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The accuracy of the radial line method
...o PHOTOGRAMMETRIA
r,.
Geod~tische A n w e n d u n g der L u l t b i l d m e s s u n g A p p l i c a t i o n s gdoddsiques de la P h o t o g r a m m d t r i e adrienne Geodetival A p p l i c a t i o n s of the a i r p h o t o g r a m m e t r y
The Accuracy of lhe Badia| Line Melhod By R. Roelofs. Communication from the Geodetic Institute at Delft.
§ I. Introduction. ........ The development of aerial photogrammetry depends to a large extent on the question whether it will be possible to improve on the existing methods or to find new and better means for the photogrammetric determination of those points necessary for locating the~ photographs or for plotting maps from them. It is therefore desirable to make a detailed researd/ of' the accuracy of the radial triangulation; this method in several variations is the most applied way. it may be remarked here that Prof. W. 8chermerhorn has formulated the problem as i* is dealt with hereafter and that the author only had the pleasure of working it out by order and with m a n y suggestions of Prof. Sd~ermerhorn. It is well known, that in radial triangulation a d~ain of rt;ombs or triangles ~s built up by measuring a n u m b e r of angles or directions i~ the plane of each photograph. One strive'~ at such a result (hereafter called the "exact" result) which would also be obtained by measuring horizontal angles or directions b y ' m e a n s of a theodolite set up over the point on the ground, corresponding with the centre of the radial-lines in the photograph. The choice of the radial point depends o n varion s circumstances; if the ground is absolutely level and the oi~tical axis of the camera exactly vertical during the flight, then the resulting photograph is a reduced b u t conform representation of the area. Angles or directions measured in any point of the image plane will be the "exact" ones. I n case of photographs obtained with a tilted camera-axis, a set of exact angles or directions will be obtained by measuring in the focal point (better: one of the two focal points or isocentres), i.e. a point situated at a distance f tg ½ i from the p r i n c i p a l point in the direction of tilt (f = local length, i = tilt). The ground being not level there is no poittt in the photograph where the exact angles or directions can be measured. There is, however, a point where angles or directions can be measured, the amount of difference between these and the exact angles m' directions being independent on the relative aliiiudes of the point. This point is the plumb point (nadir point) situated at a distance f tg i from. the principal point in the direction of tilt. It is to be expected that for nearly flat ground the measuring in the focal point has to be preferred to measuring in the p l u m b point and that, the differences in relief of the gronnd becoming larger, the results, obtai~ed in the nadir pot:at will relatively become more favourable. Thus the question is which point, focal point or nadir point, is to be preferred by a given topography and which is the a eenracy of the measured angles or directions. In ihe foregoing we have assumed: 1. that the measured tilt of the camera is errorless and that it shouhl accordingly be possible to determine the focal point or nadir point without error; 2. that no attempt i s made to deduce the horizontal directions from the measored directions by applying corrections.
42 The first-mentioned supposition ~ i l l never occur since tile tilt of the camera is an obseroed element and therefore not errorless. With regard to the second assumption the following remarks can be made. According to the above-mentioned property of plumb point measurements it is possible to deduce the horizontal directions from the measured directions by applying a correction, which is independent of the topography and therefore easily to calculate. In connection with the inaccuracy of the locating of the n a d i r point, resulting from the inaccuracy of the observed tilt, the calculated correction and therefore the corrected direction will be inexact or, in other words, will possess a mean error which shall be dependent not only on the mean error of tilt but also on the differences in altitude of the points on the ground. This raises the question with whi& method the best results can be obtained under certain circumstances: measuring in the focal point without applying any correction or measuring in the nadir point and applying corrections for tilt. Correcting focal point measurements is not so simple, since the correction depends among other things on the flying height and on the differences in height of the points on the ground. In principle it is immaterial how these heights are measured. The [lying height e. g. can be determined by means of an altimetre. Barometric methods are also suitable for measuring the differences in elevation of the points on the ground. The mean error of the corrected direction measured in the focal point depends not only on the mean error of f i l l but also on ~he mean errors of the flying altitude and of the altitude of the points on the groun d. We shall, therefore, try to give an answer to the question whether corrected focal-point-measurement or corrected n a d i r - p o i n t - m e a s u r e m e n t will give better results in certain circumstances. Summarizing the problem is as follows: Which is, in the radial line method, the mean error of a direction of wMch the focal point or the nadir point is the origin and which of the two points should be preferred in the following cases: t. corrections are not applied to the results of measurement; 2. corrections are applied to the nadir-point-measurements; the measurements remain uncorrected;
focal-point-
3. both measurements are corrected. On the basis of the answers to these questions a favourable choice between the focal point and the nadir point can be made. It will be possible to investigate the accuracy to be expected and to bring the accuracy of measuring angles or directions in accordance with the accuracy limited by the uncertainty of tilt, differences in relief and flying altitude. It will e.g. be possible to consider in whi& cases it is advisable to use the most accurate instrument for the purpose, the radial triangu!ator, and in which cases simpler instruments and methods, e. g. graphical methods, should be used. Finally our research will furnish information for an answer to the question which part of the whole process from flying to computation of rhombs or triangles needs improvement in the first place in order to increase the accuracy of radial triangulation. [t will be interesting for example to examine which is the influence of an accurate knowledge of the position of the camera-axis during the flight using one or two horizon cameras. It may be specially pointed out that this article only deals with the comparison of the focal-point- and the nadir-point-triangulation, so that the accuracy of the proper measurement of the angles or directions,, which is the same in both eases, will not be considered. "6 2. Deduction of a formula for the error Aq~ in a measured direction. The relationship will be studied between directions in two planes, i.e. the plane of the photograph and the horizontal plane. It is obvious to choose as a direction of reference
43 the direction of their line of intersection or a direction in eadt plane perpendicular to that line of intersection. For the following considerations we will apply in each tflane a system of coordinates with the axis of abscissae parallel to the above-mentioned line of intersection and the axis of ordinates p e r p e n d i c u l a r to it. The origin of the system of coordinates in the plane of the photograph is being placed in the point He' in which the directions are being measured, the origin of the coordinate 0
A .!..I'
L~
\"
Fig. 1
0
N"
He\ j~he
system in the horizontal plane is being placed in the point tfa °. This is the horizontal projection of the point on the ground Ha, whid~ corresponds to the point He' (fig. 1). Let us take in the photograph a certain point Pi', corresponding to point P~ on the ground, which has its horizontal p r o j e c t i o n in Pi °. The b e a r i n g from the origin He' to P i" is the angle Pi' between the line H e ' P { and the axis of ordinates y'. The corresponding horizontal b e a r i n g Ha ° - P ~ ° is the angle p~ b e t w e e n the line He ° - p ° and the axis of ordinates Y'. It is understood that the error Ap i in the measured bearing Pi' is the difference between Pi" and Pi: Ap = ~p~'-- Pi. ( t) In the photogrammetric literature (we refer to the list of literature at the end of this article) one finds various a p p r o x i m a t e solutions. According to an idea of Prof. Scherlnerhorn, here an absolntely exact solution will be given.. For a detailed dedtlction we refer to "Puntsbepaling door middel van fotogrammetrie" b y Prof. Schermerhorn [6]. In this article we will limit ourselves to a concise; but, we hope, sufficiently cleat' indication of the solution. With the principal point H' of the photograph and the corresponding point H on the ground as origins systems of coordinate axis x" y " z " and X" Y" Z" are applied parallel to the above-mentioned systems x ' y ' z' and X' Y" Z'. The origin /:Ie' is provisionally placed entirely a r b i t r a r i l y at a distance e' from the principal point H' in a direction q%'. The horizontal plane is placed at the same level
44 with point H. T h e f l y i n g a l t i t u d e O N o is ho, the a l t i t u d e s of H e a n d Pi a r e respectivJly h e a n d hi.
In view of the necessity to a p p l y in some eases A(p (with opposite sign) as a correction to the m e a s u r e d bearings, it is desirable to express A~o as a f u n c t i o n of elements which can be m e a s u r e d in the photo-plane. We will therefore t r y to express in (1) qoi' and ~vi in eoordinates with respect to the x ' y ' z ' - s y s t e m . The expression for
~oi' is very simple:
x/
tg %! - y;'
(2)
x i' a n d Yi' being the coordinates of the point Pi'. Before a f o r m u l a for qoi can be e o m p u t e d we m u s t o b t a i n the coordinates of f i e ° a n d p,o in the x ' y" z' - System. F o r that p u r p o s e the f o l l o w i n g o p e r a t i o n s h a v e to b e p e r f o r m e d . a) F r o m the eoordinates of He' a n d Pi' in the x ' y ' z ' - s y s t e m the coordinates in the x" y" z'-system can be o b t a i n e d b y means of a p a r a l l e l shifting of the axis from He" to H'. Together with the coordinates of I1' a n d O this results in: ~.'~
y"
z"
H' 0
0 0 0 0 0 f H e" e' sin q)e' e ' c o s q)e' . O P/ x / + e ' sin ~%' y / + e ' cos q%' 0 b) These coordinates will be t r a n s f o r m e d to coordinates in the X" l z" Z " - system. The coordinates of the origin of this system are: x Hpp = Y x =t l ° a n d z ~tp= f - - h o sect a n d the angles between the coordinate-axis of both systems:
X"
X"
y~'
Z"
0
+~.~
-~½~
Y" +}~ +i ~z+i Z" +1 2jr+ i i e) By m e a n s of the o b t a i n e d results the equations of the lines O He' a n d O P i ' in the X" Y" Z"-system can be composed. By a s s u m i n g in these e q u a t i o n s that Z " = he resp. Z" = hi, we o b t a i n the c o o r d i n a t e s of He a n d Pi. F r o m them the coordinates of He ° and p o will be o b t a i n e d b y t a k i n g Z"]j e = o a n d Z" G. = O. d) After that a b a c k w a r d following coordinates :
transformation
to the x " y ' z ' - s y s t e m
furnishes
the
XHe"o = C z ( h o - - he) e" sin ~ve' Y ~ 2 = C z (h° - - he) (e' cos ~oe' eos i -- f sin i) cos i + h o sin i Z
"o G
~
C , (h o
he) (e' cos 9)e' cos i - - f sin i) sin i - - h o cos i + f
(3)
x ~ o = C 2 ( h o - - hi) (x'~. + e' sin ~%') yfi;q = C 2 ( h o - h i ) { ( y / zfi;, = C 2 (h o -
+ e' c o s ~ e ' ) e o s i - - f
sini}eosi4-h
osini
h i) { (y/-t-e' cos ~Oe') COS i -- f sin i I sin i - - h o cos i 4. f
in which: C 1 = (e' cos q ) / s i n i + f cos 0 - 1 Cz = (C) -'l + y / s i n 0 - 1 e) A f t e r these p r e p a r a t o r y c o m p u t a t i o n s the e q u a t i o n of the line o b t a i n e d b y intersecting the p l a n e O - - H e ° - - P i ° (not indicated in e) After these p r e p a r a t o r y c o m p u t a t i o n s the e q u a t i o n of the line o b t a i n e d b y intersecting the p l a n e O - - H s o - - P i o (not indicated in horizontal plane.
