D. General Arithmetical Method for Finding the Plotted Positions
oj Terrene Points when Pictured on Vertically Exposed
With reference to Fig. 43, Plate XXIV, we have
S and 5' = the two camera stations ;
MN and MW=two photographic perspectives obtained from
5 and 5' respectively;
a and a' = two pictures of a point A ;
j = SP=S'P f = constant focal length for both pictures or plates;
D= S&A = horizontal distance from S to A ;
D' =So'A Q = horizontal distance from S' to A',
= horizontal distance between the two stations 5 and
5', the elevation of A above the horizon plane of the
station S = H and above the horizon plane of the
station S' = H f .
Finally, the horizontal angles included between B and the
principal planes that pass through the two stations S and 5' =
ao and o' respectively.
If we refer the pictured points to the principal point P of
the photographic perspective by means of the rectangular sys-
Il6 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.
tern of coordinates formed by the principal and horizon lines
(If and OO') the coordinates of a on MN will be
aa f =y t a'P=-x,
and those of a' on M'N' will be
If the camera is in perfect adjustment, if the base line B
has been measured in the field, and if the angles o and a have
been observed, we know the values of
B, a , a ', /, x t oS, y, and /
(the coordinates are measued on the negatives MN and M'N 1 }
and we can now compute:
(i) The horizontal angle 7- (or 7-') included between the prin-
cipal ray SP (or S'P') and the horizontal direction Sa'
(or S'ai) to any point A from the equation
tan r = lor tan^==y).
(2) The vertical angle /? (or /?') included between the plane
of horizon for the station S (or S') and the line of direction
Sa (or S'a') to any point A from the equation
tan 0= or
d=Vp+x* (or d'=
we may write
PHOTOTOPOGRAPHIG SURVEYING METHODS. 1 17
Of the triangle SoA So' we know the side SoSo'=B and the
angles ?-, o> fi and o'; hence
B _ _ sin (f +0:0') ___ sin <y +q / )
5~sin[i8o-(r+ao + r / +o / )]~ sin (r+ao + r' +</)'
We can now compute from
the difference in elevation between A and S (or 5'),
(or #' = >' tan /?')
. General Arithmetical Method for Finding the Plotted Positions
oj Terrene Points when Pictured on Inclined Picture Planes.
For inclined picture planes we will have to take the angles
of inclination of the plates into consideration. Under angle of
inclination of a plate we understand that angle which is included
between the optical axis of the inclined camera and the horizon
plane of the camera station (second nodal point).
Referring to Fig. 38, Plate XXII, and Fig. 44, Plate XXV,
a = horizontal angle included between the principal plane and
the vertical plane passing through the station S and the
point ^4, pictured as a;
/? = angle of elevation of the point A ;
Y = angle of inclination of the photographic plate MN;
Il8 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.
d = complement of ?- = i8o ;-;
OO' = horizon line when MN is vertical (OO f is permanently
marked on the camera);
P = principal point for the vertically exposed plate;
P7r = ;y = ordinate of a, Fig. 44, Plate XXV;
an = x = abscissa of a, very nearly = a'P', Fig. 44, Plate XXV;
1 = vanishing point ("kernel point ") for all vertical lines
pictured on MN.
From inspection of Fig. 44, Plate XXV, we find directly
aa! -KK' np Pp Px y cos f } sin f
tan p = 7 = - > = - ,
y cos Y + / sin ? y cos f / sin 7-
cos r +y sn r
For the vertically exposed plates we had found
tan B = , and tan ct = -r.
The preceding formulas for the inclined plates will assume
the form of the latter if the angle of inclination 7- is reduced
to O, as sin f will then become equal to O and cos 7- equal to i.
After the values for a and /? (or a! and /?') have been com-
puted the value for S A = D (or S 'A = D f ) and for AA'=H
(or ^4^4 / ' = 7J / ) may be obtained as follows:
Referring to Fig. 38, Plate XXII, we find
B -sin (Y <
PHOTOTOPOGRAPHIC SURVEYING METHODS. IIQ
and from = tan/?
