5.1. Solved Examples

A vertical photograph was taken at an altitude of 1200 m above msl. Find out the scale of photograph for a terrain lying at an average elevation of (i) 80 m and (ii) 300 m, if the focal length of the camera is 15 cm.

Solution: H = 1200 m, f = 15 cm, and (i) h = 80 m, (ii) h = 300 m

The scale of a photograph is given by = f / (H - h)

(i) Scale = 15 / (1200 - 80) = 1 / 7467. It means 1 cm on the photo is equal to 7467 cm on the ground.

(ii) Scale = 15 / (1200 - 300) = 1 / 6000.

In this example, we are asked to find the scale of a photograph taken from a height of 1200 meters above mean sea level (msl). The scale formula in photogrammetry relates the focal length of the camera to the difference in altitude between the camera and the terrain.

1. Explore the Variables: We identify the altitude of the photograph (H = 1200 m), the focal length of the camera (f = 15 cm), and the terrain elevations (h = 80 m and h = 300 m).

2. Apply the Formula: The scale of the photograph can be calculated using the formula: $Scale = \frac{f}{(H - h)}$. This equation calculates how the image size relates to its real-world size based on elevation.

3. Calculate the Scale:

   • For h = 80 m: $Scale = \frac{15 cm}{(1200 m - 80 m)} = \frac{15 cm}{1120 m} \approx 1 : 7467$ which tells us that 1 cm on the photo represents 7467 cm on the ground.
   • For h = 300 m: $Scale = \frac{15 cm}{(1200 m - 300 m)} = \frac{15 cm}{900 m} \approx 1 : 6000$. This means 1 cm on the photo corresponds to 6000 cm on the ground.

Thus, the calculation for elevation variations demonstrates how scale changes with differences in height.

Imagine you are trying to measure the distance of a soccer field being viewed from an airplane at a certain altitude. If you're 1200 meters high, the view is vastly different depending on where the soccer field sits. If the field is at sea level (0m), you’ll see it larger than if it’s nestled in a valley (at 300m). Similar to a zoom lens, the further you are from the ground (higher altitude), the less detail you can perceive directly, which is what affects the scale of the images you're capturing.

A camera of 152 mm focal length lens is used to take a vertical photograph from a flying height of 2780 m above mean sea level. If the terrain is flat having an elevation of 500 m above msl, determine the scale of the photograph.

Solution: f = 152 mm, H = 2780 m, h = 500 m

S = f / (H – h) = (152 / 1000) / (2780 – 500) = 0.152 / 2280 = 1 / 15000.

In this example, we're tasked with determining the scale of a photograph taken from a high altitude over flat terrain.

1. Identify Given Values: We know the focal length (f = 152 mm), the flying height (H = 2780 m), and the elevation of the flat terrain (h = 500 m).

2. Use the Scale Formula: The scale is derived from the same formula: $S = \frac{f}{(H - h)}$.

3. Convert Units: Ensure the focal length is in the same unit as the flying height, so we convert 152 mm into meters, which equals 0.152 m.

4. Calculate the Scale: Plugging the numbers in: $S = \frac{0.152}{(2780 - 500)} = \frac{0.152}{2280} \approx 1 : 15000$. Here, the scale signifies that 1 cm in the photograph depicts 15000 cm on the ground.

Think of taking a photograph of a park from a tall radio tower. The higher you go, the more the details blend together, making them appear smaller in the image. A photo taken from a height of 2780 m with minimal elevation change (like a flat terrain park) allows the camera to capture broader areas, but those areas are shown much smaller in the image compared to how they would appear from a lower viewpoint.

A camera of focal length 20 cm is used to take vertical photographs of a terrain having an average elevation of 1600 m above msl. At what height above msl must an aircraft fly to take the photographs at 1:10,000 scale.

Solution: f = 20 cm, h = 1600 m, and scale = 1:10,000

S = f / (H - h) H – h = Sf or H = Sf + h H = (20 / 100) (10,000) + 1600 m = 3600 m.

This example revolves around finding the necessary height for an aircraft to achieve a specific photographic scale.

1. Given Data: We acknowledge the focal length (f = 20 cm), average elevation of terrain (h = 1600 m), and the desired scale (1:10,000).

2. Understanding Scale Requirement: The scale indicates how much ground is represented in the photograph compared to its actual size. In this case, a scale of 1:10,000 means that 1 cm on the photograph corresponds to 10,000 cm on the actual ground.

