4.13.1. Scale of a tilted photograph

The scale of a tilted photograph changes throughout the photograph. It is too small on the high side and too large on the low side (Figure 4.23).

In a tilted photograph, the scale is not uniform. On the side of the photograph that is higher up (or tilted away from the ground), the scale is smaller, meaning that objects appear smaller than they are in reality. On the side that is lower (or tilted towards the ground), the scale is larger, making objects appear bigger than their true size.

Imagine holding a camera up while standing on a slope. The trees are taller on the side that is lower, so when you take a photo from that angle, the trees look bigger compared to when you point the camera upwards. This change in appearance depending on the angle is why scale varies in tilted photographs.

The scale also changes along the principal line (i.e., in the direction of tilt), however it does not change along any line perpendicular to the principal line.

As we measure along the principal line, which runs in the direction of the tilt, we see that the scale continues to vary. However, if we measure in a direction perpendicular to this line, the scale remains consistent. This means that understanding the orientation of your measurements is key when analyzing the photograph.

Think of a slide ruler. If you slide it along one side, you might see the numbers increase or decrease depending on the angle you are measuring at. But if you try to measure straight across the width, the measurements don’t change; they stay the same. This idea applies to tilted photographs in understanding scale.

Let this angle of rotation be θ=180°- s, where s=swing angle, P is the point lying at an elevation h above the datum, p is its image on the tilted photo.

In tilted photographs, angles play a crucial role. The rotation angle (θ) helps define how the photograph is tilted relative to the vertical. In this context, 's' represents the swing angle, which is based on how the camera was tilted. The point P represents the actual elevation of an object above a reference point (datum), while 'p' is how that elevation appears on the photograph.

Imagine standing at the bottom of a hill and looking up at a tree that’s halfway up. If you tilt your head back to see the top, the angle of your head creates a triangle with your line of sight. This triangle helps you understand how tall the tree really is compared to how it looks from your angle, similar to how the angle θ works in a tilted photograph.

Its photo-coordinates are (x, y) and measured coordinates with respect to axes defined are (x', y') by collimation marks.

For any point on the tilted photograph, there are two sets of coordinates. The original coordinates (x, y) are the straight measurements from the center point, while the transformed coordinates (x', y') consider the tilt and help in understanding how those points relate to the photograph's orientation.

Think of a map that’s been crumpled up. The coordinates you originally used (x, y) won’t help much anymore since the map is no longer flat. You need to adjust your coordinates (x', y') to get a proper representation of where you are on this new tilted 'landscape'.

From similar Δs Lkp and LNP, we can write kp/Lk = Np/LN.

By using the properties of similar triangles, we can derive the scale of the tilted photograph. It involves ratios of the sides of triangles formed by lines of sight to points on the photograph. This ratio gives us the scale at point 'p', helping us understand how size relates to elevation and tilt.

If you were to create a model building from a tilted blueprint, you'd need to know how much larger or smaller each part is compared to the real structure. This calculation, using ratios, plays a similar role in determining the scale of a photograph.

The scale of tilted photo (S) at p can be written as- S = (f sec t - y'sin t) / (H - h).

The formula for the scale of a tilted photograph shows how various factors interact. 'f' is the focal length of the camera, 't' is the tilt angle, 'H' is the flying height, and 'h' is the elevation of the point. This equation highlights how tilt influences the photograph’s scale, regardless of whether the terrain is flat.

Picture a photographer taking pictures of a mountain from different angles. The way the mountain looks (how steep or flat it appears) changes with every angle they shoot from. This formula quantifies those changing relationships, allowing one to understand exactly how perspective alters perceived size in a photograph.

--