Vx 2 + (/ cos ?- +y sin
If an ordinary surveying camera with constant focal length is
used, and it should become desirable to expose a plate in an
inclined plane, the complement d of the angle of inclination 7-
of the optical axis may be more readily (but only approximately)
determined than 7- by carefully measuring the distances AD,
Fig. 45, Plate XXV (in the direction of the line of a suspended
plumb- bob), and DB, AB, being parallel with the photographic
F. General Analytical Determination of the Elements of a
When in addition to the photographs other data obtained
by the necessary instrumental measurements are given for a
graphical determination of the focal lengths of the pictures,
their horizon lines and their principal lines, then these elements
may also be determined analytically.
A picture MN, containing the images a, b, and c of three
known points A, B, and C, may be given and the position of
the camera station (whence this picture was obtained) may be
known with reference to the three plotted points A', B', and
C', Fig. 46, Plate XXVI.
To orient the picture trace (or the ground line) gg f with
reference to the plotted station 5' and plotted points A 1 , B', and
C' the latter are preferably referred to a rectangular system of
coordinates (S'Y and S'X, Fig. 46, Plate XXVI) having the
plotted station 5' as the origin. To simplify matters one of
the axes of the system may be laid through one of the plotted
120 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.
points. In Fig. 46, Plate XXVI, the axis of abscissae S'X passes
through the point c'.
The coordinates of the points A', B', and C', measured on
the plotting- sheet, may be
XiYi, X 2 Y 2 , and X 3 respectively.
The coordinates of the orthogonal projections (on the pic-
ture trace gg f ) of the corresponding points pictured on the photo-
graph MN and located upon the radials S'A f , S'B', and S'C f
may be designated by
xiyi, xuyu, and xm respectively.
The horizontal distances measured on the photographic
plate between a and b, between b and c, and between a and c
(the same as those measured on the picture trace between a' and b f >
between b' and c', between a' and c f ) may be designated by
w 1 , m 11 , and m lu respectively.
From an inspection of Fig. 46, Plate XXVI, it will be evi-
(2) yir-xu=Y2:X 2 ;
From these five equations the five unknown quantities of
Xi, yi, xu, yii) and xm the coordinates of the points to be
located may be computed.
From the area of the triangle S'a'c',
GRAPHICAL ICONOMETRICAL PLOTTING METHODS. 121
we find the focal length
The angle of orientation 7-, included between the principal
ray S'P' and the base line S'C', may be derived from the equation
cos r = ~ or =
The principal point P f may be located upon gg by laying
off on the picture trace gg' from d the length,
P'c' =xui sin 7%
The differences in elevation between the station 5 and the
three points A, B, and C being known it will be an easy matter
to draw the horizon line upon the photograph.
n. Graphical Iconometrical Plotting Methods.
A. Col. A. Laussedafs Method (French Method).
Col. Laussedat's methods of constructing topographic maps
from (photographic) perspective views of the terrene having
been widely published, they form the groundwork for all sub-
sequent work in this field. They are chiefly of a graphical char-
acter and in harmony with the laws of perspective. Col. Laus-
sedat considers two general cases in reconnoitering expeditions
where phototopographic methods may be applied with advantage:
First. The observer may remain sufficiently long in one
locality to make a survey on a large scale, say i : 20000
and even larger for special purposes.
122 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.
Second. The explorer moves rapidly from place to place,
gathering only the most necessary data on his itinerary
to enable him to plot the topography of the traversed
country as a " running survey" on a small scale say,
i : 50000 or smaller preserving and representing only the
principal topographic features met with on the track
In the first-mentioned case the explorer will measure one
or more base lines with as great an accuracy as the means and
time at his disposal will admit. He will cover the area to be
mapped with a system of triangles connected with the base
lines, and inasmuch as the triangulation stations will also be
occupied with the surveying camera the scheme should be laid
out with due reference to the subsequent iconometric plotting
of the topographic features. .
When applying the ordinary surveying methods the tri-
angulation scheme would probably be laid out with a view toward
covering as large a territory as possible with each triangle, occupy-
ing the smallest possible number of intervisible points.
With the application of photography, however, the conditions
become somewhat changed. Every topographic feature that
is to be plotted iconometrically should be seen from two or
more camera stations, and as each camera station is to be con-
nected with the triangulation system, either directly or indirectly,
the number of triangulation points should be a relatively large
one. Often it will not be desirable that the highest peaks trigo-
nometrically laid down on the map should be occupied with
the camera, especially when fogs prevail in the higher altitudes,
and when other camera stations would answer the requirements
just as well.