3. Convert Scale to Fraction: Convert the scale into a formula: $S = \frac{20/100}{(H - 1600)}$.

4. Rearranging: Rearranging gives us the height we need: $H = Sf + h$. Plugging in the numbers: $H = (0.2) (10000) + 1600 = 3600 m$. Thus, to achieve the desired photographic scale, the aircraft must fly at an altitude of 3600 m.

Imagine you are mapping a city block in detail from the sky. If you want everything to appear clearly and measureable at a scale of 1:10,000, you must establish how high you need to fly. Think of it like a painter needing to determine the right distance to accurately portray all the tiny features of the city on a canvas. The higher the kite flies, the broader the view but lesser the detail retained!

A line AB 300 m long, lying at an elevation of 600 m above msl, measures 9 cm on a vertical photograph. The focal length of the camera lens is 30 cm. Determine the average scale of the photograph if the average elevation of the terrain is 700 m above msl.

Solution: Ground distance = 300 m, map distance = 9 cm, f = 30 cm, h = 600 m

Map distance / Ground distance = S = f / (H-h) 0.09 / 300 = 0.30 / (H - 600) H - 600 = (0.30 * 300) / 0.09 H = 1000 + 600 = 1600 m Now S = f / (H – h) av av S = 0.30 / (1600 - 700) av = 1 / 3000.

In this scenario, we examine how to establish the scale of a photograph based on specific measurements provided.

1. Analyze the Inputs: We have a line of known ground distance (300 m) that appears as 9 cm on an aerial photograph, with a camera focal length of 30 cm and an elevation of 600 m.

2. Formula Utilization: First, we set up the relationships to find flying height (H). We'll rearrange the relationship: $\frac{0.09}{300} = \frac{0.30}{(H - 600)}$.

3. Solve for H: By multiplying both sides, we find that: $H - 600 = \frac{0.30 * 300}{0.09}$ yields $H = 1600 m$.

4. Calculate Scale: Finally, with determined height, the average scale is found using $S = \frac{f}{(H - h)}$: $S = \frac{0.30}{(1600 - 700)} = 1 : 3000$.

Imagine a real estate surveyor trying to find the dimensions of a large lot using an aerial image. By knowing how long a property line is (300 m), they can measure the corresponding line segment on their photograph and use it to figure out the height at which the photograph was taken. This is akin to using a map to gauge a journey's path based on notable landmarks - you compare what's on the map with what exists on the ground!

A line AB measures 12.70 cm on an aerial photograph and the corresponding line measures 2.54 cm on a map at a scale of 1/50,000. The terrain has an average elevation of 200 m above msl. Calculate the flying height of the aircraft, above msl, if the photograph was taken with a camera of 16 cm focal length.

Solution: Photo distance = 12.70 cm, map distance = 2.54 cm, map scale = 1 / 50,000, h = 200 m, f = 16 cm

Photo distance / map distance = Photo scale / map scale 12.70 / 2.54 = Photo scale / (1 / 50000) 5.0 = 50000 * Photo scale Photo scale = 1 / 10,000.

Now S = f / (H - h) 1/ 10,000 = 0.16 / (H – 200) (H - 200) = 0.16 * 10,000 H = 1600 + 200 H = 1800 m.

This example calculates how high an airplane must fly to capture an aerial photograph with specific measurements adjusted for terrain elevation.

1. Interpret Given Quantities: The measurements established reveal a 12.70 cm line on the photo correlating to a 2.54 cm length on a map whose scale is given as 1/50,000 with an average elevation of 200 m.

2. Set Up Ratios: The relationship between photograph distance, map distance, and scales provides a means to determine the photo scale. Setting equalities shows us the photo scale: $5.0 = 50000 * Photo scale$ which simplifies to $Photo scale = 1/10,000$.

3. Final Step - Height Determination: By substituting the values back into the scale equation: $S = \frac{f}{(H - h)}$, the needed height H can be evaluated leading to the conclusion $H = 1800 m$.

When you take a picture of a family picnic from a camper, you want to ensure the whole scene fits the frame. Similarly, in aviation surveying, a pilot must calculate an optimal height to efficiently capture the terrain for mapping accurately. This process is much like a child adjusting their camera's distance to ensure Grandma's apple pie isn't cut off at the side - precise measurements matter!

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