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Lecture 7 Content

Lecture 7 Content. Photo / Map Projection Definition of common terms Errors on photo Mathematical relationship (Scale) Vertical photos. Photo / Map Projection. Definition of common terms. n. Plane I. p. o. ө. R. N. P. Plane II.

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Lecture 7 Content

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  1. Lecture 7 Content Photo / Map Projection Definition of common terms Errors on photo Mathematical relationship (Scale) Vertical photos

  2. Photo / Map Projection

  3. Definition of common terms n Plane I p o ө R N P Plane II

  4. “Plane I” represents the plane taken by the negative / photo which is called image plane • “Plane II” represents the plane taken by the terrain which is called object plane • the two planes intersect at a line called the horizontal trace • “o” is the optical center of the camera lens

  5. “p” is the principal point on the image plane, defined as the foot of the perpendicular from “o” to “Plane I” • “o” is the optical center of the lens • “P” is the ground principal point • “n” is the nadir point, which is the photo plumb point • “N” is the ground plumb point from the nadir point • “pR” is the principal line, “po” is the Principal distance of focal length. “opR” are on the Principal Plane • “ө” is the angle of tilt

  6. LEARN • Mathematical relationships from a vertical photo Scale: Photo f H Terrain 1/S = f/H where S = Scale Factor

  7. Principal Point is the point where a perpendicular projected the center of the lens intersects the photo image • Nadir Point is the point vertically beneath the camera lens at the time of exposure • Isocenter is the point on the photo that falls on a line approximately halfway between the principal point and the nadir point • Isocenter and Nadir Points

  8. Isocenter and Nadir Points

  9. Equations on the Principal Plane n Image plane i p Note: Isocenter (i) is the point in which the bisector of the angle between the plumb line and the principal axis meets the image plane. ө f o v H Object plane ө R I P N

  10. Homework: Prove the following using the previous diagram • angle nop = ө and angle poi = angle ion = ө/2 • Distance from principal point to plumb point pn = po tan ө = f tan ө • Distance from principal point to isocenter pi = po tan ө/2 = f tan ө/2 • Distance from principal point to horizon pv = po cot ө = f cot ө • PN = No tan ө = H tan ө PI = PN – NI = H tan ө - H tan ө/2 = H (tan ө - tan ө/2)

  11. Vertical photos • These are photos resulting if the axis of the camera is exactly vertical when exposure is made • Despite all precautions, small tilts, generally less than 1˚ and rarely greater than 3˚, are invariably present • Scale is determined by: • Ratio of a distance on a map to that same length on the ground, or • Ratio of the focal length to that of the flying height for a flat area or the average height of a hilly area

  12. Vertical photos

  13. Using similar triangles: Lab and LAB • Using similar triangles: • Loa and LOA • Equating the two equations and recognizing that equals photo scale at A and B and recognizing that AB is infinitesimally • short then the equation for the scale at A is: • Similar equations are generated for points B, C, and D

  14. An example: A vertical photograph was generated using a camera with a focal length of 6in at a flying height of 10,000ft. Above mean sea level. What is the photo scale at point ‘a’ if the elevation of point A on the ground is 2500ft above mean sea level Solution:

  15. Ground coordinates from a single vertical photograph

  16. photo coordinates are xa, ya and xb, yb • ground coordinates are XA, YA and XB, YB • Using similar triangles: •  LOAA΄ and Loa΄ •  Then

  17. Using similar triangles: •  LA΄A and La΄a •  Then

  18. An example: A vertical photograph was taken with a 152.4mm focal length camera from a flying height of 1385m above datum. Images ‘a’ and ‘b’ of two ground points A and B appear on the photograph, and their measured photo coordinates are xa= -52.35mm, ya= -48.27mm, xb= 40.64mm, and yb= 43.88mm. Determine the horizontal length of line AB if the elevations of points A and B are 204m and 148m above datum, respectively Solution for point A:

  19. … The End …

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