Regarding the second case, where the explorer follows a
certain route without making side excursions and never stopping
longer in one place than is absolutely necessary for his observa-
tions, the phototopographic method becomes even more valuable
than in the first case, particularly when traversing open and
GRAPHICAL ICONOMEJRICAL PLOTTING METHODS. 12$
broken country. For this kind of topographic reconnaissance it
may well be said that the photographic method surpasses all
other surveying methods regarding the amount of data which
may be collected in the field in a limited time period.
All topographic operations and instruments serve to measure
distances and vertical and horizontal angles. A photographic
perspective of which the elements are known will give all the
data needed to determine the vertical and horizontal angles of
lines of direction drawn from the point of view to all points
pictured on the photograph.
The points A and B shown on the plate MN, Fig. 47, Plate
XXVI, may represent the pictures of two mountain peaks. The
points marked a and b will be their projections upon the horizon
line HH'. The angle aSb=a will be the horizontal angle of the
lines of direction SA and SB if S is the point of view on the
distance line SP.
The vertical angles /? and 7- may be shown in horizontal plan
by revolving the vertical planes passing through SA and SB
about the lines Sa and Sb, respectively, until they coincide with
the horizon plane HH', when
The vertical angles /? and 7- may now be measured in horizontal
plan as (/?) and (/-).
To indicate in a general way Laussedat's method of icono-
metric plotting and to show how the plotted features of the terrene
may be obtained from the photographs we will refer to Figs. 48
and 49, Plate XXVII, where A, B, and C represent three camera
stations (plotted in horizontal plan, Fig. 48), whence three per-
spectives I, II, and III, Fig. 49, of the same knoll D may have
been obtained. The traces of these three pictures on the plotting-
sheet may be H A H A , H B H B , and H C H C . All three photo-
graphs having been taken with the same instrument of constant
124 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.
focal length, the distance lines P A A, P B B, and P C C will be
i. ORIENTATION OF THE PICTURE TRACES ON THE PLOTTING-SHEET.
The three stations A, B, and C are plotted either as parts of
the triangulation system or by measuring the base line AB on
the ground and observing the horizontal angles CAB and CBA,
when the sides AC and EC may be found graphically or by
computation and the triangle ABC be plotted upon the working-
Horizontal angles or directions to D having also been observed
from A, By and C, its position with reference to A, B, and C
may also be plotted.
To orient or plot the three picture traces we must know the
horizontal angles a A , a B , and a c , which are generally observed
for each picture by means of the horizontal circle attached to
These angles are plotted from A, B, and C on the lines AD,
BD, and CD with reference to the position of D on the photo-
graphs, whether to right or left of the principal line VV. The
constant focal length = / of the three negatives I, II, and III is
now laid off on the radials AP A , BP B , and CPc> Perpendiculars
erected in P Aj P B , and P c to the lines AP A) BP B , and CP C re-
spectively, will represent the picture traces H A H A , H B H B ,
and H C H C . The abscissae P A d A , P B d B) and PC&CI measured
on the negatives I, II, and III, should be made equal to the
distances P A d A) P B d B , and Pcdc on the picture traces.
The point D is termed a " reference point," and every picture
that is to be used for iconometric plotting should contain the
image of at least one such reference point of known position in
both the horizontal and vertical sense.
2. LOCATING POINTS ON THE PLOTTING-SHEET THAT HAVE BEEN IDENTIFIED
ON SEVERAL PHOTOGRAPHS.
After the picture traces have been oriented any (other) point
T of the terrene shown on two or more pictures may readily
GRAPHICAL ICONOMETRICAL PLOTTING METHODS. 125
be plotted without requiring additional instrumental measure-
ments in the field.
