Definition of common terms Errors on photo Lecture 7 Content Photo / Map Projection Definition of common terms Errors on photo Mathematical relationship (Scale) Vertical photos 4/16/2017 GEM 3366

Photo / Map Projection 4/16/2017 GEM 3366

ө Definition of common terms n Plane I p o R N P Plane II 4/16/2017 GEM 3366

the two planes intersect at a line called the horizontal trace “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 4/16/2017 GEM 3366

“o” is the optical center of the lens “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 4/16/2017 GEM 3366

Mathematical relationships from a vertical photo Scale: Photo f H LEARN Terrain 1/S = f/H where S = Scale Factor 4/16/2017 GEM 3366

Isocenter and Nadir Points 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 4/16/2017 GEM 3366

Isocenter and Nadir Points 4/16/2017 GEM 3366

ө ө Equations on the Principal Plane n i p f o v H R I N P Image plane i 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. p ө f v o H Object plane ө R N I P 4/16/2017 GEM 3366

Prove the following using the previous diagram 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) 4/16/2017 GEM 3366

Scale is determined by: 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 4/16/2017 GEM 3366

Vertical photos 4/16/2017 GEM 3366

Using similar triangles: Lab and LAB 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 4/16/2017 GEM 3366

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: 4/16/2017 GEM 3366

Ground coordinates from a single vertical photograph 4/16/2017 GEM 3366

photo coordinates are xa, ya and xb, yb ground coordinates are XA, YA and XB, YB Using similar triangles:  LOAA΄ and Loa΄  Then The figure illustrates a vertical photograph taken at a flying height H above a datum. Images ‘a’ and ‘b’ of ground points A and B appear on the photograph. The photo coordinates are xa, ya and xb, yb while the ground coordinates are XA, YA and XB, YB. 4/16/2017 GEM 3366

Using similar triangles:  LA΄A and La΄a  Then 4/16/2017 GEM 3366

An example: Solution for point A: 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: 4/16/2017 GEM 3366

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