Pi ° can fig. 1) with He o - P i o can fig. l) with
He ° -
be the be the
45 The equation of the first p l a n e is in a d e t e r m i n a n t x" .q" z" 0
0
zi~tr ~ tt
1
f
l
= o
~ f!
xp 7
. P;
or developed into an expression with the co-factors P, Q, R, a n d S of the elements of the first row: Px" + Qy" + R z " + S = o (4) The equation of the second p l a n e is: y " - - z" c o t g i + (h o sec i - - f) co|g i = o (5) The l i n e of i n t e r s e c t i o n He 0 - Pi ° is given .by the e q u a t i o n s (4) and (5). F r o m these equations the following expression is o b t a i n e d for the angle ~i--.~t.~ between this line of intersection and the S - a x i s or X"-axis: cotg @i - - ~ z) = - - tg g,i =
Qeosi+ p Rsini
(6)
From (2) a n d (6) can be deduced: --
Px~! + (Q cos i -t- R sin i) Yi' tg A~v = py,: __ (Q cos i + R sin i) .r i'
S u b s t i t u t i n g the value of co-factors P, Q, a n d R whilst a p p l y i n g (3). we a reduction of the d e t c r m i n a n t s w h i d l m a y be left to the reader: fcosi+e'cos~'sini I o y~ sin i x~.'y/(1 - - cos i) o x i" (f sin i - - e' cos q~' cos i) + Yi' e' sin 9~' ~tgA~= [ o fcosi+e'cos~¢'sini I Yi' sin i i - - ( x i ' 2 + y i ' 2 cos i) Yi (f sin i - - e" cos ~ ' cos i) - - x~ e" sin ~ '
i
obtain, after h¢--ho [ h i - h~ ho-- h i h ~ - - h o [ (7) h i - - h~ ho-- hi
i I
In the derioation of [his f o r m u l a no a p p r o x i m a t i o n is applied. The absolute height or the horizontal p l a n e being i m m a t e r i a l we will suppose that h e := o. Henceforth the accents of the various elements in the f o r m u l a (7) will be left out for easiness' sake, since there is no risk of mistal~e. In the a b o v e - m e n t i o n e d treatise of Prof. Schermerhorn [6] the form a u d the value of Aqo in some p a r t i c u l a r cases have been considered. Thus it can easily be seen that Aqo = o when the g r o u n d is level (hi = o) a n d m e a s u r i n g being performed in the focal point (e = f t g ½ i, We = o). F u r t h e r A~ is i n d e p e n d e n t of differences in level (h i:~o) m e a s u r i n g b e i n g p e r f o r m e d in the p l u m b point (e = f t g i, ~oe = o). For the rest we m a y refer tO the a b o v e - m e n t i o n e d publication. Since the exact formula (7) is too complicated to be h a n d l e d in the following considerations, we will r e p l a c e it b y an a p p r o x i m a t e f o r m u l a , s u p p o s i n g i to be relafivily small (e. g. max. 6% Moreover we will consider m e a s u r e m e n t s in the focal point or in the p l u m b point only, so that We = o a n d e = ftgqi in which q = ~ corresponds with the focal point t r i a n g u l a t i o n a n d q = 1 with the n a d i r point triangulation. A systematic a p p r o x i m a t i o n is o b t a i n e d b y e x p a n d i n g in the. f o r m u l a (7) all goniometric functions of i in ascending' powers of i and, after that, developing the d e t e r m i n a n t s as far as terms of the t h i r d o r d e r (i3)". The a d v a n t a g e of this m e t h o d over other less systematic methods lies in the fact that all a p p r o x i m a t i o n s are of the same order a n d that it is p o s s i b l e to check the neglected part.
46 Using the w e l l - k n o w n f o r m u l a in c o m m o n f o r m : sine= e---~s3+ . . . . . coss= 1 ~-~s2+ .... tgs= e +~s3+ .... we obtain, a f t e r h a v i n g s u b s t i t u t e d in (7) the r e c t a n g u l a r c o o r d i n a t e s x i a n d Yi b y p o l a r coordinates a i and ~ and after dividing numerator and denominator by ai2f (ho--hi):
in w h i d l k , ~ = f,,
(hl
h oTh,.
igd~=
k~i + kxi2 + kai3 + 1 + k 4 i + k 5 i2 +
.... ....
' a i ' %.
We ean w r i t e f o r this: tgAq~ = ( k l i + k 2 i 2 + k a i 3 + - . . . . )(t--k4i--kgi2 .... ) ~ kii+ksi2+kTi3+ or d(p = b g t g (k~ i + k'6i2 + k7i3 + . . . . ) In c o n n e c t i o n w i t h the w e l l - k n o w n f o r m u l a :
. . .
bgtgs= s--½s3+ .... we o b t a i n : z/q) = t c e i + l c 6 i a + k a i 3 + .... C o m p u t i n g the coefficients k, w e f i n a l l y o b t a i n the f o l l o w i n g e x p r e s s i o n : -
-
d ~o = i a Z sin qv,.+ i 2 a z sin qJi cos %. + i3 (a s 4- a 4 sin2 %.) sin %- + . . . .
(8)
We m a y r e f e r h e r e to t],e r e v i e w at the e n d of this treatise, giving the m e a n i n g of tile coefficients a. T h e focal p o i n t b e i n g the r a d i a l point, (q = ½), w e h a v e : ae =-~ r t a 2 = ½ t - - ¼ r2t2 a3= ~'~rt--'rt 2 + ~ r a t 3 a 4 = ½ rta - - ~= r3t3 f
h,. and r h o- - h i ' ai Since ead~ of these coefficients c o n t a i n s the f a c t o r t, it is e v i d e n t t h a t Aqo = o w h e n the g r o u n d is flat. in w h i d l t -
F o r the p l u m b p o i n t (q = 1) the c o e f f i c i e n t s are: a l ~
a 3 ~
~14 ~
o,
--
c~ 2 - -
1
o"
None of t h e m c o n t a i n s the factor t. The e r r o r is t h e r e f o r e i n d e p e n d e n t of the differences of relief. § 3" C o m p a r i s o n o f triangulation.
uncorrected
focal-point-triangulation
mith
uncorrected
W e will use t h e s q u a r e e r r o r A~o~ as a m e a s u r e for the a c c u r a c y . ~oi g e n e r a l l y can a s s u m e e v e r y v a l u e b e t w e e n o a n d 2~.
plumb-point-
In the f o r m u l a (8)
I n o r d e r to find a m e a s u r e of a c c u r a c y for a c e r t a i n area, wc Will t a k e the m e a n o f all values of t h e s q u a r e e r r o r dq~2, q~i b e i n g limited b y o a n d 2 ~ : 2:*
= 2-~fAq~ 2 . dqJ o
Squaring and integration produce: d~o2 = ~ i 2ae2-k ½ i 4(a 2 A - S a l a 3 + 6 a l a 4) F o r p l u m b - p o i n t - t r i a n g u l a t i o n this b e e o m e s : d%, 2 = a~- i4 o r dq% = -~ 0.18 i2 F o r f o c a l - p o i n t - t r i a n g u l a t i o n w e obtain, w h e n n e g l e c t i n g t h e s e c o n d t e r m : A~oZ2 = -s ~ iarat2
or
A~/=
+O.aarit
(9) (10) (t~)
47 Concerning the value of t ~
(t /~t o -/,,. V'.i n -hi)
this fornnila, the following r e m a r k s can be
made. Supposing h i to be relatively small, we can expand t in a series as follows: t2
(h~)2+ 2 (~)3
(hi/4
3+ \ho) +
.
.
.
.
F o r a fairly large area the sum of the positive values h i will almost be equal to the sum of the negative values. Thus in the formula for a m e a n value of the error the terms with odd indices must be left out. Neglecting terms with fourth and higher powers of i, we obtain : hi t=--
ho
The computation of A%, and Aq)f is facilitated by the nomogram I (fig. 2). h '°key" ("Sdfltissel") at the head of the nonmgram shows how it has to be used.
&"~.O.t8 i ~ und 1o0
2
4
6,~ ¢
~-~=o.35ri hl .~
¢
~
q"~
~
o
5"3
7
~0.20
3
5.1o
Fig 2
~ ~
~
~
~' "~ho,~..... "
"
~R ~
For nadir-point-triangulation o n l y the left part of the graph is used. At the poii/£ of the curve, corresponding with the given value of i, the value of ~ can be read. For focal-point-triangulation the entire nomogram is used with the exception or' the curve; for the rest we m a y refer to the key. If one finds that, when using the nomogram in a certain case, ~ is greater than ~ or in other words if the fixed point is situated beloro the curve in the area marked @ , then ur, eorrected focal-point-triangulation gives obviously better results than uncorrected plumb-point-triangulation. If the fixed point is situated in the area marked @ , then the plumb point has to be preferred. From this it can be concluded that the nomogram can be used in two ways: l. for computing the mean errors Aq~--"~and A%, or 2. only for deciding in w-hid~ point, f o c a l p o i n t or nadir poi~t£, measuring should be performed.
48
An e x a m p l e m a y explain this: If the flying" he_~ght hi = 1000m, the mean of the differences of relief h~ = 15 m, tile mean tilt i = 3g, a:tid the focal distance f = 21 cm, we obtain: A%~ = 2c.5 and Aq~/-= 4c.7 W h e n c o m p u t i n g by means of tile n o m o g r a m ~ we get a p o i n t in the area ( ~ . F r o m this it follows in a c c o r d a n c e w i t h the o b t a i n e d ~-alues of ~ and ~ that nadir-pointt r i a n g u l a t i o n is to be preferredl It can easily be seen, w h e n using' the n o m o g r a m , that in the a b o v e - m e n t i o n e d circumstances f o c a l - p o i n t - t r i a n g u l a t i o n gives better results only then w h e n the m e a n difference of relief does not e x c e e d 8 m. § -k
A c c u r a c y of tilt measuring by means of horizon cameras.
As long as it is only a question of accidental errors in tilt measuring, it is for the t h e o r y of errors, which will be developed in the n e x t p a r a g r a p h s , i m m a t e r i a l which m e t h o d of d e t e r m i n i n g tilt is applied. It is sufficient to k n o w file a c c u r a c y of tip and tilt a n d e v e n t u a l l y the a m o u n t of correlation b e t w e e n them. In order to give an e x a m p l e of developing the t h e o r y of errors, we will study the accuracy" of observing tilt b y horizon cameras. It is so much more interesting because the provisional results obtained w i t h horizon cameras are favourable. The horizon cameras will, no doubt, be v e r y useful for the r a d i a l - t r i a n g u l a t i o n , especially the new Zeiss camera HS 8, which shows the horizon in two p e r p e n d i c u l a r directions. We will distinguish between two cases: horizon c a m e r a and b y troo cameras.
the d e d u c t i o n
of the tilt and tip
by one
a) One horizon camera (parallel to the flying direction). The direction of the c a m e r a axis is d e t e r m i n e d b y the position of the p h o t o g r a p h e d horizon w i t h respect to the s i m i l a r l y p h o t o g r a p t l e d indices of the horizon camera. O n tlw left and on the r i g h t side of the horizon p h o t o g r a p h the distances b e t w e e n horizon and indices are m e a s u r e d b y means of a scale, which indicates directly the tilt c o r r e s p o n d i n g with the reading. The readings being 1 and r, the tilt ~o a n d the tip a are: o~=l--r = (l + ,-)c 02) c is a constant, d e p e n d i n g a m o n g other things on the focal length of the horizon c a m e r a (e. g. c = 0,382). Supposing t h a i the readings are of the same a c c u r a c y and not correlated or in other words that Ql~ = Q~.r aud Qrt = Qlr = o then by a p p l y i n g the l a w of p r o p a g a t i o n of e r r o r s to the f o r m u l a s (12) it follows: Q
Q , , = c 2 Q ...... a n d Q,o~=O
(13)
In this f o r m u l a s Q,,~ is the w e l l - k n o w n denotation for the w e i g h t n u m b e r of a. The connection b e t w e e n the weight n u m b e r and the m e a n error m , of a is: m a2 = t ~2 Qaa tt representing the m e a n error in an observation of unit weight. Q,a is the usual denotation for the a m o u n t of correlation b e t w e e n ~o and a. We also use " s y m b o l s " Qx, the m e a n i n g of which, appears f r o m : Q~ Q.r = Qx 2 ~ Q .... and Qx Qj, -= Q,:~, A .s!lrnbol has no v a l u e as a number, a product of two symbols on the other h a n d has. An a m p l e e x p l a n a t i o n of the calculating w i t h these w e i g h t m~mbers and symbols is given by Prof. T i c n s t r a [8].
49 b) T m o h o r i z o n c a m e r a s (patallel a n d p e r p e n d i c u l a r to the ilying direction). F r o m c a & of the two photographs the tilt as well as the tip can be computed: first photograph iilt:
~vj = l ~ -
second photograph
rz
~o2 = (12 + r2) c
tip: % = (l 1 + rz) c a2 = 1 2 - r 2 Supposing the m e a s u r e m e n t s in both photographs to be of the same accuracy, we read from (13) that the weights of these quantities are: g,'z : g~2 = g.'2 : g~s = 1 : e 2
Taking" these weights into account, we o b t a i n the following m e a n tilt a n d tip: -e2c°l + 0~2 a n d . ~ _ a:t -I- C2a2 ~o~ 1+c2 1+c2
(14)
I n order to c o m p u t e the weight n u m b e r of ~,, a n d a, we a p p l y the law of p r o p a g a t i o n of errors. The result is after some reduction: C2
Q~5"~ = Q ~ = i + c 2 " Q~1%"
and
QE-~= o
(15)
These formldas give insight into the p r o p o r t i o n of the a c c u r a c y of observing tilt b y means of one or t m o horizon cameras. The c o n s t a n t c being 0,382, the f o r m u l a 05) is: Qo~o, = Q ~ = 0,1273 Q~oz%" or
m E- = m~- = ~__0,36 m~oz
from (13) :
maz --= + 0,38 m~oz
Hence it :follows mE- = + 0,36 mo~ a n d m~- = ~ 0,95 m,~ f
J
Using the second horizon camera, the increase in a c c u r a c y is greater in tilt t h a n ill tip. The r e s u l t is, that lhe a c c u r a c y of observing the position of the camera-axis is the same in all directions . § 5. T h e a c c u r a c y o f d e t e r m i n i n g t h e x ' - a n d y ' - a x i s in t h e p l a n e o f the p h o t o g r a p h a n d its i n f l u e n c e on t h e a c c u r a c y o f c o o r d i n a t e s o f a p o i n t Pi.