To locate the plotted position of the point T, Fig. 48, Plate
XXVII, shown as t A and t c on two pictures I and III, Fig. 49,
Plate XXVII, the abscissae P A t A and P c t c are laid off on the
picture traces H A H A ' and H C H C ', respectively, from P A and P c ,
Fig. 48, and on the side of P corresponding with the position of
the image t with reference to the principal line VV, Fig. 49, Plate
Lines drawn from A and C, Fig. 48, through t A and t c will
represent horizontal directions to T from the stations A and C;
their intersection at T will locate the position of the tree in
3. THE ICONOMETRIC DETERMINATION OF ELEVATIONS OF PICTURED
The horizon line HH' of a perspective view, Fig. 47, Plate
XXVI, being the intersection of the horizon plane with the ver-
tical picture plane, will intersect points in the picture which in
nature have the same elevation as the optical axis SP of the
camera. All pictured points falling above the horizon line are
higher and all points falling below the horizon line are lower
in elevation than the point of view S.
The distances Sa and SA, Fig. 50, Plate XXVII, are measured
on the plotting- sheet and the ordinate (Aa, Fig. 47, Plate XXVI)
of the pictured point a (its distance from the horizon line) is
taken from the negative. Perpendiculars to SA are then erected
in a and A (on the plotting- sheet, Fig. 50, Plate XXVII), and
the one in a is made equal to the ordinate (A a, Fig. 47, Plate
XXVI) of the pictured point = a(a), Fig. 50, Plate XXVII.
If we now draw the line S(a) to its intersection with the per-
pendicular to SA in A, the triangle Sa(a) and 5(^4)^4 will be
similar and the angle ^45(^4) will represent the vertical angle
of the visual ray from S to A, revolved about 5^4 into the plane
126 PHOTOTOPOGRAPHIC METHODS AND ' INSTRUMENTS.
of the horizon. From the similar triangles Sa(a) and SA(A)
we derive the proportional equation
a(d) is measured on the negative; SA and Sa are taken from
the plotting-sheet. A (A) measured on the plotting-scale will
give the difference in elevation between 5 and A.
In practical work the elevations of the camera stations are
known and by adding the height of the instrument including the
value for A (A ) to the elevation of the camera station the absolute
elevation of the geodetic point A is found, which, however, is still
to be corrected for curvature and refraction.
A second value for the elevation of the geodetic point A is
found in the same manner from another negative containing
an image of A and obtained from another station. The mean
of several such determinations is adopted for the final value for
the height of A.
4. DRAWING THE PLAN INCLUDING HORIZONTAL CONTOURS.
After some little practice, points pictured on different nega-
tives but representing identical geodetic points will readily be
identified by the observer and he will select characteristic points
to reproduce the watercourses, water-sheds, roads, shore lines,
etc., on the plotting-sheet.
After these principal guide lines are well located on the chart
the buildings, outlines of woods, marshes, etc., are plotted, includ-
ing everything that is to be shown on the finished map.
Enough points must be plotted iconometrically to form a
good control for a correct delineation of the relief. Should the
number of points determined on the plan be sufficient only to
GRAPHICAL ICONOMETRICAL PLOTTING METHODS. 127
give an adequate control for the delineation in the horizontal
sense, additional points should be plotted from the photographs
to obtain an equally good control of the terrene in the vertical
The planimetric work completed, elevations of as many of
the plotted points as seem necessary are determined and inscribed
on the chart. Horizontal equidistant contours may then be
drawn by interpolation to harmonize with the elevations suffixed
to the points of control on the chart, conforming the courses of
the contours between the determined points to the configuration
of the terrene as it is shown on the panorama views.
It cannot be denied that a certain amount of study and
practical application are requisite to enable the iconometric
draughtsman to interpret forms correctly when shown in per-
spective. Yet it should also be admitted that such transla-
tion or conversion of the configuration of the terrene into hori-
zontal projection may be accomplished far more accurately
at one's leisure in the office by means of geometrically correct
perspectives than could be accomplished by sketching in the
When topographic features as seen from one direction are
sketched by the plane-tabler, their forms will often be found to
have been misconceived when they are again seen from another
point of view. Of course, forms sketched on the plane-table
sheet may then be corrected, in a measure at least, but many
details are sketched that will not be seen again from other sta-
tions, and even those that are again observed from other stations
may not be modified to conform with their true shapes unless
the original station whence they were first seen and sketched
could again be occupied to verify the suggested changes, and,
generally speaking, topographers regard a second occupation
of a station with little favor, considering it too great a loss of
time, retarding progress and considerably increasing the cost
of the work.'
