on the strip) will fall upon their corresponding radials; a line
drawn along the edge of the paper strip while in this position
will represent the oriented picture trace, as indicated by the
If we now draw a perpendicular line (Si PI) to h\h\ from the
plotted station S\, the point PI will be the horizontal projection
of the principal point P and SiPi=f will be the distance line
for the perspective MN.
Should the positions of the points A , B, C . . . with refer-
ence to the station 5 be not known, it will become necessary
to observe the horizontal angles A SB, BSC, CSD . . . instru-
mentally from the station S and plot them in their proper order
upon a sheet of paper (AiSiBi, BiSiCi . . . ) and adjust the
paper strip hh upon these radials in the same manner as just
B. Determination of the Position of the Horizon Line on the
When the elevations AA f , BB', CC' . . . (Fig. 13, Plate VII)
of the points A, B, C . . . above the horizon plane SOO' of the
DETERMINATION OF PRINCIPAL AND HORIZON LINES. 53
station S are known, the position of the horizon line OO r on
the perspective MN may be found by computing the ordinates
aa', W t cd . . . from the equations:
The distances Sa', 5^, Sc f . . . are taken from the plotting-
sheet. The horizontal distances SA', SB', SC r . . . and the
differences in elevations A A', BE', CC' ... are known.
For example, the difference in elevation between A and
A' = 100 m., the distance of A' from the station S = 1000 m.,
and the distance Sa', measured on the plotting-sheet, =0.05 m.,
then we will have
aa r = y = ^ =0.005 m.
The horizon line OO f on the negative will be 5 mm. ver-
tically below the pictured point a (measured in a direction parallel
with the pictured plumb line w). A line OO' drawn through a'
at right angles with the pictured plumb line w will locate the
horizon line. The computed ordinates bt/=yi, cc f =y 2 ...of
the other pictured points &, c . . . will serve to check the position
of the horizon line OO' ; it should be tangent to the arcs described
with aa', bV> ccf . . . about a, b, c . . . respectively as centers.
54 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.
III. Graphic Method for Determining the Positions of the Prin-
cipal and Horizon Lines on the Perspectives.
The following method for orienting the picture trace, pub-
lished by Prof. F. Schiffner, in 1887, and mentioned by Prof.
Steiner, leads to the same result graphically as the preceding
one does arithmetically.
The horizontal projections AI, BI, C\, and Si of three points
A, B, C, and station S, Fig. 14, Plate VII, may be given. From
Si, as center, radials are drawn through A i, BI, and C\. Through
a point a on the radial Si<4i a parallel to SiCi is drawn and
the distance a'b' taken from the negative MN, not shown
in the figure is laid off from a = abi upon this parallel line,
while the distance b'c' is laid off upon the same line from bi
Parallels to the radial 5i^4i are then drawn through the
points bi and c\ and produced to intersect with the radials
SiBi and SiCi. The line h'h' connecting these two points
of intersection will be parallel with the direction of the picture
The same distances a'b' and b'c f taken from the negative
are laid off upon this line h'h' from a 2 = a 2 b 2 and from b 2 = b 2 c 2
respectively. The parallels to the radial SiAi, drawn through
these points b 2 and c 2 , are brought to intersections with the radials
SiBi and SiCi, when the line hh, passing through these inter-
sections b' and c', will represent the picture trace, oriented with
reference to Si, AI, BI, and Ci.
The distance Si PI of Si from hh represents the distance line
(focal length) of the picture MN, and the point PI will be the
horizontal projection of the principal point of the perspective.
After having transferred PI (with reference to a', b', and c')
to the perspective MN by means of a strip of paper, a parallel
to the pictured plumb line vu drawn through the point P L will
locate the principal line upon the negative.
HI. The " Five-point Problem " (by Prof. F. Steiner), or Locating
the Plotted Position of the Camera Station by Means of the
Perspective when Five Triangulation Points are Pictured
on the Same Photographic Perspective.
In the methods considered until now it had been assumed
that the position of the camera station Si on the plotting-sheet
was known with reference to the plotted triangulation points
AI, BI, Ci . . . .
In case the panorama pictures have been taken from a camera
station Si of unknown position and a series of known points
are pictured upon the panorama views, both the position of the
camera station may be found (with reference to the positions
of the surrounding points of known positions) and the picture
trace may be oriented by means of Prof. F. Steiner's " five-point
problem," if one of the panorama views contains the pictures
of five or more points of known positions.