F o r the following considerations we i n t r o d u c e an a d d i t i o n a l system of coordinates a n d ~7, the origin of which coincides with the p r i n c i p a l point a n d of which the axis passes through the collimating marks. Let us suppose that it is possible to determine this system of coordinates w i t h o u t error. The position of the origin of the x ' y ' - s y s t e m (focal point or p l u m b point) a n d the, direction of its axis are o b t a i n e d b y means of the o b s e r o e d tilt a n d tip a n d are therefore not errorless. The tilt a n d tip being ~o a n d a, the tilt i of the camera proceeds from i 2 ~ o)2 + a 2 whilst the direction of tilt follows from O=bgtg~ Therefore b y a p p l y i n g the law of p r o p a g a t i o n of errors:
1
Q oo = - ~ (a2 Q~o) + co2 Q~, - - 2 coa Q,o~ )
Qo, = ~ { c o a ( Q ~ - Qo,~o)+ (~2-a2)Q~,~,}
(16)
5O For one horizon camera we have, according to (13):
07) ,e(.. = -
p
-~
Q,oo,
in which p 2 = 1 - - c 2 For two horizon cameras: Qii = Qgg,~, 1
Os)
Qoi = o
The formulas (18) obviously proceed frmn (17) simply by putting p = o and replacing Q~o~o b y Q,~,~,.
The x' y ' - system being placed in the direction of tilt and its origin being at a distance e = f tg q i from the principal point, it is evident that the x ' y ' - s y s t e m can be obtained by turning the @-system through the angle ~9 and by removing it in the y'-direction uver a distance f tg q i (fig. 3).
g
I
.I'"
..... ~ ......
.i
S #~
jl/
?~e
~t
,/"
/'
/ /
4"-
/ / / / Fig. 3
The relation between the coordinates ~ ~i in the $~-system und xi 9i i n t h e x" y ' - system is therefore given b y the usual transformation formulas: xi = ~i c o s 69 - - *]i s i n 0 .t/; = ~i s i n O + ~ i c o s
O -- e
from whi& follows: •Qx i = ( - - y ; -~ e) q o Q.~;; = x¢ Q o - - f q sec~ q i. Qi
(1.9)
The relations between the rectangular coordinates x i and yi a n d the polar coordinates a~ and ~i are: x'; a,.~ = xi 2 + .G.2; %- = bg tg - Y;
By applying the law of propagation of errors~ we obtain after having substituted (19) and after some reduction, in symbolic form:
Q"i = - - e sin %.. Q o - - fq sec2 q i cos 9~i. Qi
Q~,.= ( - t - - ~ c oes m , .
) Qo+
[~--7 qsec2qisin%.Q i
(20)
For the sake of brevity we write:
Q~. = A Q o + BQ,. Q
Q~e
=
-- (20
(20 (22)
§ 6. Accuracy of Aq; inasmuch as it is dependent on the accuracy of tilt i. ......... Since we may suppose the measurements of tilt / , h e i g h t of the ground and the flying altitude to be uneorrelated, the influences of the mean errors of these quantities on the accuracy of Aq~ can be treated separately. According to (8) is:
~t~-= f (a~., 0%, i, ~e) Denoting the partial derivatives of this function with respect to a~, q)i, i and q~e, respectively fl, 7, 6, and e, we obtain b y applyillg the law of propagation of errors in symbolic form:
Qaqo = flQ"i + 7Q~i + 6Qi + eQ~e In connection with the formulas (21), (22), and (2:1) we can write:
Q&o= (~A + yC--e) Qo + (fib + yD + d) Qi
(23)
(24)
or introducing other denotations: •
O~
= F Oo
+ G O,.
(25)
:Ill ram-symbolic form: QzqJztm = F2Qoo + G2Qil + 2 F G Q o , Substitu.ting the values of the weight numbers according to (17), we obtain:
=
P ~-/r + i J/Q
(26)
(27)
In case of two horizon cameras p = O whilst Q~o
(28)
Sinee p does not oeeur in this formula it is applicable (save eventual substitution of Q~oo~by Q ~ ) not only when using one hut also when using tloo horizon cameras. However, in the first ease it represents a m a x i m u m value. Heneeforth we shall use for the sake of simplicity the formula (28) only, without risk of overestimating the accuracy. The eomputation of the partial derivatives fl, 7, and 6, proeeeding from the formula (8), is merely a matter of algebraical teelmics; it is of speeial importance to continue computation until just sufficiently high powers of i are obtained. We will not fill too much space by giving the development of the formulas to the full extent, b u t only mention the final result. The computation of the derivative e with respeet to qJe cannot be commenced from the formula (8) sinee in this formula +Pe = o. It is, therefore, neeessary to start from the general formula (7). Denoting in (7) numerator and denominator D~ and D.2, we obtain: tg
dqo = -~-
~2 f r o m w h i & it follows t h a t
( Odg~ ~
- = --
f
t
(D ODz
&=o
D OD2 ~ I
=-;C-
Jr%_o
I t is, t h e r e f o r e , s u f f i c i e n t to c o m p u t e t h e v a l u e s of Ds, D2 a n d t h e i r d e r i v a t i v e s w i t h r e s p e c t to rpe a n d to s u b s t i t u t e t h e results, a f t e r h a v i n g p u t r; c = o in t h e a b o v e - m e n t i o n e d formula. T h e c o m p u t a t i o n of fl, 7, 6, a n d e a n d t h e m u l t i p l i c a t i o n w i t h t h e k n o w n q u a n t i t i e s A, B, C, a n d D (20 a n d 21) give t h e f o l l o w i n g f i n a l r e s u l t : F i - Tz cos qvi + i (T 2 + 1"3 cos 2qJi) + i 2 (T 4 + T s cos 2%.) cos %. (29) - - G -- T 6 sin qv,. + i 7 7 sin 2%. q- i 2 (Ts + T~ cos 2mi) sin %. T h e d e n o t a t i o n s T h a v e b e e n e x p l a i n e d in t h e r e v i e w of d e n o t a t i o n s a t t h e e n d of this article. B y m e a n s of t h e f o r m u l a s (29) it is e a s y to c o m p u t e t h e v a h t e of t h e c o e t f i c i e n t of Q ..... i n t h e f o r m u l a (28). F2 i~- + G~ = f (%) T h e v a l u e of % is l i m i t e d b y O a n d 2 n. J u s t like in § 3 we will c o m p u t e h e r e a m e a n v a l u e : 2~
1
¢1 F2
+
G21d%.~. ,
(30)
......
By i n t e g r a t i o n w e o b t a i n : [t2 = ½ (T12 -4- T62 ) + ½ i2 (,2 T22 + T32 + T72 + 2 T s T4 + T s T s + 2 T 6 7'~ - - T 6 Tg) F o r t w o of t h e m o s t f r e q u e n t cases we c o m p u t e d t h e coefficients T, viz. for c a m e r a s w i t h focal l e n g t h s of I0 a n d 2t c e n t i m e t r e s , t h e size of t h e p h o t o g r a p h b e i n g 1S X 18 cm. B y this size t h e d i s t a n c e s ai are a l m o s t 6 ~l 10 era, t h u s r = off to 1,5 a n d 3. hi
H is o b t a i n e d a s a f u n c t i o n of i a n d t -- h e - - h i "
i averagely and ronnded a~.
S i n c e it h e r e a g a i n c o n c e r n s a mean
v a l u e f o r a f a i r l y l a r g e area, j u s t like in § 3, we c a n w r i t e a g a i n t = r e a s o n t e r m s c o n t a i n i n g o d d p o w e r s of t a r e oInitted.
hi
~oo' F o r a s i m i l a r
Finally we obtain the following expressions: H 2 = 1,406 t 2 -~ (0,094 q- 1,38 t2q- 1,25 ]~4) i 2 H 2 = 1,125 t 2 -4- (0,375 -t- 1,69 t 2) i 2 H 2 = 5,625 t 2 + (0,094 + 7,97 t 2 + 20,50 t 4) i 2 H 2 = 4,500 t 2 + (0,375 -4- 5,62 t 2) i 2 (a a n d b: f = l O c m ; c a n d d: f = 21 cm; a a n d c: f o c a l - p o i n t - t r i a n g u l a t i o n ; nadir-point-triangulation).
(31a) (31b) (31c) (3ld) b a n d d:
W h e n t a n d i a r e s m a l l t h e t e r m s of the coefficients of i c o n t a i n i n g t m a y b e o m i t t e d , so that the formulas have the simple form: hi 2 H2=c 1~ + c3 i 2 (32) F o r t h e c o m p u t a t i o n of H we d e s i g n e d t h e n o m o g r a m I I (fig. 4) w h i c h r e q u i r e s no f u r t h e r e x p l a n a t i o n besides t h e " k e y " w h i c h is d r a w n a t t h e h e a d of it. W e will o n l y p o i n t o u t t h a t u s i n g t h e f o r n m l a (30) i n t h e f o l l o w i n g f o r m :
m2A~ = H2m2o.
(33)
we d e r i v e f r o m t h e n o m o g r a m such v a l u e s of H as c o r r e s p o n d w i t h mare e x p r e s s e d i n c e n t e s i m a l seconds, rno~ b e i n g e x p r e s s e d i n c e n t e s i m a l minutes.
53 § 7. Accuracy of Aq; inasmuch as it depends on the accuracy of hi and ho. The formula (8) shows that when measuring in the plumb point, Ag is independent of
hi and h o. Therefore, the following considerations only concern focal-point-triangulatiom h~ and ho being uncorrelated, we obtain: Qa~,~= \Oh i! QGG + Oho l QGG By computing the derivatives in this formula and taking the mean of all values 9 over the field for which c& lies between O and 2 z, we obtain:
0.......= -~-i2r2+ ,,',~ i4(3 + 2r2-- '8r2t+ J2r4t2) "(t~o2 /,,,/,,. + ~ T.
Schliissel:
NOMOGRAMM
/Or
H~=
h'
_
o
/=Were ,
i
FOalPt.
~ /
~ -~
Y/
'1="
H
Nod Pt
\
\
---
H
H
"~
=4_=_2>
/
. /
~
/
~
i - - - -
_ _
1
"~
Fob. Pt
_.~
H2,
/~/~ 7
cm
[-2, Nod Pt.
\
"•'
QG/,. )
~
~
ho~......
~
6-0
___
2
4
2
6-0
2
4
6-0
\
2
Fig. 4
6
~P
If i and t are small, the term containing i4 may be neglected,
~,,~o,~ = ('- ir ~-~)2QG./,, + (½ ir ~f2)2Q/~oG or denoting the coefficients K and L: m2~q~= K2m2ai + L2m2ao
(34)
The nomogram ]II (fig. 5) gives K and L. In order to obtain m a t expressed in centesimal seconds, it is necessary that in (34) m G. und rn G are expressed in metres. Combining (33) and (34), we obtain tile following complete formulas: foe. point: m'-2a~= Hf 2 moo2 + K 2 m~ + L2 ink2 (35) had. point: m2A~o= H,? m J (.36) For clearness' sake we have added to the coefficients H in these formulas the indices f and n.
§ 8. Comparison of uncorrected focal-point-triaugulatiou mith corrected nadir-point'" triangulation. As a measure of accuracy for the uncorrected focal-point-triangulation we used in § 3: he
54 In the p r e c e d i n g p a r a g r a p h has been expressed b y :
the a c c u r a c y of the corrected n a d i r - p o i n t - t r i a n g u l a t i o n 7~t2~m = H~,, m2o~
(36)
In spite of the dissimilarity of these measures of a c c u r a c y - - the f o r m e r being the mean v a l u e of the systemalie error and the lafler being the m e a n accidental error - we can use them w i t h sufficient c e r t a i n t y as a base for our choice between the two methods of triangulation. W e will illustrate this with an e x a m p l e : c a m e r a f = lOcm, size of p h o t o g r a p h ~8)< I S c m ; m e a n tilt: i = 3"~;
~. IVOMOORAMM
K.o.,rt
"'::>'"" .-1°
,,,,
4k ~. 45 90
~0 80
\
":'\
.
