John Adolphus Flemer. # An elementary treatise on phototopographic methods and instruments, including a concise review of executed phototopographic surveys and of publicatins on this subject online

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Stations

whence They

were Derived.

Elevations of

Stations.

Difference of

Elevations.

Elevation of

Point.

Remarks.

D. General Arithmetical Method for Finding the Plotted Positions

oj Terrene Points when Pictured on Vertically Exposed

Picture Planes.

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',

d =

= 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

and as

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' +</)'

whence

sn

We can now compute from

H

tan/?=-p,

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,

we have

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-

~

and

=

cos r +y sn r

For the vertically exposed plates we had found

y oc

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

D sing'-ft

B -sin (Y <

hence

sin

PHOTOTOPOGRAPHIC SURVEYING METHODS. IIQ

H

we obtain

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

plate.

F. General Analytical Determination of the Elements of a

Photographic Perspective.

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-

dent that

(1) yiixi-YnXi-,

(2) yir-xu=Y2:X 2 ;

(3)

(4)

(5)

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

f yi

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

survey.

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

a(A)=aA,

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

equally long.

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-

plan.

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

Laussedat's phototheodolite.

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

XXVII.

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

horizontal plan.

3. THE ICONOMETRIC DETERMINATION OF ELEVATIONS OF PICTURED

TERRENE POINTS.

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(A):SA=a(a):Sa,

whence

Sa

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

sense.

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

field.

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

panorama.

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

PERSPECTIVE.

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

tan 30'

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

negative;

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

map construction.

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

station /;

1 11 = base line measured between the two camera stations /

and II.

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,

cO=f,

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.

whence They

were Derived.

Elevations of

Stations.

Difference of

Elevations.

Elevation of

Point.

Remarks.

D. General Arithmetical Method for Finding the Plotted Positions

oj Terrene Points when Pictured on Vertically Exposed

Picture Planes.

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',

d =

= 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

and as

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' +</)'

whence

sn

We can now compute from

H

tan/?=-p,

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,

we have

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-

~

and

=

cos r +y sn r

For the vertically exposed plates we had found

y oc

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

D sing'-ft

B -sin (Y <

hence

sin

PHOTOTOPOGRAPHIC SURVEYING METHODS. IIQ

H

we obtain

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

plate.

F. General Analytical Determination of the Elements of a

Photographic Perspective.

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-

dent that

(1) yiixi-YnXi-,

(2) yir-xu=Y2:X 2 ;

(3)

(4)

(5)

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

f yi

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

survey.

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

a(A)=aA,

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

equally long.

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-

plan.

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

Laussedat's phototheodolite.

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

XXVII.

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

horizontal plan.

3. THE ICONOMETRIC DETERMINATION OF ELEVATIONS OF PICTURED

TERRENE POINTS.

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(A):SA=a(a):Sa,

whence

Sa

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

sense.

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

field.

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

panorama.

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

PERSPECTIVE.

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

tan 30'

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

negative;

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

map construction.

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

station /;

1 11 = base line measured between the two camera stations /

and II.

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,

cO=f,

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.