A. Determination oj the Principal Point and Distance Line.
A panorama view MN may contain the images a, b, c, d, e
of the triangulation points A, B, C, D, E, already plotted on
the working-plan, and also the picture of a suspended plumb
line or other vertical (or horizontal) line sufficiently long to be
used for drawing parallel lines to the principal (or horizon)
The points a, b, c, d, and e of the negative MN are projected
upon the straight edge of a strip of paper = a\, bi, c\, di, and e\.
Radials are now drawn from one AI, Fig. 15, Plate VIII
of the five plotted points as center to the other four points, BI,
Ci, DI, and EI. The paper strip is then placed over the radials
AiBi, AiDi, and AiEi, that bi falls upon AiBi, d\ upon AiDi,
and e\ upon AiEi, when the strip will have the position ai,
b\, Cij di, e\. The line drawn through AI and ai (the latter
having been transferred to the sheet by means of the paper strip)
56 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.
will be tangent in A to the ellipse E\ (which passes through A\>
BI, DI, and EI and through the plotted station Si).
The paper strip is now placed over the radials AiBi, AiCi,
and AiDi, that bj falls upon A\B\, c\ upon AiCi, and d\ upon
AiD\, when the strip will have the position indicated by a 2 ,
b 2 , 2, d 2y 2, and the line Aia 2 will be the tangent in A\ to the
ellipse E 2 (passing through the points AI, BI, Ci, DI, and the
plotted station point Si).
The plotted position of the station point Si with reference
to the five plotted points AI, BI, C\, DI, and EI will be at the
fourth point of intersection Si of the two ellipses EI and E 2 .
After drawing the radials SiAi, Si-Bi, SiCi, SiDi, and SiEi
the paper strip is placed over these radials in such manner that a\
falls upon SiAi, bi upon SiBi, . . . and e\ upon Si-Ei, in the
position indicated by a, b, c, d, e=HH, when HH will be the
plotted picture trace.
The perpendicular upon HH passing through Si = Si PI
represents the distance line and PI is the principal point of the
negative projected into the horizontal plan, which, in order
to locate the principal line, may now be transferred to the per-
spective by means of the paper strip in the manner already
B. Simplified Construction for Locating the Plotted Position o)
the Camera Station by Means oj the " Five- point Problem."
The method just described being rather complicated, Prof.
SchifTner recommends the following construction, Fig. 16, Plate IX,
in which the drawing of the two ellipses EI and E 2 is avoided:
The plotted positions of the same five points A, B, C, D^
and E, together with a negative containing the images a, b, c, d,
and e, of these points may be given. It is desired to find the
fourth point of intersection Si of the two ellipses EI arid E 2
without actually drawing their perimeters.
The two tangents b s Bi and M*i to the ellipses EI and E 2
in BI are located in precisely the same manner as the two tan-
gents aiAi and a 2 Ai for the point A\ were found in Fig. 15,
Plate VIII. The intersections R\ and R 2 of the tangent pairs
aiAi, bzBi, and a 2 Ai, bBi, Fig. 16, Plate IX (belonging respec-
tively to the ellipses EI and E 2 ), are situated on a line QX, form-
ing one side of the polar triangle QXT, common lo both ellipses.
This line RiR 2 = QX intersects the diagonal AiD\ in X and
the quadrilateral side BiDi in the point Q. The lines drawn
through Q from A\ and through X from B\ will intersect each
other in the fourth point of intersection Si of the two ellipses.
This method may also appear rather complicated in view
of the many lines that have to be drawn before the picture trace
H H and the position of the camera station may be plotted.
C. Application of the " Five- point Problem "to the Special Case,
where the Five Points range themselves into a Triangle on the
The application of the five-point problem becomes very
much simplified when the five points A, B, C, D, and E form
a triangle of which two sides A\C\ and C\E\ 9 Fig. 17, Plate X,
contain three points each.
If we place the strip of paper upon the radials, drawn from AI,
that e\ falls upon A\E\,d\ upon AiDi, and c\ upon A\C\> it will
have the position indicated by a 2 , b 2 , c 2t d 2 , e 2 , and the first ellipse
will resolve into the lines C\E\ and A\a 2 . If we now place the
paper strip #1, bi, c\, d\, e\ upon the radials drawn from E\ to AI,
to BI and to Ci, that #1 falls upon E\Ai, bi upon EiBi, and c\
upon EiCi, it will assume the position a\ t bi, c\, d\, e\, and the
second ellipse will resolve into the lines A\C\ and EI^I.