70 70
g ~o L
30 60 25 50 I
,
20 4O
\
~5 30
¢o
20
5
~
~0 0
0
2
o
~
~
~
~
-
~
~ i
~
~
~00
~
0
6
Fig. 5
i,c,.,~o.,)
/IR
flying altitude: ho = 4000 m; m e a n h e i g h t of the relief: hi = 5 0 m ; one horizon c a m e r a : m . , = +- 12~; According' to n o m o g r a m I:
~ : ) = + 2c.o and a c c o r d i n g to n o m o g r a m II: H,, = 3,4 therefore:
mzlq.
= ~__3;4X 12 = "4-41 cc
E v i d e n t l y corrected p l u m b - p o i n t - t r i a n g u l a t i o n has to be p r e f e r r e d in this case. The f o c a l - p o i n t - t r i a n g u l a t i o n as c o m p a r e d w i t h p l u m b - p o i n t - t r i a n g u l a t i o n gains in a c c u r a c y a c c o r d i n g to the ground b e c o m i n g flatter. I n the a b o y e - m e n t i o n e d e x a m p l e the mean h e i g h t being m e r e l y 10 metres: d q J / = q 40 cc and m--,l~,= ± 36 cc In t h i s case we can save ourselves the trouble of correcting n a d i r - p o i n t - t r i a n g u l a t i o n a n d ratt~er p e r f o r m m e a s u r i n g in the focal point w i t h o u t a p p l y i n g corrections.
55 i t is understood, that one has to consider not only the a c c u r a c y but also the economy of the two methods. In order to a v o i d the complication of applying' corrections one shall have to tolerate as a rule some decrease in a c c u r a c y b y m e a s u r i n g in the focal point w i t h o u t a p p l y i n g corrections.
§ 9. Comparison of corlected [ocal-point-triangulaticn ...... triangulalion.
roith corrected nadir-point-
This subject requires only a little e x p l a n a t i o n ; the complete f o r m u l a s (35) and (30) h a v e to be used here. Using the data of the f i r s t e x a m p l e of the preceding pai'agraph, we obtain 1)7 means of the n o m o g r a m s II and i I I : H / : = 2.2, H,~ = 3,4, K = 6, L = 0,15 t h e r e f o r e , a c c o r d i n g to (35) and (36): t-72j = 4.84 m2o) + 36 m2G. + 0,0225 m2/q, Zf Itt2Aq ,z ~ l t , 5 6 m2(~
s u b s t i t u t i n g trG= _q2_12 c and m G = q~ 50 m, we o b t a i n :
nl~l,,f = 1259 + 36 m2 G. m2z~,~ = 1665 T h u s m239~/
[ m,%. I < 3,4 metre.
U n d e r this condition alone the a c c u r a c y
of f o c a l - p o i n t - t r i a n g u l a t i o n is s u p e r i o r to that of nadir-point-t1"iangulation. In this case the increase of a c c u r a c y is v e r y little and negligible in c o m p a r i s o n with other sources of errors; even w h e n m G = o, the dffferenee between both errors is v e r y s m a l h 1~a~f= ! 3 b cc and n-~l(; = + 4 t cc In this case n a d i r - p o i n M r i a n g u l a t i o n has to be p r e f e r r e d its correction being much t~impler than that of f o c a l - p o i n t - t r i a n g u l a t i o n . G e n e r a l l y speaking, the merits of corrected f o c a l - p o i n t - t r i a n g u l a t i o n are evident only in suda cases where a great difference between H, and H~ exists, a n d the v a l u e of m ~, is r a t h e r large, or in other words (consult n o n m g r a m II) mhen the a m o u n t of tilt is large and
only a p p r o x i m a t e l y knomn. Example: c a m e r a : f = 10 era; size of p h o t o g r a p h : 18 ;K 18 cm; m e a n tilt: i = 6~; m e a n error of tilt: m.,, = ± 50~: flying altitude: h o = 4000 m, mL, = --+ 50 m; m e a n relief: h i ~ 120 m, rnG = +--5 m: Thus from the n o m o g r a m s II and I I I : H / =' 4,6, /-/,~ = 6,6, K = 11, L = 0,35 h-ore w h i & , a c c o r d i n g to (35 and (36):
mdqvf = ~ 2c,4 and m3~,~ = 2~ 3 c.3, I n this case the increase of a c c u r a c y by m e a s u r i n g in the focal point instead of in the p l u m b point is i m p o r t a n t . T r i a n g u l a t i o n w i t h o u t a p p l y i n g corrections w o u l d give, a c c o r d i n g to the nomogvam [: zq~.f = _+ 9c,6, d-~,~= + t0c,0.
§ lo:..Applping corrections. It has been b r i e f l y m e n t i o n e d in § 8 that it is necessary to take into accotmt not oMy the accuracy, b u t also the speed, simplicity, and e c o n o m y of the methods. This specially concerns those cases, w h e r e the a c c u r a c y - d a t a lead np to corrected foeal-point-triangulath,~.. F o r c o m p u t i n g this correction it is necessary, indeed, to i n d i c a t e a great n , m b e r of
56
elements ~ tilt, bearings and lengths of the radial lines, differences of relief on the ground, flying height - - and moreover the formula is rather complicated. The much simpler correction of plumb-point-triangulation needs only the indication of tilt and the hearing of the radial lines. In both cases the usefulness of the method will depend to a high degree on the possibility of furnishing the necessary data and of constructing appliances (e. g. tables or nomograms) for simplifying the computation of corrections. The few foregoing examples show clearly that under certain circumstances an important increase in accuracy can be obtained, so that it is worth while going further into these questions. By means of the nomograms the accuracy of the various methods to be expected can easily be estimated. In principle this accnray could be considered in each photograph and the most favourable method of triangulation be chosen. However, in consideration of the continuity of work it is to be preferred that a n u m b e r of consecutive photographs are treated in the same manner. In general this can be done on account of the circumstances (tilt, reliability of horizon-photographs, topography) being indeed constant to some degree. It is possible and recommendable, however, to examine those paris o f a r u n separately which show evident differences with respect to the amount and the accuracy of tilt or to the topography and e v e n t u a l l y to deal with them in different ways. Some
denotMions,
a z = (1-- q) rt q)2 r 2 t 2 r t - - 2 a sa 2-az3
a 2 = - - ~ + (1 - - q ) (1 + t) - - (1 - -
as =--~(1--3q+2q3) c~4 = ~ a l 3 + 2 a l a 2
a s = qrt
a6 =½q(] +t) a 7 = ½q(l+t)+(1--q)qr2t 2 a~, = ~ q a r t - - ( 1 @2qr3t a' -
-
a 9 =½qrt--2(1--q)(l+t)qrl+(l--q)Zqrat
3
hi t
--
-
-
h o - - h,. f ai
q = ½(focal point) o r 1 (nadir point) Tl
=
cTtI +
aS
T2 = a s T s = r q a / + a~ + a 7 T 4 = r q a l 2 -F a 3 + 1,5 ~/4 +//~' T s = -- rqap
+ rqa 2 --
1,5 a 4 + a 9
T 6 = + az T 7 = rqaz + ~j
T~ = - - r q a z 2 + 3a 3 + 1,5 a 4 1,5 a 4
T 9 = -- rqaz2 + rqa 2-
Literature.
Bildpolygonierung hei gleiehmafliger Nadirdistanz und Gclgndeneigung. Festsehrift Ed. Dole2al, 1932. 2. B u c h h o l t z : Etude sur la polygonation a~rienne. 3. K i n t : Anwendung der Radialtriangulation in Niederl~ndisch-Indien. Bildmessung und Luftbildwesen, 1935. 4. K o p p m a i r : Nadirtriangulicrung. Allgemeine Vermessungs-Nadlrichten, 1929. 1.
Buchholtz:
57 5. Rehu: Fehleruntersudmngen zur Nadirpuakttriangulation. Bildmessung und Luftbildwesen, 1929. 6. S d w r m e r h o r n : Puntsbepaling door middel van fotograrmnetrie. Fotogramlnetrie, /938. 7. Schmeizer: Untersu&ung und praktische Durehftihrung einer Radialtriangulation in
Htigelland. Dissertation, Stuttgart. Het rekenen met gewiehtsgetallen. kunde, 1934.
S. Tier~stra:
Tijdschr. v. Kadaster en Lalldmeet-
Genauigkeilsunlersuthungen der Radiallriangulalion Ton t?. ~oelofs Mitteilung des Geod~itise~en /nstituts in Delft. j (Zusammenfassun g.) Die Entwicklung der Aerophotogrammetrie h~ngt in starkem Mal~e mit der Frage zusammen, inwieweit es gelingt, brauchbare l~ethoden zu finden oder ~chon vorhandene Methoden zu verbessern, um aus d e n Laftbil~taufnahmen jene fiir die Lokalisierung' und Kartierung notwendige Anzahl yon Paltpunkte~ zu gewinnen. Es erscheint deshalb angezeigt, die t/adiOltriangulation als die Methode, die in ver~ sehiedenen Variatioffen wohl am meisten a ngewendet wird, einer eingehenden fehlertheoretischen Untersuehung zu unterwerfen, i Es mtige vorausgestellt sein, dal~ das beJ~andelte Problem in seiner vorliegenden Fassung yon Prof ir. W. Sehermerhorn aufgestellt wui'de, die Ansarbeitung desselben jedoeh dem Verfasser tibertragen wurde. Wie bekannt, wird bet der Radialtriangulation eine Reihe yon Rauten oder Dreieeken aneinandergereiht, wobei in den Bildebenen ider einzelnen Aufnahmen eine Anzahl yon Winkeln oder tlichtungen gemessen werden, i In der Bildebene kiSnnen zwei Punkte angegeben werden, die als tladialpunkte besonders geeignet stud, n~mlieh der Fokalpunkt und der Nadirpunkt. Die im Fokalpunkt gemessenen Ridltungen entspreehen direkt der~ im iibereinstimmenden Gel/~ndepunkt gemessenen horizontalen Richtungen, vorausgesetzt, da~ das Gel/inde eben ist. Die im Nadirpunkt gemessenen Riehtungen weiehen dagegen yon den horizontalen Riehtungen um Betr/ige ab, die jedoeh unabh~ingig sind yon Gel/indehbhennnterschieden. Man kann daher erwarten, dal3 bet bewegtem Gelgnde die im Nadirpunkt erhaltenen Messungen relativ genauer sind. In naehfolgender Abhandlung ~erden Mittel nnd W e g e angegeben, die erlauben, eine richtige und begriindete Wahl zwlschen~' Fokalpunkt- nnd Nadirpunktmessung zu vollziehen. Es ist wetter mtiglieh, an den gemessenen Richtun~;en Korrektionen anzubringea, um sie mit den wahren Riehtungen in Uberemstnnmm~g zu bringen. Infolge der Ungenauigkeit, mit der die Plattenneigung bestimmt werden kann, sowie der damit verbundenen Ungenauigkeit in der iFestlegung der Fokal- bzw. Nadirpunkte ergibt sieh ohne weiteres, dal~ die bereehneten Korrektionsgr/313en aueh nieht fehlerlos seia k6nnen. In dem vorliegel~den Artikel werden die mittleren Fehler der verbesserten Ridltungen der Fokalpunkt- und der Nadirpunkttriangulation untersueht bzw. berechnet. Die Berechnungen werden dutch einige beigegebene Nomogramme wesentlieh vereinfa&t, so da/~ in allen in der Praxis x'orkommenden F/illen di~ giinstigste Arbeitsmethode gew/ihlt und die zu erwartende Genauigkeit festgestellt werde~ kann. Ausdrtieklidl set jedoeh bemerkt, dal~ es hi~r nt~r um eine V e r g 1 e i e h u n g der Nadirpunkt- bzw. Fokalpnnkttriangulation geht uJ}d somit die reinen Messnngsfehler, die ftir beide F/ille gleich sind, aul~er Betrachtung bletben. Der § 2 der Originalarbeit befal3t sidl mi~. der Bereehnung des Fehlers A~ in den gemessenen Itichtungen, wie er dm'eh die Neigling der Kammeraehse usw. verursa&t wird. Die Literatnr bietet zu dieser Aufgabe eine Anzahl yon gen~iherten Ltisungeu, in der vor.~
.
