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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To plot the direction from 5' to a point A, Fig. 36, Plate XX,.

pictured in MN as a, we first locate the orthogonal projection a a

of the pictured point a in the ground plane (plotting-plane).

We project the image point a, Fig. 37, Plate XXI, upon //' or

upon PP =o:, and describe a circle about P with P a=P (a)

to locate the position (a) of the projected point on the principal

line (/)(/')> revolved into the ground plane. (The positions

of the pictured points a in Figs. 36 and 37 do not correspond;

both should be on the same side of //' in the picture planes,

The perpendicular to 5'P , Fig. 37, Plate XXI, in a and the

vertical that passes through a intersect each other in a . The

point a , Fig. 36, Plate XX, is located on the plotting-sheet as the

intersection of (a )o (perpendicular to 5'P through ( ) -)

and (a)a Q (parallel with 5'P or with (/)(/') through (a) -).

S'a , Fig. 36, Plate XX, will be the horizontal projection

8o PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

(in the plan) of the line of direction (or radial) from S' to the

point A to be plotted.

C. Determination of the Altitudes oj Points pictured on an

Inclined Plate.

It is desired to find the elevation H of the point A, pictured

in MN as a, above the ground plane GG. With reference to

Fig. 36, Plate XX, the elevation aa =aa , in Fig. 37, Plate XXI,

corresponds to (a)ao.

If D= horizontal distance of the plotted point A from the

station S f (taken from the plotting-sheet), h=aao = aao = (a)aQ,

H = elevation of A above GG, and d=S'a (Fig. 36), taken from

the plotting-sheet, then the elevation H of the point A may be

found, either graphically from a diagram, Fig. 39, Plate XXIII,

>r it may be computed from the relation

D. Applications of Prof. Guido Hauck's Method.

The constructions described for locating the horizontal direc-

tions to points photographed on inclined plates may be greatly

simplified by applying Prof. Hauck's method, utilizing the prop-

erties of the " kernel points" of two photographs obtained from

different stations, but covering the same terrene.

In Fig. 38, Plate XXII, S and S' may represent the two camera

stations, S and So are the foot points of S and S' respectively,

MN and M'N' may represent the inclined picture planes, both

containing the images a and a', respectively, of a point A and the

pictures s' and 5 of the stations S' and S. The orthogonal pro-

jections of the pictured points a and a' in the ground plane are

o and (/ ^o is the orthogonal projection of A into the ground

plane GG. We had seen that 2, s', and n are "kernel points "

PHOTOGRAPHS ON INCLINED PLATES. 8l

for the picture plane MN and 2', s, and T^ are the "kernel

points " for M'N'.

The line connecting a and s* in MN and the line a's in M'N'

intersect each other in the same point Q of the line of inter-

section of the two pictures planes (MN and M'N'), and they

also intersect the ground lines gg' in n and K' respectively.

All lines in MN connecting s' with pictured points and those

in M'N' connecting s with the images in M'N' of the same points

will intersect each other in points Q of the line of intersection

("perspective axis ") of the picture planes.

The points 2 and 2 r (the intersections of the verticals passing

through the camera stations S and 5' with the inclined picture-

planes MN and M'N') are the vanishing points for the pictures

of all verticals shown in the negatives. Whenever the pictures

contain images of vertical lines, the intersections of their pictures-

would locate I and 2 r on MN and M'N' respectively ; still, when

the picture plane is inclined in such a way that the principal

line of the same would coincide with that of the vertically ex-

posed plate (if the former were revolved about a line as axis

passing through the second nodal point and being parallel with

the horizon line OO', or HH f ), the kernel point 2 may more

readily be located upon //', as previously shown for 5 in Fig. 37,

Plate XXI.

The horizontal direction SoA (S 'A Q ) intersects the ground

line gg' of MN (M'N') in a (respectively in a '), Fig. 38, Plate

XXII, In order to locate the position of A with reference

to a on MN (to a' on M'N') we connect a and 2 (also a' with

2 f )j which line locates a (and a ') upon the ground line gg f of

the picture plane MN (and M'N' respectively).

The interesction A of the lines Soa and S'a ' will now give

the plotted position in the ground plane GG of the point A.

CHAPTER VI.

PHOTOTOPOGRAPHIC SURVEYING METHODS.

FROM' the preceding chapters we find that in order to utilize

for iconometric purposes the data contained in a photographic

perspective we, should know:

First. The three constants or elements of the perspective,

which are the focal length, together with the horizon and

the principal line, or the focal length and the principal

point, together with either the horizon line or the principal

line of the perspective.

Second. The position of the picture plane with reference to

fixed points of the terrene, which means the elements for

the orientation of the picture trace in the plotting-plane.

To plot the position of any geodetic point in both the horizontal

and in the vertical sense, we should know, or be able to ascertain,

First. The horizontal angles included between the principal

plane and the lines of direction from two or more stations

to the geodetic point.

Second. The angle of elevation (or depression) which is the

vertical angle included between the horizon plane and the

line of direction to the geodetic point.