The intersection Si, of the two lines Aia 2 and 1^1, locates
the position of the plotted station point Si with reference to the
five given points A\, B\, Ci, D\, E\. By placing the paper
strip upon the radials SiA 1} SiBi, SiCi, SiA, and SiEi in
58 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.
such manner that ai falls upon Si^A, upon SiJBi, c\ upon
SiC\ . . . , its edge HH will locate the picture trace, PI ' will be
the horizontal projection of the principal point P, and S\Pi will
be the distance line.
D. To find the Elevation of the Camera Horizon for a Station
that has been located by means of the "Five- point Problem."
To ascertain the elevation of the station S, plotted after one
of the preceding methods, it will be necessary to know the ele-
vations of at least two of the five given points. The elevation
of the station horizon SOO', Fig. 13, Plate VII, above the datum
or ground plane S'OiOi', may be designated by X, H and HI
may be the elevations of A and B respectively, both supposed
to be known. The ordinates of the pictured points a and b
are aa'=y and bb'=y.
From the relation S'ai' -.S'A^ = aa f :AA'
we find y = ^7 (HX),
and yi = 7 (Hi - X).
As the difference between y and y\ may be found by direct
measurements made on the negative, y yi=m will be known
and the value for X may be computed from the equation
since the measures for Sa', SA', 56', and SB' may be obtained
from the plotting-sheet, measured in the scale of the map.
THE " THREE-POINT PROBLEM." 59
The above equation may be written in the general form:
H-X H-X i Sa' i SV
m = ? where * = S^ and 7 = 5^ ;
m-n-p H-p HI n
By substitution of this value in the equations
the numerical values for the ordinates y and yi (governing the
position of the horizon line) may be found.
V. The " Three-point Problem."
If the triangulation points are not sufficiently close together
that five or more points may be pictured on one photographic
perspective, and if stations are occupied with the camera that
are not directly connected with the trigonometric system, it will
become necessary to employ other means than those hereto-
fore considered for locating the position of such detached camera
stations with reference to the triangulation system.
To connect detached camera stations with the triangulation
by observations made at the camera station, at least three tri-
angulation points should be visible from such station. When
the camera party is in advance of the triangulation party many
camera stations will be located by the triangulation party by
observing upon a signal left at the camera station, if such signal
be visible from two or more triangulation stations (the camera
station will be a " concluded point " of the triangulation system).
60 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.
The determination of the position of a detached camera
station by observing upon three fixed and known points (pro-
vided with signals) is generally known as the " three-point prob-
lem " (station-plotting, station-pointing, etc.), or " Pothenot's
method," although Snellius was probably the first to use this
method (in his trigonometric surveys in the Netherlands in the
second decade of the seventeenth century).
A. Mechanical Solution oj the " Three- point Problem"
(using a Three-arm Protractor or Station- pointer).
The simplest solution of the three-point problem is purely
mechanical in application. The two observed angles M and N
are laid off upon a three-arm protractor (" station-pointer ")
or upon a sheet of tracing-paper, and the three arms or lines S\Ai,
SiB ly and SiCi, Fig. 18, Plate X, are placed over the three fixed
and plotted points AI, BI, and C\ in such manner that the three
lines of direction 5i^4i, S\B\-> S\C\ pass through their respective
points AI, BI, and C\, the point S being transferred to the
working-sheet while holding the two horizontal angles M and N
in unchanged position.
B. Graphic Solution of the " Three- point Problem."
i. USING THE SO-CALLED "TWO-CIRCLE METHOD."
Theoretically the best graphic method is probably that by
which the position of the fourth, or station, point is located at
the intersection of two circles, one passing through A i and BI
and having over AiB, as chord, the angles of circumference
=AiSiBi=M, Fig. 1 8, Plate X, the second circle passing through
BI and Ci and having over the chord B\Ci the angles of cir-
cumference equal to BiSiCi=N.
From the plotted triangle side AiB\ we lay off at A\ and B\
the angles BiAiCi and AiBiCiy each equal to
iSo-2(A l S l B l ) o* n u
= 90 4iSi>i=9o M,
and about the point Ci, thus obtained, we describe a circle
,4i#iSi with the radius = CiAi=CiBi. The observed angle
AiSiBi=M will then be an angle of circumference over AiBi,
and the point Si will be located somewhere on the arc over A\B\.