.
(
r,.
Geod~tische A n w e n d u n g der L u l t b i l d m e s s u n g A p p l i c a t i o n s gdoddsiques de la P h o t o g r a m m d t r i e adrienne Geodetival A p p l i c a t i o n s of the a i r p h o t o g r a m m e t r y
The Accuracy of lhe Badia| Line Melhod By R. Roelofs. Communication from the Geodetic Institute at Delft.
§ I. Introduction. ........ The development of aerial photogrammetry depends to a large extent on the question whether it will be possible to improve on the existing methods or to find new and better means for the photogrammetric determination of those points necessary for locating the~ photographs or for plotting maps from them. It is therefore desirable to make a detailed researd/ of' the accuracy of the radial triangulation; this method in several variations is the most applied way. it may be remarked here that Prof. W. 8chermerhorn has formulated the problem as i* is dealt with hereafter and that the author only had the pleasure of working it out by order and with m a n y suggestions of Prof. Sd~ermerhorn. It is well known, that in radial triangulation a d~ain of rt;ombs or triangles ~s built up by measuring a n u m b e r of angles or directions i~ the plane of each photograph. One strive'~ at such a result (hereafter called the "exact" result) which would also be obtained by measuring horizontal angles or directions b y ' m e a n s of a theodolite set up over the point on the ground, corresponding with the centre of the radial-lines in the photograph. The choice of the radial point depends o n varion s circumstances; if the ground is absolutely level and the oi~tical axis of the camera exactly vertical during the flight, then the resulting photograph is a reduced b u t conform representation of the area. Angles or directions measured in any point of the image plane will be the "exact" ones. I n case of photographs obtained with a tilted camera-axis, a set of exact angles or directions will be obtained by measuring in the focal point (better: one of the two focal points or isocentres), i.e. a point situated at a distance f tg ½ i from the p r i n c i p a l point in the direction of tilt (f = local length, i = tilt). The ground being not level there is no poittt in the photograph where the exact angles or directions can be measured. There is, however, a point where angles or directions can be measured, the amount of difference between these and the exact angles m' directions being independent on the relative aliiiudes of the point. This point is the plumb point (nadir point) situated at a distance f tg i from. the principal point in the direction of tilt. It is to be expected that for nearly flat ground the measuring in the focal point has to be preferred to measuring in the p l u m b point and that, the differences in relief of the gronnd becoming larger, the results, obtai~ed in the nadir pot:at will relatively become more favourable. Thus the question is which point, focal point or nadir point, is to be preferred by a given topography and which is the a eenracy of the measured angles or directions. In ihe foregoing we have assumed: 1. that the measured tilt of the camera is errorless and that it shouhl accordingly be possible to determine the focal point or nadir point without error; 2. that no attempt i s made to deduce the horizontal directions from the measored directions by applying corrections.
42 The first-mentioned supposition ~ i l l never occur since tile tilt of the camera is an obseroed element and therefore not errorless. With regard to the second assumption the following remarks can be made. According to the above-mentioned property of plumb point measurements it is possible to deduce the horizontal directions from the measured directions by applying a correction, which is independent of the topography and therefore easily to calculate. In connection with the inaccuracy of the locating of the n a d i r point, resulting from the inaccuracy of the observed tilt, the calculated correction and therefore the corrected direction will be inexact or, in other words, will possess a mean error which shall be dependent not only on the mean error of tilt but also on the differences in altitude of the points on the ground. This raises the question with whi& method the best results can be obtained under certain circumstances: measuring in the focal point without applying any correction or measuring in the nadir point and applying corrections for tilt. Correcting focal point measurements is not so simple, since the correction depends among other things on the flying height and on the differences in height of the points on the ground. In principle it is immaterial how these heights are measured. The [lying height e. g. can be determined by means of an altimetre. Barometric methods are also suitable for measuring the differences in elevation of the points on the ground. The mean error of the corrected direction measured in the focal point depends not only on the mean error of f i l l but also on ~he mean errors of the flying altitude and of the altitude of the points on the groun d. We shall, therefore, try to give an answer to the question whether corrected focal-point-measurement or corrected n a d i r - p o i n t - m e a s u r e m e n t will give better results in certain circumstances. Summarizing the problem is as follows: Which is, in the radial line method, the mean error of a direction of wMch the focal point or the nadir point is the origin and which of the two points should be preferred in the following cases: t. corrections are not applied to the results of measurement; 2. corrections are applied to the nadir-point-measurements; the measurements remain uncorrected;
focal-point-
3. both measurements are corrected. On the basis of the answers to these questions a favourable choice between the focal point and the nadir point can be made. It will be possible to investigate the accuracy to be expected and to bring the accuracy of measuring angles or directions in accordance with the accuracy limited by the uncertainty of tilt, differences in relief and flying altitude. It will e.g. be possible to consider in whi& cases it is advisable to use the most accurate instrument for the purpose, the radial triangu!ator, and in which cases simpler instruments and methods, e. g. graphical methods, should be used. Finally our research will furnish information for an answer to the question which part of the whole process from flying to computation of rhombs or triangles needs improvement in the first place in order to increase the accuracy of radial triangulation. [t will be interesting for example to examine which is the influence of an accurate knowledge of the position of the camera-axis during the flight using one or two horizon cameras. It may be specially pointed out that this article only deals with the comparison of the focal-point- and the nadir-point-triangulation, so that the accuracy of the proper measurement of the angles or directions,, which is the same in both eases, will not be considered. "6 2. Deduction of a formula for the error Aq~ in a measured direction. The relationship will be studied between directions in two planes, i.e. the plane of the photograph and the horizontal plane. It is obvious to choose as a direction of reference
43 the direction of their line of intersection or a direction in eadt plane perpendicular to that line of intersection. For the following considerations we will apply in each tflane a system of coordinates with the axis of abscissae parallel to the above-mentioned line of intersection and the axis of ordinates p e r p e n d i c u l a r to it. The origin of the system of coordinates in the plane of the photograph is being placed in the point He' in which the directions are being measured, the origin of the coordinate 0
A .!..I'
L~
\"
Fig. 1
0
N"
He\ j~he
system in the horizontal plane is being placed in the point tfa °. This is the horizontal projection of the point on the ground Ha, whid~ corresponds to the point He' (fig. 1). Let us take in the photograph a certain point Pi', corresponding to point P~ on the ground, which has its horizontal p r o j e c t i o n in Pi °. The b e a r i n g from the origin He' to P i" is the angle Pi' between the line H e ' P { and the axis of ordinates y'. The corresponding horizontal b e a r i n g Ha ° - P ~ ° is the angle p~ b e t w e e n the line He ° - p ° and the axis of ordinates Y'. It is understood that the error Ap i in the measured bearing Pi' is the difference between Pi" and Pi: Ap = ~p~'-- Pi. ( t) In the photogrammetric literature (we refer to the list of literature at the end of this article) one finds various a p p r o x i m a t e solutions. According to an idea of Prof. Scherlnerhorn, here an absolntely exact solution will be given.. For a detailed dedtlction we refer to "Puntsbepaling door middel van fotogrammetrie" b y Prof. Schermerhorn [6]. In this article we will limit ourselves to a concise; but, we hope, sufficiently cleat' indication of the solution. With the principal point H' of the photograph and the corresponding point H on the ground as origins systems of coordinate axis x" y " z " and X" Y" Z" are applied parallel to the above-mentioned systems x ' y ' z' and X' Y" Z'. The origin /:Ie' is provisionally placed entirely a r b i t r a r i l y at a distance e' from the principal point H' in a direction q%'. The horizontal plane is placed at the same level
44 with point H. T h e f l y i n g a l t i t u d e O N o is ho, the a l t i t u d e s of H e a n d Pi a r e respectivJly h e a n d hi.
In view of the necessity to a p p l y in some eases A(p (with opposite sign) as a correction to the m e a s u r e d bearings, it is desirable to express A~o as a f u n c t i o n of elements which can be m e a s u r e d in the photo-plane. We will therefore t r y to express in (1) qoi' and ~vi in eoordinates with respect to the x ' y ' z ' - s y s t e m . The expression for
~oi' is very simple:
x/
tg %! - y;'
(2)
x i' a n d Yi' being the coordinates of the point Pi'. Before a f o r m u l a for qoi can be e o m p u t e d we m u s t o b t a i n the coordinates of f i e ° a n d p,o in the x ' y" z' - System. F o r that p u r p o s e the f o l l o w i n g o p e r a t i o n s h a v e to b e p e r f o r m e d . a) F r o m the eoordinates of He' a n d Pi' in the x ' y ' z ' - s y s t e m the coordinates in the x" y" z'-system can be o b t a i n e d b y means of a p a r a l l e l shifting of the axis from He" to H'. Together with the coordinates of I1' a n d O this results in: ~.'~
y"
z"
H' 0
0 0 0 0 0 f H e" e' sin q)e' e ' c o s q)e' . O P/ x / + e ' sin ~%' y / + e ' cos q%' 0 b) These coordinates will be t r a n s f o r m e d to coordinates in the X" l z" Z " - system. The coordinates of the origin of this system are: x Hpp = Y x =t l ° a n d z ~tp= f - - h o sect a n d the angles between the coordinate-axis of both systems:
X"
X"
y~'
Z"
0
+~.~
-~½~
Y" +}~ +i ~z+i Z" +1 2jr+ i i e) By m e a n s of the o b t a i n e d results the equations of the lines O He' a n d O P i ' in the X" Y" Z"-system can be composed. By a s s u m i n g in these e q u a t i o n s that Z " = he resp. Z" = hi, we o b t a i n the c o o r d i n a t e s of He a n d Pi. F r o m them the coordinates of He ° and p o will be o b t a i n e d b y t a k i n g Z"]j e = o a n d Z" G. = O. d) After that a b a c k w a r d following coordinates :
transformation
to the x " y ' z ' - s y s t e m
furnishes
the
XHe"o = C z ( h o - - he) e" sin ~ve' Y ~ 2 = C z (h° - - he) (e' cos ~oe' eos i -- f sin i) cos i + h o sin i Z
"o G
~
C , (h o
he) (e' cos 9)e' cos i - - f sin i) sin i - - h o cos i + f
(3)
x ~ o = C 2 ( h o - - hi) (x'~. + e' sin ~%') yfi;q = C 2 ( h o - h i ) { ( y / zfi;, = C 2 (h o -
+ e' c o s ~ e ' ) e o s i - - f
sini}eosi4-h
osini
h i) { (y/-t-e' cos ~Oe') COS i -- f sin i I sin i - - h o cos i 4. f
in which: C 1 = (e' cos q ) / s i n i + f cos 0 - 1 Cz = (C) -'l + y / s i n 0 - 1 e) A f t e r these p r e p a r a t o r y c o m p u t a t i o n s the e q u a t i o n of the line o b t a i n e d b y intersecting the p l a n e O - - H e ° - - P i ° (not indicated in e) After these p r e p a r a t o r y c o m p u t a t i o n s the e q u a t i o n of the line o b t a i n e d b y intersecting the p l a n e O - - H s o - - P i o (not indicated in horizontal plane.