In iconometric plotting, however, it would be an easy matter
128 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.
to refer back to the panorama views obtained from some other
station, and the plotting of topographic details should not be
attempted without having first made a careful study of (and a
close comparison between) the various pictures representing
identical areas but seen from different points of view.
B. Dr. A. Meydenbaur's Method (German Method).
The pantoscopic lens (made by E. Bush in Rathenow, Prussia)
of Dr. Meydenbaur's surveying camera commands an angle of
about 100. By excluding the external rays of the effective
field of these lenses by means of diaphragms (within the camera)
pictures are obtained subtending a horizontal angle of but 60
(irrespective of the 5 mm. wide margins with which two adjoining
plates lap over each other) requiring six plates for a complete
After the camera has been adjusted over a station the pano-
rama is photographed by exposing six plates in succession, each
successive turn of the camera in azimuth covering an angle of 60,
two adjoining plates lapping over each other by a margin of 3
in arc, Fig. 51, Plate XXVIII. These common margins, con-
taining identical sections of the panorama view, may well serve
to find the value for the focal length of the negatives.
From the panorama set of six plates exposed from one sta-
tion objects or geodetic points may be selected on the middle
lines of the common margins of adjoining plates that must be
equidistant from the principal lines of adjoining plates.
i. DETERMINATION OF THE FOCAL-LENGTH VALUE FOR THE PHOTOGRAPHIC
After having selected a series of such reciprocal points, using
a magnifier of low power if needed, on all six plates, we shall
have twelve determinations (represented by the length /) of
the positions of the principal line and the greatest discrepancy
between any two values should not exceed 0.2 mm. if the instru-
ment is well adjusted. The sum 2/ of two such distances
GRAPHICAL ICONOMETRICAL PLOTTING METHODS. I2O,
represents the effective length of one picture, or the length
of one side of a regular hexagon, with an inscribed circle of
the radius equal to the constant focal length =/ of the negatives.
The value for the focal length may be found graphically or it
may be computed from the formula
When "positive prints" are to be used in the iconometric
map construction this focal length often will have to be changed
to correspond with changes that may have taken place in the
dimensions of the prints compared with their negatives. The
total linear changes in a print, measured in the direction of the
principal and horizon lines, may readily be found by comparing
the distances between the " teeth " (metal plates permanently
marking the principal and horizon lines in the image plane of
the camera) on the negative with those included between their
contact prints on the positive.
With reference to Fig. 52, Plate XXVIII, we have:
ab = original length of horizon (or principal) line between the
teeth of the camera or between their imprints on the
a'b' = the corresponding length measured on the positive;
CO = / = constant focal length of the camera or negative.
The focal length c'O of the contracted (or expanded) posi-
tive may be found graphically by drawing the triangle abO,
placing the line a'b' (measured on the positive) parallel with ab
and moving it (maintaining its direction parallel to ab) towards
(or from) O until a' falls upon aO and b' upon bO.
c'O will be the focal length to be applied when considering
the horizontal angles deduced from the positive. Had ab been
measured in the direction of the principal line, c'O would be
the focal length for the positive to be considered when deducing
vertical angles from the point. The focal lengths c'O should be
130 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.
ascertained for every print that is to be used in the iconometric
The topographic map proper is constructed iconometrically
from the negatives and positives in a manner very similar to
that described under Col. Laussedat's method.
Referring to Fig. 53, Plate XXIX, we have:
I and 11 = negatives of plates exposed at stations / and // re-
spectively. I shows the image of a signal at station
//, and negative II shows the image of a signal at
1 11 = base line measured between the two camera stations /
Both negatives show the image t of the same tower T.
2. ORIENTATION OF THE PICTURE TRACES ON THE PLOTTING-SHEET.
After the base line / //, Fig. 53, Plate XXIX, has been plotted
in reduced scale we describe circles about / and // with the radius
equal to the constant focal length of the negatives,
and produce the line / // beyond both station points, make
/ // = O //o (Plate I)
and ///o=0 7 (Plate II),
describe arcs from // as center with HQC=X\ I (Plate I) and
from /o as center with I c = x 1 } (Plate II) as radius.