By means of the angle BiS\C=N another circle BiCSi is
described over the triangle side B^C, in a similar manner, about
the point C% as center, having C<iB\=C<2.C as radius. The
observed second angle B\S\C=N will be an angle of circum-
ference over the chord B\C and the point Si will be on the arc
over BiCy hence its true position is at the (second) point of inter-
section Si of the two circles.
2. USING THE METHOD OF BOHNENBERGER AND BESSEL.
The following method, by Bohnenberger and Bessel, is readily
applied and simple in construction. If we describe a circle
through two of the given points, through A\ and B\, Fig. 19,
Plate XI, and through the station Si, the angles designated by M
and those designated by AT" in the figure will be respectively
equal, being angles of circumference over the same arcs 4iZ>i
and DiCi respectively.
Hence if we lay off the observed horizontal angle N on AiCi
at A i, and the other observed horizontal angle M on the line
^4iCi at Ci, the point of intersection D\ of their convergent
sides CiZ>i and A^D\ will fall upon the line connecting the third
plotted triangulation point BI with the station point Si.
After having thus determined the direction of the line BiDi
or BiSi the position of the point sought may be found as follows :
At any point x on the produced line BiDi the observed angles
M and N are laid off to either side of #iA, in the sense in which
62 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.
they were observed at S. Lines AiSi and CiSi, drawn through
A i and Ci, parallel to xy and ^2 respectively, will locate the
plotted position of the station point 5*1 (upon D\B\) with refer-
ence to the three plotted points A\, B\, and C\.
This solution is recommended only when B\D\ is sufficiently
long (in Fig. 19, Plate XI, it evidently is too short) to assure a
correct prolongation toward S\.
The picture trace HH, containing the horizontal projections
of the pictured points a, b, c, may now be oriented in the known
manner by adjusting the paper strip, having the three points a\,
bi, and c\ marked on its edge, over the radials S\A\, S\B\> and
Sid to bring a\ on SiAi, bi on SiBi, and c\ on S\C\.
VI. The Orientation of Picture Traces, Based on Instrumental
Measurements Made in the Field.
When no points of the area to be mapped phototopographic-
ally are known, the elements (horizon line, principal point, and
distance line) of the photographic perspectives can no longer
be determined from the photographs alone. Instrumental obser-
vations will have to be made at the camera stations in the
field to supply the data needed for their determination. This
method, among others, having been adopted by Capt. Deville,
will be described in the chapter giving the description of the
VII. Relations between Two Perspectives of the Same Object,
Viewed from Different Stations.
'(Prof. Guido Hauck's Method.)
A more .general application of photogrammetric methods
dates j from the publication of Prof. G. Hauck's investigations
and results regarding the relationship between trilinear systems
of different planes (Guido Hauck, " Theorie der trilinearen
TWO PERSPECTIVES OF THE SAME OBJECT. 6j
Verwandtschaft ebencr Systeme," Journal fuer reine und ange-
wandte Mathematik, L. Kronecker und Weierstrass, Bd. 95,
1883). In this publication Prof. Hauck discusses the relation-
ship between the projections of the same object upon three differ-
ent planes. The practical value of his theoretical deductions
was fully established and tested practically by the students of
the Royal Technical High School of Berlin who attended ProL
Hauck's lectures and exercises connected with the course in
descriptive geometry in 1882.
In the discussion of the relation between three perspectives
of the same object Prof. Hauck refers to some properties of
decided value in iconometric plotting. The principal law (as
deduced by Prof. Hauck) with reference to phototopographj
may be stated as follows:
If an object be projected from three different points as cen-
ters upon three different planes that may have any position in
space, one of these projections (perspectives) can be evolved
from the other two by means of graphic construction. Or,
expressed in terms more suited to our case, if an object has
been photographed on three plates exposed from different
stations, any one of these photographic perspectives may be
evolved graphically from the remaining two. A topographic
map (the orthogonal projection of the terrene) may be regarded
as a central projection or perspective in a horizontal plane,
having its center of projection (point of view) at infinite dis-
tance, and we may state Prof. Hauck's law as follows :
From two photographs MN and M'N' of the same terrene,
taken from different stations 5 and S', the orthogonal horizontal
projection of the terrene may be obtained graphically by means
of rays emanating from the so-called " kernel points " (" Kern-
punkte ") as centers.