Pi ° can fig. 1) with He o - P i o can fig. l) with
He ° -
be the be the
45 The equation of the first p l a n e is in a d e t e r m i n a n t x" .q" z" 0
0
zi~tr ~ tt
1
f
l
= o
~ f!
xp 7
. P;
or developed into an expression with the co-factors P, Q, R, a n d S of the elements of the first row: Px" + Qy" + R z " + S = o (4) The equation of the second p l a n e is: y " - - z" c o t g i + (h o sec i - - f) co|g i = o (5) The l i n e of i n t e r s e c t i o n He 0 - Pi ° is given .by the e q u a t i o n s (4) and (5). F r o m these equations the following expression is o b t a i n e d for the angle ~i--.~t.~ between this line of intersection and the S - a x i s or X"-axis: cotg @i - - ~ z) = - - tg g,i =
Qeosi+ p Rsini
(6)
From (2) a n d (6) can be deduced: --
Px~! + (Q cos i -t- R sin i) Yi' tg A~v = py,: __ (Q cos i + R sin i) .r i'
S u b s t i t u t i n g the value of co-factors P, Q, a n d R whilst a p p l y i n g (3). we a reduction of the d e t c r m i n a n t s w h i d l m a y be left to the reader: fcosi+e'cos~'sini I o y~ sin i x~.'y/(1 - - cos i) o x i" (f sin i - - e' cos q~' cos i) + Yi' e' sin 9~' ~tgA~= [ o fcosi+e'cos~¢'sini I Yi' sin i i - - ( x i ' 2 + y i ' 2 cos i) Yi (f sin i - - e" cos ~ ' cos i) - - x~ e" sin ~ '
i
obtain, after h¢--ho [ h i - h~ ho-- h i h ~ - - h o [ (7) h i - - h~ ho-- hi
i I
In the derioation of [his f o r m u l a no a p p r o x i m a t i o n is applied. The absolute height or the horizontal p l a n e being i m m a t e r i a l we will suppose that h e := o. Henceforth the accents of the various elements in the f o r m u l a (7) will be left out for easiness' sake, since there is no risk of mistal~e. In the a b o v e - m e n t i o n e d treatise of Prof. Schermerhorn [6] the form a u d the value of Aqo in some p a r t i c u l a r cases have been considered. Thus it can easily be seen that Aqo = o when the g r o u n d is level (hi = o) a n d m e a s u r i n g being performed in the focal point (e = f t g ½ i, We = o). F u r t h e r A~ is i n d e p e n d e n t of differences in level (h i:~o) m e a s u r i n g b e i n g p e r f o r m e d in the p l u m b point (e = f t g i, ~oe = o). For the rest we m a y refer tO the a b o v e - m e n t i o n e d publication. Since the exact formula (7) is too complicated to be h a n d l e d in the following considerations, we will r e p l a c e it b y an a p p r o x i m a t e f o r m u l a , s u p p o s i n g i to be relafivily small (e. g. max. 6% Moreover we will consider m e a s u r e m e n t s in the focal point or in the p l u m b point only, so that We = o a n d e = ftgqi in which q = ~ corresponds with the focal point t r i a n g u l a t i o n a n d q = 1 with the n a d i r point triangulation. A systematic a p p r o x i m a t i o n is o b t a i n e d b y e x p a n d i n g in the. f o r m u l a (7) all goniometric functions of i in ascending' powers of i and, after that, developing the d e t e r m i n a n t s as far as terms of the t h i r d o r d e r (i3)". The a d v a n t a g e of this m e t h o d over other less systematic methods lies in the fact that all a p p r o x i m a t i o n s are of the same order a n d that it is p o s s i b l e to check the neglected part.
46 Using the w e l l - k n o w n f o r m u l a in c o m m o n f o r m : sine= e---~s3+ . . . . . coss= 1 ~-~s2+ .... tgs= e +~s3+ .... we obtain, a f t e r h a v i n g s u b s t i t u t e d in (7) the r e c t a n g u l a r c o o r d i n a t e s x i a n d Yi b y p o l a r coordinates a i and ~ and after dividing numerator and denominator by ai2f (ho--hi):
in w h i d l k , ~ = f,,
(hl
h oTh,.
igd~=
k~i + kxi2 + kai3 + 1 + k 4 i + k 5 i2 +
.... ....
' a i ' %.
We ean w r i t e f o r this: tgAq~ = ( k l i + k 2 i 2 + k a i 3 + - . . . . )(t--k4i--kgi2 .... ) ~ kii+ksi2+kTi3+ or d(p = b g t g (k~ i + k'6i2 + k7i3 + . . . . ) In c o n n e c t i o n w i t h the w e l l - k n o w n f o r m u l a :
. . .
bgtgs= s--½s3+ .... we o b t a i n : z/q) = t c e i + l c 6 i a + k a i 3 + .... C o m p u t i n g the coefficients k, w e f i n a l l y o b t a i n the f o l l o w i n g e x p r e s s i o n : -
-
d ~o = i a Z sin qv,.+ i 2 a z sin qJi cos %. + i3 (a s 4- a 4 sin2 %.) sin %- + . . . .
(8)
We m a y r e f e r h e r e to t],e r e v i e w at the e n d of this treatise, giving the m e a n i n g of tile coefficients a. T h e focal p o i n t b e i n g the r a d i a l point, (q = ½), w e h a v e : ae =-~ r t a 2 = ½ t - - ¼ r2t2 a3= ~'~rt--'rt 2 + ~ r a t 3 a 4 = ½ rta - - ~= r3t3 f
h,. and r h o- - h i ' ai Since ead~ of these coefficients c o n t a i n s the f a c t o r t, it is e v i d e n t t h a t Aqo = o w h e n the g r o u n d is flat. in w h i d l t -
F o r the p l u m b p o i n t (q = 1) the c o e f f i c i e n t s are: a l ~
a 3 ~
~14 ~
o,
--
c~ 2 - -
1
o"
None of t h e m c o n t a i n s the factor t. The e r r o r is t h e r e f o r e i n d e p e n d e n t of the differences of relief. § 3" C o m p a r i s o n o f triangulation.
uncorrected
focal-point-triangulation
mith
uncorrected
W e will use t h e s q u a r e e r r o r A~o~ as a m e a s u r e for the a c c u r a c y . ~oi g e n e r a l l y can a s s u m e e v e r y v a l u e b e t w e e n o a n d 2~.
plumb-point-
In the f o r m u l a (8)
I n o r d e r to find a m e a s u r e of a c c u r a c y for a c e r t a i n area, wc Will t a k e the m e a n o f all values of t h e s q u a r e e r r o r dq~2, q~i b e i n g limited b y o a n d 2 ~ : 2:*
= 2-~fAq~ 2 . dqJ o
Squaring and integration produce: d~o2 = ~ i 2ae2-k ½ i 4(a 2 A - S a l a 3 + 6 a l a 4) F o r p l u m b - p o i n t - t r i a n g u l a t i o n this b e e o m e s : d%, 2 = a~- i4 o r dq% = -~ 0.18 i2 F o r f o c a l - p o i n t - t r i a n g u l a t i o n w e obtain, w h e n n e g l e c t i n g t h e s e c o n d t e r m : A~oZ2 = -s ~ iarat2
or
A~/=
+O.aarit
(9) (10) (t~)
47 Concerning the value of t ~
(t /~t o -/,,. V'.i n -hi)
this fornnila, the following r e m a r k s can be
made. Supposing h i to be relatively small, we can expand t in a series as follows: t2
(h~)2+ 2 (~)3
(hi/4
3+ \ho) +
.
.
.
.
F o r a fairly large area the sum of the positive values h i will almost be equal to the sum of the negative values. Thus in the formula for a m e a n value of the error the terms with odd indices must be left out. Neglecting terms with fourth and higher powers of i, we obtain : hi t=--
ho
The computation of A%, and Aq)f is facilitated by the nomogram I (fig. 2). h '°key" ("Sdfltissel") at the head of the nonmgram shows how it has to be used.
&"~.O.t8 i ~ und 1o0
2
4
6,~ ¢
~-~=o.35ri hl .~
¢
~
q"~
~
o
5"3
7
~0.20
3
5.1o
Fig 2
~ ~
~
~
~' "~ho,~..... "
"
~R ~
For nadir-point-triangulation o n l y the left part of the graph is used. At the poii/£ of the curve, corresponding with the given value of i, the value of ~ can be read. For focal-point-triangulation the entire nomogram is used with the exception or' the curve; for the rest we m a y refer to the key. If one finds that, when using the nomogram in a certain case, ~ is greater than ~ or in other words if the fixed point is situated beloro the curve in the area marked @ , then ur, eorrected focal-point-triangulation gives obviously better results than uncorrected plumb-point-triangulation. If the fixed point is situated in the area marked @ , then the plumb point has to be preferred. From this it can be concluded that the nomogram can be used in two ways: l. for computing the mean errors Aq~--"~and A%, or 2. only for deciding in w-hid~ point, f o c a l p o i n t or nadir poi~t£, measuring should be performed.
48
An e x a m p l e m a y explain this: If the flying" he_~ght hi = 1000m, the mean of the differences of relief h~ = 15 m, tile mean tilt i = 3g, a:tid the focal distance f = 21 cm, we obtain: A%~ = 2c.5 and Aq~/-= 4c.7 W h e n c o m p u t i n g by means of tile n o m o g r a m ~ we get a p o i n t in the area ( ~ . F r o m this it follows in a c c o r d a n c e w i t h the o b t a i n e d ~-alues of ~ and ~ that nadir-pointt r i a n g u l a t i o n is to be preferredl It can easily be seen, w h e n using' the n o m o g r a m , that in the a b o v e - m e n t i o n e d circumstances f o c a l - p o i n t - t r i a n g u l a t i o n gives better results only then w h e n the m e a n difference of relief does not e x c e e d 8 m. § -k
A c c u r a c y of tilt measuring by means of horizon cameras.
As long as it is only a question of accidental errors in tilt measuring, it is for the t h e o r y of errors, which will be developed in the n e x t p a r a g r a p h s , i m m a t e r i a l which m e t h o d of d e t e r m i n i n g tilt is applied. It is sufficient to k n o w file a c c u r a c y of tip and tilt a n d e v e n t u a l l y the a m o u n t of correlation b e t w e e n them. In order to give an e x a m p l e of developing the t h e o r y of errors, we will study the accuracy" of observing tilt b y horizon cameras. It is so much more interesting because the provisional results obtained w i t h horizon cameras are favourable. The horizon cameras will, no doubt, be v e r y useful for the r a d i a l - t r i a n g u l a t i o n , especially the new Zeiss camera HS 8, which shows the horizon in two p e r p e n d i c u l a r directions. We will distinguish between two cases: horizon c a m e r a and b y troo cameras.
the d e d u c t i o n
of the tilt and tip
by one
a) One horizon camera (parallel to the flying direction). The direction of the c a m e r a axis is d e t e r m i n e d b y the position of the p h o t o g r a p h e d horizon w i t h respect to the s i m i l a r l y p h o t o g r a p t l e d indices of the horizon camera. O n tlw left and on the r i g h t side of the horizon p h o t o g r a p h the distances b e t w e e n horizon and indices are m e a s u r e d b y means of a scale, which indicates directly the tilt c o r r e s p o n d i n g with the reading. The readings being 1 and r, the tilt ~o a n d the tip a are: o~=l--r = (l + ,-)c 02) c is a constant, d e p e n d i n g a m o n g other things on the focal length of the horizon c a m e r a (e. g. c = 0,382). Supposing t h a i the readings are of the same a c c u r a c y and not correlated or in other words that Ql~ = Q~.r aud Qrt = Qlr = o then by a p p l y i n g the l a w of p r o p a g a t i o n of e r r o r s to the f o r m u l a s (12) it follows: Q
Q , , = c 2 Q ...... a n d Q,o~=O
(13)
In this f o r m u l a s Q,,~ is the w e l l - k n o w n denotation for the w e i g h t n u m b e r of a. The connection b e t w e e n the weight n u m b e r and the m e a n error m , of a is: m a2 = t ~2 Qaa tt representing the m e a n error in an observation of unit weight. Q,a is the usual denotation for the a m o u n t of correlation b e t w e e n ~o and a. We also use " s y m b o l s " Qx, the m e a n i n g of which, appears f r o m : Q~ Q.r = Qx 2 ~ Q .... and Qx Qj, -= Q,:~, A .s!lrnbol has no v a l u e as a number, a product of two symbols on the other h a n d has. An a m p l e e x p l a n a t i o n of the calculating w i t h these w e i g h t m~mbers and symbols is given by Prof. T i c n s t r a [8].
49 b) T m o h o r i z o n c a m e r a s (patallel a n d p e r p e n d i c u l a r to the ilying direction). F r o m c a & of the two photographs the tilt as well as the tip can be computed: first photograph iilt:
~vj = l ~ -
second photograph
rz
~o2 = (12 + r2) c
tip: % = (l 1 + rz) c a2 = 1 2 - r 2 Supposing the m e a s u r e m e n t s in both photographs to be of the same accuracy, we read from (13) that the weights of these quantities are: g,'z : g~2 = g.'2 : g~s = 1 : e 2
Taking" these weights into account, we o b t a i n the following m e a n tilt a n d tip: -e2c°l + 0~2 a n d . ~ _ a:t -I- C2a2 ~o~ 1+c2 1+c2
(14)
I n order to c o m p u t e the weight n u m b e r of ~,, a n d a, we a p p l y the law of p r o p a g a t i o n of errors. The result is after some reduction: C2
Q~5"~ = Q ~ = i + c 2 " Q~1%"
and
QE-~= o
(15)
These formldas give insight into the p r o p o r t i o n of the a c c u r a c y of observing tilt b y means of one or t m o horizon cameras. The c o n s t a n t c being 0,382, the f o r m u l a 05) is: Qo~o, = Q ~ = 0,1273 Q~oz%" or
m E- = m~- = ~__0,36 m~oz
from (13) :
maz --= + 0,38 m~oz
Hence it :follows mE- = + 0,36 mo~ a n d m~- = ~ 0,95 m,~ f
J
Using the second horizon camera, the increase in a c c u r a c y is greater in tilt t h a n ill tip. The r e s u l t is, that lhe a c c u r a c y of observing the position of the camera-axis is the same in all directions . § 5. T h e a c c u r a c y o f d e t e r m i n i n g t h e x ' - a n d y ' - a x i s in t h e p l a n e o f the p h o t o g r a p h a n d its i n f l u e n c e on t h e a c c u r a c y o f c o o r d i n a t e s o f a p o i n t Pi.