If the constants or the elements of the perspective are known,

the geodetic elements (the horizontal and vertical a'ngles) needed

for plotting the position of the geodetic point may be ascertained

either graphically or arithmetically.

Phototopographic methods being generally applied with a view

toward obtaining a graphic record of the measurements in the

82

PHOTOTOPOGRAPHIC SURVEYING METHODS. 83

form of cartographic representation of the terrene, we shall give

in these pages principally graphic solutions of the more important

problems met with in phototopography.

I. Analytical or Arithmetical Phototopographic Methods.

A. Method oj Pro}. Jordan.

In Chapter I, section IH, mention has been made of Prof.

Jordan's map of the oasis "Dachel " and village "Gassr-Dachel,"

based on Remele's photographs. Care was exercised to expose

the plates in vertical plane, and horizontal directions to at least

three points of each photograph were measured instrumentally

to obtain the required data for the orientation of the pictures.

Vertical angles to at least two such points for every picture were

also observed to give the means for locating the horizon lines

of the pictures, thus enabling the draughtsman to deduce the

elevations of other points pictured on the photographs. With

reference to Fig. 40, Plate XXIII, we have:

OO' = horizon line of photographic perspective MN\

//' = principal line ;

P = principal point ;

5 = second nodal point (focus) of camera lens;

SP = } = focal length of picture MN = principal ray;

a, b, and c = images of three points A, B, and C;

i, 2 > and a 3 = horizontal angles a'SP, b'SP, and c'SP\

SN= direction of the meridian passing through the

station 5;

<j>\ fa, and (f>3 = azimuthal angles NSa', NSb', and NSc' re-

spectively;

HI, H 2 , and H 3 = elevations of the points A, B, and C above the

plane of reference or ground plane.

The photographic plate MN having been exposed in vertical

plane, it will be evident that for the three points a, b, and c

respectively the abscissae #1, x 2 , and x 3 should be

84 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

Xi=j tan i,

x 2 = t tan a 2

xz = j tan 0:3,

f sin (0:20:1)

or x 2 #i=/(tano:2 tanai)=; - -

1 coscti cosa 2

, , /<t x , sin -

and * 3 -* 2 = /(tan a 3 -tan ,)/.

The values for (x 2 Xi) and (#3 #2) may be scaled off directly

on the negative, MN, and the values for (a. 2 ai) and (0:3 0:2)

may be taken from the field records of the observed horizontal

angles, when the value for ^ may be computed by means

of the formula

%2Xi cos 0:3 sin (a 2 -fli)

#3^2 cos ai sin (0:3 a 2 )'

cos 0:3

If we substitute tan r for - , and as

cos a i

i +tan

we may write

tan (45 + r)

_ cos o; 3 cos ai- cos o: 3

cos i

cos 5I5L cos 21=2!

0:1+0:3 (OL\ a3\

hence tan =cot (45+7*) cot 1 ).

PHOTOTOPOGRAPHIC SURVEYING METHODS. 8$

From this equation i +0:3 may be computed.

By inspection we find from Fig. 40, Plate XXIII,

3 2 = 03 - 02 = 2,

2~ 0:1 =02 01 = l.

By adding these two equations we obtain

i -0:3 = 02 03-

Knowing (i +ct 3 ) and (i 3 ) we can readily find i and 3 ;

also ct2 =

or =0:3 2-

We had found

. sin (2 i) t sm *i

X 2 Xi=f- - /- ;

7 cos 0:1 cos a 2 cos i cos az

(x2%i) cos 0:1 cos o: 2

hence * = : ,

sm 1

, sin (0:3-0:2) 9 sin 2

and ^a~^

whence /=

COS 3 COS 2 COS O: 3 COS 2

(^3^2) cos 0:3 cos a 2

sm

Thus two elements of the perspective MN, the focal length /,

and the principal line //' (given by the abscissae Xi, x 2 , and #3),

may be found.

With the aid of the observed vertical angles /?, the third ele-

ment, the horizon line OO', may now be located on the photo-

graph.

86 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

The vertical angle fo = cSc' having been observed at S to

the point C we find

CC ' = y 3 = Sc' tan #, - - - tan /? 3 ,

and for the point a the vertical distance to the horizon line would

be

aai = y l = Sa f tan ft = ^^- tan ft.

The horizon line OO f will be the common tangent to two

circles, one described with the radius = - tan ft about a

cosai

and the other with a radius = - tan 3 3 about c.

cos a: 3

At least two vertical angles having been observed for each

exposed plate, the horizon line OO' may thus be located and

marked upon the negative, when the principal point P may

also be located on OO' by means of the principal line //', the

latter being tangent to the three circles described about a, b t

and c with the radii x\, x 2 , and #3 respectively.

B. Method oj Dr. Le Bon.

Dr. G. Le Bon (who used his instrument chiefly for the plot-

ting of ancient buildings and monuments in India) provided

the ground-glass plate of his camera with a net of squares, each

square having i cm. sides, one set of the latter being drawn

parallel with the horizon, while the second set of lines is paral-

lel with the principal line of the perspective. The lines repre-

senting the horizon and principal lines are again subdivided into

millimeters.

