PRACTICE PROBLEMS

Part I: General

6.2 In question 6.1, state the advantages and disadvantages of each type of aerial photograph in respect to the others.

6.3. Define the following:
 

• Focal length
• Principal points
• Nadir point
• Isocenter
• Flying height
• Datum
• Plumb line
• Optical axis
• Fiducial marks
• Ground elevation

6.5. What are the major sources of distortions and displacements affecting aerial photographs?

6.6. Describe relief displacement and its effect on a vertical aerial photograph and state its benefits and drawbacks.

6.7. What is image motion and what causes it?

6.8. Suppose we measured the distances between the fiducial marks of a vertical photograph to be 23.37 cm along x-axis and 23.35 cm along y-axis. The calibrated measurement on the same camera used to take the photograph were found to be 23.15 cm and 23.17 cm, respectively along x-axis and y-axis. Find the correct values for the following measured photocoordinates of points a, b, c, and d:

x_a = -8.93 cm x_b = -10.15 cm x_c = 9.37 cm x_d = 6.92 cm
y_a = -9.25 cm y_b = 7.67 cm y_c = 10.13 cm y_d = -10.41 cm

6.9. Assume that we know the correct photocoordinates of the following points on a vertical photograph:

x_a = 83.74 cm x_b = -76.35 cm x_c = -24.83 cm x_d = 50.52 cm
y_a = -90.92 cm y_b = 86.80 cm y_c = 5.09 cm y_d = 54.27 cm

The photocoordinates of these points were measured on the photograph as:

x_a = 84.08 cm x_b = -76.67 cm x_c = -24.96 cm x_d = 50.74 cm
y_a = -91.47 cm y_b = 87.33 cm y_c = 5.19 cm y_d = 54.62 cm

If the distances between the fiducial marks of the photograph were measured to be 233.6 mm along x-axis and 233.4 mm along y-axis, what are the corresponding calibrated distances on the camera?

Part II: Relief and tilt displacement

6.11. In question 7.8, which flying height would be more suitable to take better photographs, 2000 m or 3500 m above datum? Why?

6.12. Draw a diagram showing relief displacement of an electric pole extending above datum and a ditch extending below datum. On the photographic positive image, draw the displaced image of each feature and use an arrow to indicate the direction of displacement in respect to the photo center (assume a truly vertical photograph).

6.13. Select the correct word(s) in the brackets:
 

1. Topographic displacement varies [directly, inversely] with the focal length of the camera, [directly, inversely] with the flying height of the aircraft above datum, [directly, inversely] with the height of the object, and [directly, inversely] with the distance of the object from the [principal point, nadir, isocenter].
2. Objects above datum are displaced radially [inward toward, outward away from] the [principal point, nadir, isocenter] and those below datum are displaced radially [inward toward, outward from] the [principal point, nadir, isocenter].
3. Tilt displacement is radial from [the principal point, nadir, isocenter].

6.15. Draw a 5 cm by 5 cm vertical photograph, place the four fiducial marks, and find the principal point of the photograph. Suppose an aircraft was flying in the East-West direction (East is to the right on your 5x5 cm photo) with the left wing being up and the right wing being down. Sketch the position of the nadir point and the isocenter with respect to the principal point, then draw the tilt axis and indicate the up and down sides of the tilt.

6.16. From problem 6.13, assume a tall tower was located on the up side of the tilted photograph. Sketch the actual position of the tower on the photograph as affected by the combined effect of relief and tilt and sketch the true position of this tower had the photograph been free of tilt and relief displacement (see figure 6.25 for help).

6.17. Assume you have located the principal point on an aerial vertical photograph. Assume also that the photograph was taken from an aircraft 3000 m above datum and using a 152.4-mm focal length camera tilted 2^o from the vertical. If a 100-m tower is located 9.8 cm from the principal point, on the down side of the tilt, and on the line passing through the principal point, what is the image relief displacement on the photograph?

Solution:

 Because relief displacement is radial from the nadir and because all measurements are from the nadir point, we first need to find the position of the nadir. We know that the nadir is always on the down side of tilt from the principal point as illustrated by the figure above. Since the tower is on the same line connecting the p and the nadir (n), the position of the nadir can be determined from the right triangle onp as:

np = op.tan2^o, where op is the flying height above the datum (3000 m)

np = 3000 m (0.03492) = 104.76 m, which is the distance between the principal point and the nadir.

