Pythagoras revisited Measuring distance on a map

On the map in Figure 15.9(a), the x and y axes of the S coordinate frame are drawn in the east-west and south-north directions. (The sides of the grid squares are approximately 100 km and we are assuming that the map is flat, i.e. ignoring the earth's curvature.) In Figure 15.9(b) a coordinate frame has t

Figure 15.9 Invariant interval in space-time.

been chosen (S' frame) which has been rotated and displaced by an arbitrary amount.

The distance between Dublin and Edinburgh is of course independent of the grid, but the components of that interval in the directions of the axes will depend on the orientation of the grid used.

Applying the theorem of Pythagoras,

The actual components in the diagram are (approximately)

S frame Ax = 2.7, Ay = 2.7 S' frame Ax' = 1.6, Ay' = 3.5 ^ AS = 3.82 ^ AS = 3.85

in good agreement with the measurement error.

Extending the theorem of Pythagoras to three dimensions

The formula applies equally well to three dimensions, the distance between any two points being expressed in terms of the sum of the squares of three perpendicular components.

Figure 15.10 Pythagoras' theorem in three dimensions.

AB2 = OA2 + OB2 and OB2 = Ax2 + Ay2 fi AS2 = Az2 + Ax2 + Ay2

AS is an invariant interval in three-dimensional Euclideanf space.

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