1. فصل 9 : مشخصات و هندسه عکس مایل
33 Slide

2. DEFINITIONS Nadir point (n) Ground nadir point (N) Principal plane
Principal line
Azimuth (α)
Tilt (t)
Swing (s)
Church angles (α, t, s)

3. DEFINITIONS (cont.)
Isocenter (i)
Axis of tilt

4. AUXILIARY COORDINATE SYSTEM
In tilt-swing-azimuth system, auxiliary coordinate system required for some computations
Two stages:
Rotate about principal point through an angle θ
Translate along principal line from principal point to nadir point
Origin at nadir point, y’ axis coincides with the principal line, x’ axis is perpendicular to principal line at the nadir point, clockwise 90°

5. AUXILIARY COORDINATE SYSTEM (cont.)

6. AUXILIARY COORDINATE SYSTEM (cont.)
Rotation angle
θ = s −180°
Transformed coordinates after rotation:
x" = x cos θ − y sin θ
y" = x sin θ + y cos θ

7. AUXILIARY COORDINATE SYSTEM (cont.)
Translate origin
Final auxiliary coordinate values
x' = x cos θ − y sin θ
y' = x sin θ + y cos θ + f tan t

8. AUXILIARY COORDINATE SYSTEM (cont.)
Recognize that sin θ = − sin s and cos θ = − cos s
Then auxiliary coordinate system can be expressed as
x' = − x cos s + y sin s
y' = − x sin s − y cos s + f tan t

9. AUXILIARY COORDINATE SYSTEM (cont.)
Note: if rotation angle defined as θ = 180° − s
Then auxiliary coordinates become
x' = x cos θ − y sin θ
y' = x sin θ + y cos θ + f tan t
Do not mix formulas from the two different approaches

10. SCALE OF TILTED PHOTO

11. SCALE OF TILTED PHOTO (cont.)
In following diagram
wk is perpendicular to line Ln and is therefore a horizontal line
wp is perpendicular to the principal line and is also a horizontal line
Plane kwp is horizontal plane
Plane NWP is horizontal plane

12. SCALE OF TILTED PHOTO (cont.)
Scale between planes kwp & NWP found by similar triangles
Since Lk = L − kn = f sec t − y'sin t
Then scale becomes

13. GROUND COORDINATES FROM TILTED PHOTO
On ground, the Y-axis lies principal plane
In figure, wp = x’ and WP = X, then from scale:

14. GROUND COORDINATES FROM TILTED PHOTO (cont.)
Similarly,

15. EXAMPLE: SCALE ON A TILTED PHOTO
Given the following known values:
Compute the scale at points a and b

16. Solution : SCALE ON A TILTED PHOTO
Compute auxiliary coordinates
– Rotation Angle:
θ = s − 180°
= 272 ° − 180 °
= 92 °

17. Solution : SCALE ON A TILTED PHOTO (cont.)
Auxiliary coordinates:

18. Solution : SCALE ON A TILTED PHOTO (cont.)

19. EXAMPLE: GROUND COORDINATES FROM A TILTED PHOTO
Compute coordinates of points A and B
Solution

20. Solution : GROUND COORDINATES FROM A TILTED PHOTO

21. Solution : GROUND COORDINATES FROM A TILTED PHOTO
Compute the ground distance between A and B

22. RELIEF DISPLACEMENT
LN – vertical line then planes LNAA and LNA’ are vertical planes
Points n, a’ and a lie in same vertical plane and on same line
Points n, b’, and b lie in same line
Therefore, relief displacement radial from nadir point

23. RELIEF DISPLACEMENT (cont.)
Amount function of:
Flying height
Distance from nadir point to image point
Elevation of ground point
Position of point w.r.t. principal line and axis of tilt
Relief displacement
Will be less on upward side of photo
Identical along axis of tilt
Greater on downward half of photo

24. RELIEF DISPLACEMENT (cont.)
Amount computed with sufficient accuracy using:
Distances r and r’ measured from nadir point

25. TILT DISPLACEMENT

26. TILT DISPLACEMENT (cont.)
Amount of displacement given as
y* is distance in y’ direction from isocenter to the point
Approximate formula

27. TILT DISPLACEMENT (Example.)
Example: Given the following data,
f = 12.0 mm
t = 2° 15’
s = 135° 00’
compute the tilt displacement for a point given the measured photo coordinates
x = mm
y = mm

28. TILT DISPLACEMENT (Solution.)
Solution: the angle of rotation is:
θ = (135° 00')−180° = −45°
Transform the y-coordinate to “tilted” photo coordinate system and compute translation in y’ direction from isocenter to give y*
Tilt displacement is:

29. Image Geometry- Scale Image truly vertical only at the principle point
Scale only truly valid at the principle point
Obliquity increases away from the principle point

30. مقياس در عکس های تیلت دار

31. خلاصه این فصل مقياس در عکسهای تيلت دار
تغييرات مقياس نسبت به تيلت و ناهمواری زمين

32. منابع درسی این فصل INTRODUCTION TO PHOTOGRAMMETRY
Robert Burtch (Center for Photogrammetric Training) Ferris State University

33. پایان فصل 9