 From similar triangles on'p' and onp, we can write:

therefore, the distance n'p' on the photograph may be calculated as:
 

 n'p' = (f . np)/H_D = (0.1524 x 104.76)/3000 = 5.32 mm

r' = tp - np = 98 mm - 5.32 mm = 92.68 mm

6.19. A truly vertical photograph was taken at a flying height of 6000 m above datum. A 110-m- tall building was located on the photograph and the distance between its base and the principal point was measured to be 8 cm. Calculate the displacement of the building on the photograph and state the direction of this displacement in respect to the photo center.

6.20. In the previous problem, another tower was identified on the photograph and the distances between the photo center and the base and the top of the tower were measured to be 7.53 cm and 7.95 cm respectively. Find the height of the tower.

6.21. The distance between the top and the base of a 150-m-high tower was found to be 5.7 mm. If the distance from the nadir to the base of the tower is 8.8 cm and the ground elevation of the top of the tower is 500 m above mean sea level, how high was the aircraft flying above datum at the moment of exposure?

6.22. Find the displacement of a tree that is 100 m high if the photographic distance from its base to the nadir is 7.6 cm and if the photograph was taken vertically at a flying height of 6000 m above datum.

6.23. The distance, on a vertical photograph, from the nadir to the top of a hill is 9.1 cm and the hill is 200 m higher that datum on the ground. If the photograph was taken from an aircraft flying at 5500 m above datum at the time of exposure, by how much is the top of the hill displaced from its base?

6.24. A photograph was taken with a 305 mm focal length from an aircraft flying at 6500 m above datum. By how much would the summit of a 130-m-tall tower be displaced from its base if its base is at 2 kilometers from the nadir on the ground?

6.25. On a truly vertical photograph, a 300-m-high hill was photographed at 8.5 cm from the photo center. If the photograph was taken at a flying altitude of 3000 m above the base of the hill, what would be the amount of relief displacement of the summit of the hill from its base?

6.26. The radial distance between the nadir and the top of building is measured on a photograph as 6.8 cm. The distance between the top and the bottom of the building is 0.95 cm. What is the height of the building if the photograph was taken from an altitude of 4500 m above datum.

6.27. Two trees with the same height (85 m) are 1800 m and 1200 m from the nadir on the ground. The photograph was taken from a flying altitude of 3000 m above datum with a 152.4 mm focal length. Which tree is displaced more and by how much compared to the other tree?

6.28. Assume a tower, at a distance R from the nadir on the ground, was photographed from the same altitude using simultaneously a 152.4-mm and a 305-mm focal lengths. Which focal length will cause a larger displacement and by how much compared to the other focal length?

6.29. Assume a vertical photograph was taken at 2900 m above mean sea level with a 152.4-mm focal length. What is the relief displacement of the top of a hill that extends 750 m above mean sea level and whose image on the photograph is 9.45 cm from the photo center?

6.30. A vertical photograph was taken at a flying altitude of 2000 m above datum. The summit of a hill, which extends 400 m above datum, was imaged at 8.25 cm from the photo center. By how much and in what direction is the top of the hill displaced from its planimetric position? Where would be the true position of the top of the hill on the photograph?

6.31. The figure below shows a parcel that was imaged on a vertical photograph taken at a flying height of 2000 m above datum with a 152.4-mm focal length. The corners a, b, c, and d of the parcel are respectively 3.12 cm, 9.91 cm, 12.15 cm, and 10.98 cm from the photo center. The corresponding ground points A and C are respectively 420 m and 315 m above datum and B and D are respectively 460 m and 500 m below datum. Find the amount of relief displacement for the four corners and draw the correct positions a', b', c', and d'.
 

6.32. A vertical photograph was taken at a flying height of 2500 m above mean sea level. A pine tree was identified on the photograph and the distance between its bottom and its top was measured to be 6.25 mm on the photograph. the radial distance from the photo center to the base of the tree was found to be 9.16 cm. If the base of the tree is 1200 m above mean sea level, how tall is this tree?

6.33. A 210-mm focal length camera was used to take a vertical aerial photograph at 2500 m above datum. A tall building was located on the photograph and the distance between its bottom and the photo center was measured to be 7.53 cm and that between its top and the photo center was measured to be 8.15 mm. How tall is the tree?

