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VERTICAL
PHOTOGRAPHY
Photographic Scale Calculations Over Flat Terrain
Given: distance on the photograph and coordinates of corresponding points on the ground
ab := 3.0833in ab (^) m ab 25.4⋅ mm in
:= ab (^) m =0.078m
XA := 3910451.51m YA :=244219.02m
XB := 3910138.12m YB :=243867.07m
AB := (^) ( XB −XA)^2 +(Y (^) B −YA)^2 AB =471.256m
The denominator of the scale is (s):
s AB ab
:= (^) s = 6017
The scale of the photograph is 1:
Computing Flying Height on a Vertical Photograph
The basic equation for finding the flying height on a vertical photograph
where the elevations of the points are not equal is
AB^2 = ( XB−XA)^2 +( YB−YA)^2
where AB is the ground distance between two points and X i , Y i are the
ground coordinates of the two points at the end of the line. Recall that the
ground coordinates can be determined from the photo measurements using
the relationships:
y
f
x Y H h
f
X =H−h = −
where H is the flying height above the datum, h is the elevation of the ground
point, f is the focal length of the camera, and x, y are the photographic
coordinates of point i.
Center for Photogrammetric Training Vertical Photography
Substitute the values for X and Y into the distance equation yields
2 a
A b
B
2 a
A b
2 B y
f
H h
y
f
H h
x
f
H h
x
f
H h
AB ⎟
+⎛^ − − −
=⎛^ − − −
Rearranging,
( ) ( )
2 [(^ b a)^ (^ A a B b)]^22 [(^ b a)^ (^ A a B b)]^2
2 2 b B b a A a
2 2 b B b a A a
2
y y H h y h y
f
x x H h x h x^1
f
Hy h y Hy h y
f
Hx h x Hx h x^1
f
AB^1
Let:
( ) ( ) ( (^) A a B b) ( (^) A a B b)
b a b a
n h x h x q h y h y
m x x p y y
Then the ground distance can be represented by
( ) ( )
2 [(^22 )^2 (^ )^ (^22 )]
2 2
2 2
2
m p H 2 mn 2 pgH n q
f
pH q
f
mH n^1
f
AB^1
Moving the distance squared (AB^2 ) to the right hand side gives us the
quadratic form of the equation. Let
( ) ( ) ( )
( )( ) ( )( )
( ) ( ) (^2) 2
2 A a B b
2 (^2) A a B b 2
2 2
2
b a A a B b b a A a B b 2
2
2 b a
2 b a 2
2 2
AB
f
h x h x h y h y
AB
f
n q
c
f
2 x x h x h x 2 y y h y h y
f
2 mn 2 pq
b
f
x x y y
f
m p
a
The solution is found by solving the quadratic equation in the form of
2 a
b b 4 ac
H
− ±^2 −
Center for Photogrammetric Training Vertical Photography
AB =553.
h (^) Avg
Z A +Z B
:= h (^) Avg =293. Compute initial estimate of the flying height
H 1 AB
ab
:= ⋅f +h (^) Avg H 1 =1219.
Check the initial estimate of H:
XA
H 1 −Z A
f
:= ⋅xa YA
H 1 −Z A
f
:= ⋅y (^) a
YB
H 1 −Z B
f X := ⋅y^ b B
H 1 −Z B
f
:= ⋅xb
AB 1 := ( XB −XA)^2 +(Y (^) B −YA)^2 AB 1 =555.
Diff := AB −AB 1 Diff =−2. Adjust the flying height
H 2 AB
AB 1
:= ⋅ ( H 1 −h (^) Avg)+h (^) Avg H 1 := H 2 H 1 =1215.
Check the current estimate of H:
Calculation of flying height over variable terrain
Iterative Method
Given quantities: (^) f :=152. x 1 := 69 y 1 := − 174 xa
x 1 60
:= ⋅25.4 y (^) a
y (^1) 60
x 2 := − 28 y 2 := 19 xb
x 2 60
:= ⋅25.4 y (^) b
y (^2) 60
XA := 3910451.51 YA := 244219.02 Z A :=287.
XB := 3909949.05 YB := 243987.10 Z B :=298.
Compute photo distance and ground distance
ab := ( xb −xa)^2 +( y (^) b −y (^) a)^2 ab =91.
AB := ( XB −XA)^2 +( YB −YA)^2
Center for Photogrammetric Training Vertical Photography
X A
H 1 −Z A
f
:= ⋅xa Y (^) A
H 1 −Z A
f
:= ⋅y (^) a
Y B
H 1 −Z B
f X := ⋅y^ b B
H 1 −Z B
f
:= ⋅xb
AB 1 := (^) ( X (^) B − X (^) A)^2 + ( Y (^) B − Y (^) A)^2 AB 1 =553.
Diff := AB −AB 1 Diff =− 0
RCB
