onto H, giving BA and BC. Lines B’A B’C and BA BC intersect in point BAC.

The line through B0 and BAC is the intersection of planes H and E.

In our case, plane H is the surface of the earth, and plane E is the shadowcone periphery defined by the sun’s movement across the sky on a given day.

The shadowcone is the cone traced on the celestial globe by the sun’s movement across the sky on a given day. The surface of the earth and the daily shadowcone intersect on an East-West line.