6.34. Two points a and b were identified on a vertical photograph taken at a flying height of 2000 m above datum. Point a is located on the bisector line of the top right quadrant defined by the +x and +y axes of the photograph and at 9.25 cm from the photo center. Point b is located on the bisector line of the bottom left quadrant defined by the +x and -y axes of the photograph and at 7.65 cm from the photo center. The corresponding ground elevations are h_A = 300 m above datum and h_B = 150 m below datum. Find the relief displacement of points a and b and, after correcting both points for the displacement, compute the angle formed at the photo center between the two lines radial from a and b.

Part III: Ground coordinates from Vertical and tilted photo measurements

6.36. The ground elevations of points A, B, and C are 320 m, 80 m, and 150 m, respectively. The photocoordinates of their images on a vertical photograph, after correction for film shrinkage, are x_a = 4.61 cm, y_b = 7.15 cm, x_b = 5.83 cm, y_b = -5.20 cm, x_c = -5.73 cm, and y_c = -8.45 cm. If the photograph were taken at 2800 m above mean sea level with a 210-mm focal length camera, what would be the horizontal distances of lines AB, AC, and BC?

6.37. Points a, b, and c on a vertical photograph are imaged positions for ground points A, B, and C. The photograph was taken from a flying height of 4500 m above datum with a 152.4-mm focal length camera. The photocoordinates of the three points are x_a = -52.25 mm, y_a = 22.53 mm, x_b = 55.15 mm, y_b = 89.75 mm, x_c = 59.15 mm, and y_c = -81.34 mm. The ground elevations are hA = 200 m, hB = 100 m, and hC = 300 m. Find the ground coordinates for points A, B, and C, the ground horizontal distances of AB, AC, and BC, and the horizontal angles at A, B, and C.

6.38. Images a and b, of two ground points A and B, were identified on a vertical photograph taken at a flying altitude of 2000 m above datum and with a 152.4-mm focal length. The photocoordinates of a and b, measured with respect to the x and y axes determined by the fiducial marks, are x_a = -6.25 cm, y_a = -5.72 cm, x_b = 3.14 cm, and y_b = 5.28 cm. What is the horizontal ground distance between A and B if A is 200 m and B is 150 m above datum?

6.39. Three ground features, A, B, and C were imaged at a, b, and c on a vertical photograph taken at 2500 m above datum with a 152.4-mm focal length camera. Ground elevations for points A, B, and C were found to be 450 m, 400 m, and 500 m, respectively. The photocoordinates of a, b, and c, measured with respect to x and y axes defined by the fiducial marks, are x_a = 5.17 cm, y_a = 4.05 cm, x_b = 4.10 cm, y_b = -6.25 cm, x_c = -7.12 cm, and y_c = -6.31 cm. Find the horizontal ground distances of AB, AC, and BC.

6.40. Solve problem 6.32 using equations 6.19 and 6.20.

6.41. Solve problem 6.35 assuming that the photograph is not vertical but tilted at 3^o20' from the vertical with a swing angle of 260^o.

6.42. Two ground points A and B were imaged at points a and b on an aerial photograph taken with a 152.4-mm focal length lens, a 2^o 85' tilt, and a swing angle of 67^o. The photocoordinates of points a and b are x_a = -5.65 cm, y_a = 7.12 cm, x_b = -6.15 cm, and y_b = -3.56 cm. The ground elevations of points A and B are h_A = 300 m and h_B = 600 m and the horizontal distance between A and B is 1500 m. Find the flying height above mean sea level from which the photograph was taken.

6.43. An aerial photograph was taken with a 210-mm focal length camera inclined at 2^o95' from the vertical and with a swing of 115^o. Find the photocoordinates of the nadir and the isocenter with respect to the axes defined by the fiducial marks.

Part IV: Ground coordinates from oblique photograph measurements

6.45. Use the information from problem 6.42 to compute the distances between the principal point and: the true horizon, the nadir, and the isocenter.

6.46. A 210-mm focal length camera was used to take an oblique photograph at a flying height of 2000 m above datum. the average ground elevation of the photograph is 380 m above datum and the distance between the principal point and the apparent horizon, measured along the principal line, is 72.5 mm. Find the true depression angle.

6.47. Solve problem 6.44 if the flying height is 1200 m above datum, the focal length is 152.4 mm, the average ground elevation is 200 m above datum, and the distance between the principal point and the apparent horizon, measured along the principal line, is 67.3 mm.