The line of intersection of the two photographs (the two
planes of projection or perspective planes) MN and M'N 9 ,
Fig. 20, Plate XI, will be the " perspective axis."
To better illustrate the connection between two different
64 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.
photographs we will first refer to the simple case of two vertical
perspective planes or photographs MN and M'N', Fig. 20,
A. " Kernel Points " and " Kernel Planes."
Let S and S' represent the two camera stations (centers of
projection or points of view for the two vertical photographic
perspectives MN and M'N'), let s f be the picture in MN of S',
and s be the picture of S in M'N', then these two pictured sta-
tions s and s' will be so-called " kernel points " (" Kernpunkte ").
The two picture planes MN and M'N' intersect each other
in the line IQ, the so-called " perspective axis."
Planes passing through the line SS' (base line) will contain
the " kernel points " 5 and s'; they are termed " kernel planes "
(" Kernebenen ").
A kernel plane M 2 N 2 , laid through any point A, pictured
as a and a', will intersect the first picture plane MN in the line as'
and the second picture plane M'N' in the line sa'. These lines
of intersection (as' and a's) will intersect the " perspective axis "
IQ in the same point Q] they will contain the pictures a and a'
of the point A, and they will pass through the picture s and s'
of the two camera stations S and S'.
The lines S'A, SA, SS', as', and a's fall within the " kernel
plane " M 2 N 2 . All lines as' for all points pictured in MN
will pass through the pictured station point s' (image of the second
camera station S'), and all lines as for the picture plane M'N'
will pass through the pictured point s of the camera station S.
Furthermore, all pairs of lines (as' and a's) joining the per-
spectives (a and a') of identical points (A) with their corre-
sponding pictured station points (" kernel points " s' and s)
will intersect the " perspective axis " (IQ) of the two pcture
planes (MN and M'N') in identical points (Q).
From two photographs of the same object which also con-
tain the pictures of the two reciprocal stations peculiar advan-
TWO PERSPECTIVES OF THE SAME OBJECT. 65
tages may be gained for the iconometric plotting, inasmuch as
such pictured stations s f and s will be " kernel points."
The perspective axis of the picture planes may also play an
important part in iconometric plotting, not only for pictures
exposed in vertical planes, but even more so for inclined picture
If two photographs MN and M'N' are given (in Fig. 21,
Plate XII, their traces are represented by the lines HH and H'H')
representing the same object, viewed from the two stations S
and S' without containing the pictures of the stations, the posi-
tions of the pictures s and s' of the corresponding camera sta-
tions S and S' may be located upon the picture planes (out-
side of the actual field of the photograph) by construction.
The horizontal projections Si and Si' of the " kernel points "
s and s* are identical with the points of intersection of the plotted
base line SS' and the picture traces HH and H'H', Fig. 21,
Plate XII. Hence, if we revolve the picture planes MN and
M'N' about their ground lines, until they coincide with the
ground plane, the line IQ, common to both picture planes (the
" perspective axis "), will be represented by the two lines i(I),
and the " kernel points " 5 and s f of the revolved planes MN
and MN will fall upon the lines si(Si) and V(Si') respectively
(these lines are perpendiculars upon the picture traces in the
horizontal projections si and Si' of the kernel points 5 and s f )..
To find the lengths si(Si) and s^ (Si')' (the ordinates of
the " kernel points " in the picture planes above the ground
lines) we erect perpendiculars to the base line in S and S' with
lengths equal to the elevations of the camera stations above
the ground plane =S(S) and S'(S') respectively.
The line (S)(S') the vertical plane passing through the base
line S has been revolved about the horizontal projection of the
base line into the ground plane to coincide with the latter
will intersect the lines $i(Si") and Si'(Sn') they are perpen-
diculars to SS' in the " kernel points " 5 and s and the lengths
$i(Si") and Ji'(Sn') will be equal to the ordinates of the " ker-
66 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.
nel points " J and s f above the ground lines of MN and
The " kernel points " s and s' , Fig. 20, Plate XI, may be
located after this manner in the picture planes of any two photo-
graphs, provided such picture planes are not parallel (or even
nearly so) with the base line SS'.
B. Use of the "Perspective Axis " (Line of Intersection) of two
Picture Planes that show identical Objects viewed from
If a series of characteristic points of the terrene, pictured
in a vertical plane 'MN, Fig. 22, Plate XII, are connected with
the " kernel point " s by straight lines, these will, when pro-