F o r the following considerations we i n t r o d u c e an a d d i t i o n a l system of coordinates a n d ~7, the origin of which coincides with the p r i n c i p a l point a n d of which the axis passes through the collimating marks. Let us suppose that it is possible to determine this system of coordinates w i t h o u t error. The position of the origin of the x ' y ' - s y s t e m (focal point or p l u m b point) a n d the, direction of its axis are o b t a i n e d b y means of the o b s e r o e d tilt a n d tip a n d are therefore not errorless. The tilt a n d tip being ~o a n d a, the tilt i of the camera proceeds from i 2 ~ o)2 + a 2 whilst the direction of tilt follows from O=bgtg~ Therefore b y a p p l y i n g the law of p r o p a g a t i o n of errors:
1
Q oo = - ~ (a2 Q~o) + co2 Q~, - - 2 coa Q,o~ )
Qo, = ~ { c o a ( Q ~ - Qo,~o)+ (~2-a2)Q~,~,}
(16)
5O For one horizon camera we have, according to (13):
07) ,e(.. = -
p
-~
Q,oo,
in which p 2 = 1 - - c 2 For two horizon cameras: Qii = Qgg,~, 1
Os)
Qoi = o
The formulas (18) obviously proceed frmn (17) simply by putting p = o and replacing Q~o~o b y Q,~,~,.
The x' y ' - system being placed in the direction of tilt and its origin being at a distance e = f tg q i from the principal point, it is evident that the x ' y ' - s y s t e m can be obtained by turning the @-system through the angle ~9 and by removing it in the y'-direction uver a distance f tg q i (fig. 3).
g
I
.I'"
..... ~ ......
.i
S #~
jl/
?~e
~t
,/"
/'
/ /
4"-
/ / / / Fig. 3
The relation between the coordinates ~ ~i in the $~-system und xi 9i i n t h e x" y ' - system is therefore given b y the usual transformation formulas: xi = ~i c o s 69 - - *]i s i n 0 .t/; = ~i s i n O + ~ i c o s
O -- e
from whi& follows: •Qx i = ( - - y ; -~ e) q o Q.~;; = x¢ Q o - - f q sec~ q i. Qi
(1.9)
The relations between the rectangular coordinates x i and yi a n d the polar coordinates a~ and ~i are: x'; a,.~ = xi 2 + .G.2; %- = bg tg - Y;
By applying the law of propagation of errors~ we obtain after having substituted (19) and after some reduction, in symbolic form:
Q"i = - - e sin %.. Q o - - fq sec2 q i cos 9~i. Qi
Q~,.= ( - t - - ~ c oes m , .
) Qo+
[~--7 qsec2qisin%.Q i
(20)
For the sake of brevity we write:
Q~. = A Q o + BQ,. Q
Q~e
=
-- (20
(20 (22)
§ 6. Accuracy of Aq; inasmuch as it is dependent on the accuracy of tilt i. ......... Since we may suppose the measurements of tilt / , h e i g h t of the ground and the flying altitude to be uneorrelated, the influences of the mean errors of these quantities on the accuracy of Aq~ can be treated separately. According to (8) is:
~t~-= f (a~., 0%, i, ~e) Denoting the partial derivatives of this function with respect to a~, q)i, i and q~e, respectively fl, 7, 6, and e, we obtain b y applyillg the law of propagation of errors in symbolic form:
Qaqo = flQ"i + 7Q~i + 6Qi + eQ~e In connection with the formulas (21), (22), and (2:1) we can write:
Q&o= (~A + yC--e) Qo + (fib + yD + d) Qi
(23)
(24)
or introducing other denotations: •
O~
= F Oo
+ G O,.
(25)
:Ill ram-symbolic form: QzqJztm = F2Qoo + G2Qil + 2 F G Q o , Substitu.ting the values of the weight numbers according to (17), we obtain:
=
P ~-/r + i J/Q
(26)
(27)
In case of two horizon cameras p = O whilst Q~o
(28)
Sinee p does not oeeur in this formula it is applicable (save eventual substitution of Q~oo~by Q ~ ) not only when using one hut also when using tloo horizon cameras. However, in the first ease it represents a m a x i m u m value. Heneeforth we shall use for the sake of simplicity the formula (28) only, without risk of overestimating the accuracy. The eomputation of the partial derivatives fl, 7, and 6, proeeeding from the formula (8), is merely a matter of algebraical teelmics; it is of speeial importance to continue computation until just sufficiently high powers of i are obtained. We will not fill too much space by giving the development of the formulas to the full extent, b u t only mention the final result. The computation of the derivative e with respeet to qJe cannot be commenced from the formula (8) sinee in this formula +Pe = o. It is, therefore, neeessary to start from the general formula (7). Denoting in (7) numerator and denominator D~ and D.2, we obtain: tg
dqo = -~-
~2 f r o m w h i & it follows t h a t
( Odg~ ~
- = --
f
t
(D ODz
&=o
D OD2 ~ I
=-;C-
Jr%_o
I t is, t h e r e f o r e , s u f f i c i e n t to c o m p u t e t h e v a l u e s of Ds, D2 a n d t h e i r d e r i v a t i v e s w i t h r e s p e c t to rpe a n d to s u b s t i t u t e t h e results, a f t e r h a v i n g p u t r; c = o in t h e a b o v e - m e n t i o n e d formula. T h e c o m p u t a t i o n of fl, 7, 6, a n d e a n d t h e m u l t i p l i c a t i o n w i t h t h e k n o w n q u a n t i t i e s A, B, C, a n d D (20 a n d 21) give t h e f o l l o w i n g f i n a l r e s u l t : F i - Tz cos qvi + i (T 2 + 1"3 cos 2qJi) + i 2 (T 4 + T s cos 2%.) cos %. (29) - - G -- T 6 sin qv,. + i 7 7 sin 2%. q- i 2 (Ts + T~ cos 2mi) sin %. T h e d e n o t a t i o n s T h a v e b e e n e x p l a i n e d in t h e r e v i e w of d e n o t a t i o n s a t t h e e n d of this article. B y m e a n s of t h e f o r m u l a s (29) it is e a s y to c o m p u t e t h e v a h t e of t h e c o e t f i c i e n t of Q ..... i n t h e f o r m u l a (28). F2 i~- + G~ = f (%) T h e v a l u e of % is l i m i t e d b y O a n d 2 n. J u s t like in § 3 we will c o m p u t e h e r e a m e a n v a l u e : 2~
1
¢1 F2
+
G21d%.~. ,
(30)
......
By i n t e g r a t i o n w e o b t a i n : [t2 = ½ (T12 -4- T62 ) + ½ i2 (,2 T22 + T32 + T72 + 2 T s T4 + T s T s + 2 T 6 7'~ - - T 6 Tg) F o r t w o of t h e m o s t f r e q u e n t cases we c o m p u t e d t h e coefficients T, viz. for c a m e r a s w i t h focal l e n g t h s of I0 a n d 2t c e n t i m e t r e s , t h e size of t h e p h o t o g r a p h b e i n g 1S X 18 cm. B y this size t h e d i s t a n c e s ai are a l m o s t 6 ~l 10 era, t h u s r = off to 1,5 a n d 3. hi
H is o b t a i n e d a s a f u n c t i o n of i a n d t -- h e - - h i "
i averagely and ronnded a~.
S i n c e it h e r e a g a i n c o n c e r n s a mean
v a l u e f o r a f a i r l y l a r g e area, j u s t like in § 3, we c a n w r i t e a g a i n t = r e a s o n t e r m s c o n t a i n i n g o d d p o w e r s of t a r e oInitted.
hi
~oo' F o r a s i m i l a r
Finally we obtain the following expressions: H 2 = 1,406 t 2 -~ (0,094 q- 1,38 t2q- 1,25 ]~4) i 2 H 2 = 1,125 t 2 -4- (0,375 -t- 1,69 t 2) i 2 H 2 = 5,625 t 2 + (0,094 + 7,97 t 2 + 20,50 t 4) i 2 H 2 = 4,500 t 2 + (0,375 -4- 5,62 t 2) i 2 (a a n d b: f = l O c m ; c a n d d: f = 21 cm; a a n d c: f o c a l - p o i n t - t r i a n g u l a t i o n ; nadir-point-triangulation).
(31a) (31b) (31c) (3ld) b a n d d:
W h e n t a n d i a r e s m a l l t h e t e r m s of the coefficients of i c o n t a i n i n g t m a y b e o m i t t e d , so that the formulas have the simple form: hi 2 H2=c 1~ + c3 i 2 (32) F o r t h e c o m p u t a t i o n of H we d e s i g n e d t h e n o m o g r a m I I (fig. 4) w h i c h r e q u i r e s no f u r t h e r e x p l a n a t i o n besides t h e " k e y " w h i c h is d r a w n a t t h e h e a d of it. W e will o n l y p o i n t o u t t h a t u s i n g t h e f o r n m l a (30) i n t h e f o l l o w i n g f o r m :
m2A~ = H2m2o.
(33)
we d e r i v e f r o m t h e n o m o g r a m such v a l u e s of H as c o r r e s p o n d w i t h mare e x p r e s s e d i n c e n t e s i m a l seconds, rno~ b e i n g e x p r e s s e d i n c e n t e s i m a l minutes.
53 § 7. Accuracy of Aq; inasmuch as it depends on the accuracy of hi and ho. The formula (8) shows that when measuring in the plumb point, Ag is independent of
hi and h o. Therefore, the following considerations only concern focal-point-triangulatiom h~ and ho being uncorrelated, we obtain: Qa~,~= \Oh i! QGG + Oho l QGG By computing the derivatives in this formula and taking the mean of all values 9 over the field for which c& lies between O and 2 z, we obtain:
0.......= -~-i2r2+ ,,',~ i4(3 + 2r2-- '8r2t+ J2r4t2) "(t~o2 /,,,/,,. + ~ T.
Schliissel:
NOMOGRAMM
/Or
H~=
h'
_
o
/=Were ,
i
FOalPt.
~ /
~ -~
Y/
'1="
H
Nod Pt
\
\
---
H
H
"~
=4_=_2>
/
. /
~
/
~
i - - - -
_ _
1
"~
Fob. Pt
_.~
H2,
/~/~ 7
cm
[-2, Nod Pt.
\
"•'
QG/,. )
~
~
ho~......
~
6-0
___
2
4
2
6-0
2
4
6-0
\
2
Fig. 4
6
~P
If i and t are small, the term containing i4 may be neglected,
~,,~o,~ = ('- ir ~-~)2QG./,, + (½ ir ~f2)2Q/~oG or denoting the coefficients K and L: m2~q~= K2m2ai + L2m2ao
(34)
The nomogram ]II (fig. 5) gives K and L. In order to obtain m a t expressed in centesimal seconds, it is necessary that in (34) m G. und rn G are expressed in metres. Combining (33) and (34), we obtain tile following complete formulas: foe. point: m'-2a~= Hf 2 moo2 + K 2 m~ + L2 ink2 (35) had. point: m2A~o= H,? m J (.36) For clearness' sake we have added to the coefficients H in these formulas the indices f and n.
§ 8. Comparison of uncorrected focal-point-triaugulatiou mith corrected nadir-point'" triangulation. As a measure of accuracy for the uncorrected focal-point-triangulation we used in § 3: he
54 In the p r e c e d i n g p a r a g r a p h has been expressed b y :
the a c c u r a c y of the corrected n a d i r - p o i n t - t r i a n g u l a t i o n 7~t2~m = H~,, m2o~
(36)
In spite of the dissimilarity of these measures of a c c u r a c y - - the f o r m e r being the mean v a l u e of the systemalie error and the lafler being the m e a n accidental error - we can use them w i t h sufficient c e r t a i n t y as a base for our choice between the two methods of triangulation. W e will illustrate this with an e x a m p l e : c a m e r a f = lOcm, size of p h o t o g r a p h ~8)< I S c m ; m e a n tilt: i = 3"~;
~. IVOMOORAMM
K.o.,rt
"'::>'"" .-1°
,,,,
4k ~. 45 90
~0 80
\
":'\
.
