# Calculation of Height of Objects in Areal Photos Using Map Projections: Radial Displacement Method and Shadow Length Method (Especially for GATE-Geospatial 2022)

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The height of objects in airphotos can be calculated using two different methods: the radial displacement method and the shadow length method.

The logic of the radial displacement method is illustrated in below figure. The vertical line PQ represents an object, e. g. a building, whose height (h) we want to calculate. On the image, the building appears as the line aq. Because of radial displacement of objects in the image, the top of the building is displaced outwards on the image relative to the base of the building. D is the length of the side of the building on the image. R is the distance from the nadir (n) of the image to the top of the building on the image. In this example, we are assuming that we have a vertical airphoto so the nadir and principal point are the same. The triangles ONA, PQA and Ona are similar triangles (same proportions but different sizes) . Therefore, h/H = D/R. Thus, the height of the building can be calculated using: h = H ⚹ D/R.

h = height of object

H = flying height above ground

D = object length on photo

R = distance from nadir to top of the object on the photo

## Shadow Method for Calculating Height

• An alternative method for calculating object heights is based on the shadows cast by the object on the image. Objects that are located very close to the nadir will have little radial displacement, making an estimation of height by the radial displacement method prone to measurement errors.
• However, objects on images taken under clear sky conditions do cast shadows, regardless of their position on the image. Although we cannot directly measure the height of objects on t ′ , -) e image, we can measure the length of their shadows and can use the shadow length of the object to calculate its height provided that we know the solar angle at the time the image was taken.
• Air photos include a clock that gives the time of exposure in Greenwich Mean Time. We can convert this into local time if we know the longitude of the object. To get the solar angle, we would need to know the latitude of the object and the date of the image. There are now several websites that have calculators that determine sun angle for a given location and time of day.
• The tangent of the solar angle is equal to the height of the object divided by the length of its shadow. Thus, if we know the solar angle (a) and the length of the shadow (I) , we can calculate the height (h) of the object as h = l ⚹ tan (a) (Figure 4.39) .