70 70
g ~o L
30 60 25 50 I
,
20 4O
\
~5 30
¢o
20
5
~
~0 0
0
2
o
~
~
~
~
-
~
~ i
~
~
~00
~
0
6
Fig. 5
i,c,.,~o.,)
/IR
flying altitude: ho = 4000 m; m e a n h e i g h t of the relief: hi = 5 0 m ; one horizon c a m e r a : m . , = +- 12~; According' to n o m o g r a m I:
~ : ) = + 2c.o and a c c o r d i n g to n o m o g r a m II: H,, = 3,4 therefore:
mzlq.
= ~__3;4X 12 = "4-41 cc
E v i d e n t l y corrected p l u m b - p o i n t - t r i a n g u l a t i o n has to be p r e f e r r e d in this case. The f o c a l - p o i n t - t r i a n g u l a t i o n as c o m p a r e d w i t h p l u m b - p o i n t - t r i a n g u l a t i o n gains in a c c u r a c y a c c o r d i n g to the ground b e c o m i n g flatter. I n the a b o y e - m e n t i o n e d e x a m p l e the mean h e i g h t being m e r e l y 10 metres: d q J / = q 40 cc and m--,l~,= ± 36 cc In t h i s case we can save ourselves the trouble of correcting n a d i r - p o i n t - t r i a n g u l a t i o n a n d ratt~er p e r f o r m m e a s u r i n g in the focal point w i t h o u t a p p l y i n g corrections.
55 i t is understood, that one has to consider not only the a c c u r a c y but also the economy of the two methods. In order to a v o i d the complication of applying' corrections one shall have to tolerate as a rule some decrease in a c c u r a c y b y m e a s u r i n g in the focal point w i t h o u t a p p l y i n g corrections.
§ 9. Comparison of corlected [ocal-point-triangulaticn ...... triangulalion.
roith corrected nadir-point-
This subject requires only a little e x p l a n a t i o n ; the complete f o r m u l a s (35) and (30) h a v e to be used here. Using the data of the f i r s t e x a m p l e of the preceding pai'agraph, we obtain 1)7 means of the n o m o g r a m s II and i I I : H / : = 2.2, H,~ = 3,4, K = 6, L = 0,15 t h e r e f o r e , a c c o r d i n g to (35) and (36): t-72j = 4.84 m2o) + 36 m2G. + 0,0225 m2/q, Zf Itt2Aq ,z ~ l t , 5 6 m2(~
s u b s t i t u t i n g trG= _q2_12 c and m G = q~ 50 m, we o b t a i n :
nl~l,,f = 1259 + 36 m2 G. m2z~,~ = 1665 T h u s m239~/
[ m,%. I < 3,4 metre.
U n d e r this condition alone the a c c u r a c y
of f o c a l - p o i n t - t r i a n g u l a t i o n is s u p e r i o r to that of nadir-point-t1"iangulation. In this case the increase of a c c u r a c y is v e r y little and negligible in c o m p a r i s o n with other sources of errors; even w h e n m G = o, the dffferenee between both errors is v e r y s m a l h 1~a~f= ! 3 b cc and n-~l(; = + 4 t cc In this case n a d i r - p o i n M r i a n g u l a t i o n has to be p r e f e r r e d its correction being much t~impler than that of f o c a l - p o i n t - t r i a n g u l a t i o n . G e n e r a l l y speaking, the merits of corrected f o c a l - p o i n t - t r i a n g u l a t i o n are evident only in suda cases where a great difference between H, and H~ exists, a n d the v a l u e of m ~, is r a t h e r large, or in other words (consult n o n m g r a m II) mhen the a m o u n t of tilt is large and
only a p p r o x i m a t e l y knomn. Example: c a m e r a : f = 10 era; size of p h o t o g r a p h : 18 ;K 18 cm; m e a n tilt: i = 6~; m e a n error of tilt: m.,, = ± 50~: flying altitude: h o = 4000 m, mL, = --+ 50 m; m e a n relief: h i ~ 120 m, rnG = +--5 m: Thus from the n o m o g r a m s II and I I I : H / =' 4,6, /-/,~ = 6,6, K = 11, L = 0,35 h-ore w h i & , a c c o r d i n g to (35 and (36):
mdqvf = ~ 2c,4 and m3~,~ = 2~ 3 c.3, I n this case the increase of a c c u r a c y by m e a s u r i n g in the focal point instead of in the p l u m b point is i m p o r t a n t . T r i a n g u l a t i o n w i t h o u t a p p l y i n g corrections w o u l d give, a c c o r d i n g to the nomogvam [: zq~.f = _+ 9c,6, d-~,~= + t0c,0.
§ lo:..Applping corrections. It has been b r i e f l y m e n t i o n e d in § 8 that it is necessary to take into accotmt not oMy the accuracy, b u t also the speed, simplicity, and e c o n o m y of the methods. This specially concerns those cases, w h e r e the a c c u r a c y - d a t a lead np to corrected foeal-point-triangulath,~.. F o r c o m p u t i n g this correction it is necessary, indeed, to i n d i c a t e a great n , m b e r of
56
elements ~ tilt, bearings and lengths of the radial lines, differences of relief on the ground, flying height - - and moreover the formula is rather complicated. The much simpler correction of plumb-point-triangulation needs only the indication of tilt and the hearing of the radial lines. In both cases the usefulness of the method will depend to a high degree on the possibility of furnishing the necessary data and of constructing appliances (e. g. tables or nomograms) for simplifying the computation of corrections. The few foregoing examples show clearly that under certain circumstances an important increase in accuracy can be obtained, so that it is worth while going further into these questions. By means of the nomograms the accuracy of the various methods to be expected can easily be estimated. In principle this accnray could be considered in each photograph and the most favourable method of triangulation be chosen. However, in consideration of the continuity of work it is to be preferred that a n u m b e r of consecutive photographs are treated in the same manner. In general this can be done on account of the circumstances (tilt, reliability of horizon-photographs, topography) being indeed constant to some degree. It is possible and recommendable, however, to examine those paris o f a r u n separately which show evident differences with respect to the amount and the accuracy of tilt or to the topography and e v e n t u a l l y to deal with them in different ways. Some
denotMions,
a z = (1-- q) rt q)2 r 2 t 2 r t - - 2 a sa 2-az3
a 2 = - - ~ + (1 - - q ) (1 + t) - - (1 - -
as =--~(1--3q+2q3) c~4 = ~ a l 3 + 2 a l a 2
a s = qrt
a6 =½q(] +t) a 7 = ½q(l+t)+(1--q)qr2t 2 a~, = ~ q a r t - - ( 1 @2qr3t a' -
-
a 9 =½qrt--2(1--q)(l+t)qrl+(l--q)Zqrat
3
hi t
--
-
-
h o - - h,. f ai
q = ½(focal point) o r 1 (nadir point) Tl
=
cTtI +
aS
T2 = a s T s = r q a / + a~ + a 7 T 4 = r q a l 2 -F a 3 + 1,5 ~/4 +//~' T s = -- rqap
+ rqa 2 --
1,5 a 4 + a 9
T 6 = + az T 7 = rqaz + ~j
T~ = - - r q a z 2 + 3a 3 + 1,5 a 4 1,5 a 4
T 9 = -- rqaz2 + rqa 2-
Literature.
Bildpolygonierung hei gleiehmafliger Nadirdistanz und Gclgndeneigung. Festsehrift Ed. Dole2al, 1932. 2. B u c h h o l t z : Etude sur la polygonation a~rienne. 3. K i n t : Anwendung der Radialtriangulation in Niederl~ndisch-Indien. Bildmessung und Luftbildwesen, 1935. 4. K o p p m a i r : Nadirtriangulicrung. Allgemeine Vermessungs-Nadlrichten, 1929. 1.
Buchholtz:
57 5. Rehu: Fehleruntersudmngen zur Nadirpuakttriangulation. Bildmessung und Luftbildwesen, 1929. 6. S d w r m e r h o r n : Puntsbepaling door middel van fotograrmnetrie. Fotogramlnetrie, /938. 7. Schmeizer: Untersu&ung und praktische Durehftihrung einer Radialtriangulation in
Htigelland. Dissertation, Stuttgart. Het rekenen met gewiehtsgetallen. kunde, 1934.
S. Tier~stra:
Tijdschr. v. Kadaster en Lalldmeet-
Genauigkeilsunlersuthungen der Radiallriangulalion Ton t?. ~oelofs Mitteilung des Geod~itise~en /nstituts in Delft. j (Zusammenfassun g.) Die Entwicklung der Aerophotogrammetrie h~ngt in starkem Mal~e mit der Frage zusammen, inwieweit es gelingt, brauchbare l~ethoden zu finden oder ~chon vorhandene Methoden zu verbessern, um aus d e n Laftbil~taufnahmen jene fiir die Lokalisierung' und Kartierung notwendige Anzahl yon Paltpunkte~ zu gewinnen. Es erscheint deshalb angezeigt, die t/adiOltriangulation als die Methode, die in ver~ sehiedenen Variatioffen wohl am meisten a ngewendet wird, einer eingehenden fehlertheoretischen Untersuehung zu unterwerfen, i Es mtige vorausgestellt sein, dal~ das beJ~andelte Problem in seiner vorliegenden Fassung yon Prof ir. W. Sehermerhorn aufgestellt wui'de, die Ansarbeitung desselben jedoeh dem Verfasser tibertragen wurde. Wie bekannt, wird bet der Radialtriangulation eine Reihe yon Rauten oder Dreieeken aneinandergereiht, wobei in den Bildebenen ider einzelnen Aufnahmen eine Anzahl yon Winkeln oder tlichtungen gemessen werden, i In der Bildebene kiSnnen zwei Punkte angegeben werden, die als tladialpunkte besonders geeignet stud, n~mlieh der Fokalpunkt und der Nadirpunkt. Die im Fokalpunkt gemessenen Ridltungen entspreehen direkt der~ im iibereinstimmenden Gel/~ndepunkt gemessenen horizontalen Richtungen, vorausgesetzt, da~ das Gel/inde eben ist. Die im Nadirpunkt gemessenen Riehtungen weiehen dagegen yon den horizontalen Riehtungen um Betr/ige ab, die jedoeh unabh~ingig sind yon Gel/indehbhennnterschieden. Man kann daher erwarten, dal3 bet bewegtem Gelgnde die im Nadirpunkt erhaltenen Messungen relativ genauer sind. In naehfolgender Abhandlung ~erden Mittel nnd W e g e angegeben, die erlauben, eine richtige und begriindete Wahl zwlschen~' Fokalpunkt- nnd Nadirpunktmessung zu vollziehen. Es ist wetter mtiglieh, an den gemessenen Richtun~;en Korrektionen anzubringea, um sie mit den wahren Riehtungen in Uberemstnnmm~g zu bringen. Infolge der Ungenauigkeit, mit der die Plattenneigung bestimmt werden kann, sowie der damit verbundenen Ungenauigkeit in der iFestlegung der Fokal- bzw. Nadirpunkte ergibt sieh ohne weiteres, dal~ die bereehneten Korrektionsgr/313en aueh nieht fehlerlos seia k6nnen. In dem vorliegel~den Artikel werden die mittleren Fehler der verbesserten Ridltungen der Fokalpunkt- und der Nadirpunkttriangulation untersueht bzw. berechnet. Die Berechnungen werden dutch einige beigegebene Nomogramme wesentlieh vereinfa&t, so da/~ in allen in der Praxis x'orkommenden F/illen di~ giinstigste Arbeitsmethode gew/ihlt und die zu erwartende Genauigkeit festgestellt werde~ kann. Ausdrtieklidl set jedoeh bemerkt, dal~ es hi~r nt~r um eine V e r g 1 e i e h u n g der Nadirpunkt- bzw. Fokalpnnkttriangulation geht uJ}d somit die reinen Messnngsfehler, die ftir beide F/ille gleich sind, aul~er Betrachtung bletben. Der § 2 der Originalarbeit befal3t sidl mi~. der Bereehnung des Fehlers A~ in den gemessenen Itichtungen, wie er dm'eh die Neigling der Kammeraehse usw. verursa&t wird. Die Literatnr bietet zu dieser Aufgabe eine Anzahl yon gen~iherten Ltisungeu, in der vor.~
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