This arrangement enables the operator to obtain the measure-

ments of objects directly by inspection of the image on the gradu-

ated ground-glass plate.

PHOTOTOPOGRAPHIC SURVEYING METHODS. 87

To determine the dimensions of the front of a building Dr. Le

Bon measures a certain length directly upon the same and then

takes a picture by exposing a photographic plate in vertical

plane and parallel with the base of the front (facade) of the

building.

For example, to find

First. The distance D of an object, the height H of which

is not known, Fig. 41, Plate XXIII:

Two stations S and S' are occupied on a base line B (which

is measured directly in the field) laid off in a direction perpen-

dicular to the base of the object.

If the height of the image measured on the ground glass

at the first station is h, at the second station h', and if the focal

length for both exposures be the same and=/, then

and for the second station 5'

h and h' being known they may be measured directly on the

negative or on the ground-glass plate we find, after dividing

the second equation by the first,

D+B h

B h h-h'

or

whence

D~W h'

Bh'

88 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

Second. The height H of an object is to be found when

the fractional length H' has been obtained by direct

measurement (Fig. 42, Plate XXIV).

On the image of the object on the graduated ground-glass

plate the lengths for the heights h and hf may be read off directly,

and as H r is also known, we find H from the equation

H'

-u. h ,.

C. Method of L. P. Paganini (Italian Method).

This method has been extensively used for the new topo-

graphic survey of the kingdom of Italy and for the Colonial

possessions in East Africa (" Eritrea").

COMPUTATION OF THE ELEMENTS OF THE PHOTOGRAPHIC

PERSPECTIVE WHICH ARE NEEDED FOR THE

ICONOMETRIC PLOTTING.

i. DETERMINATION OF THE FOCAL LENGTH or THE PHOTOGRAPHIC

PERSPECTIVE.

(a) When the Reference Point is Bisected by the Principal Line of the

Perspective.

A triangulation point S, Fig. 121, Plate LX, may be visible

from the camera station V. The camera is directed toward 5

in such manner that the image s of the distant peak S is bisected

by the vertical thread //'.

V camera station or the point of view of the perspective

MN-

PI = principal point of the photograph;

VPi = focal length or distance line of the perspective, denoted

by/;

PHOTOTOPOGRAPHIC SURVEYING METHODS. 89

5' = orthogonal projection of 5 in the horizontal plane which

passes through F;

F5i' = horizontal distance from V to 5, designated by D\

55' = apparent difference in elevation between V and 5, des-

ignated by L.

After having carefully measured the ordinate Ps=y on the

negative, we can determine the focal length from the equation

Example No. I. The station V may be occupied over the

centre of the triangulation point Reale Accampamento and the

bisected point 5 be the signal upon Cian del Lei. The camera

having been leveled and adjusted over Reale Accampamento is

turned in azimuth until the signal Cian del Lei is bisected by the

vertical thread //' and the first plate is then exposed (Fig. 122,

Plate LXI).

The focal distance, read off on the scale a of the lens tube

= 244.5 mm -> an d the values for D and L, taken from the records

of the new trigonometrical survey of Italy, are

D= distance from Reale Accampamento (signal) to

Punta Cian del Lei (signal) =3270.7 m.

Elevation of station mark at Punta Cian del Lei =2811.72 m.

Elevation of camera horizon at Reale Accampamento = 2 191.80 m.

Difference of true elevation = 619.92 m.

The ordinate y, carefully measured on the negative (from the

principal point P to the image s of the point Cian del Lei) gave

46.25 mm.

Computation of L :

True difference in elevation =619.92 m.

Correction for curvature and refraction = 0.72 m.

Z, = apparent difference in elevation = 619. 20 m.

90 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

Computation of /:

logZ> = log 3270.7 =3.5146407

log y= log 0.04625=8.6651117

colog L = colog 619.20 =7.2081691

Scale-reading for / was = 244. 50

Difference = 0.20 mm.

(b) The Image of the Reference Point Falls to either Side of the Principal

Line of the Photographic Perspective.

If the image 5 of the reference point S be to either side of

the vertical thread //' of the perspective MN, Fig. 123, Plate

LX, the principal ray VP making an angle e with the horizontal

direction VS' to the reference point s, then the value VP = f may

be found as follows :

d= horizontal distance Vs f (Fig. 123);

y and x coordinates of the image s\

D = horizontal distance between the camera station V and the

reference point 5, and

L= apparent difference in elevation between V (or s f ) and s.

From the similar triangles VSS' and Vss' we find

_

D d'

hence

PHOTOTOPOGRAPHIC SURVEYING METHODS. 91

From the horizontal triangle s'P V we find

d= :

hence /= cos e.

Example No. II. In the panorama (series of ten photographic

perspectives) obtained September 21, 1884, vertically above the

trigonometrical point (of the new Italian geodetic triangulation

system), near Reale Accampamento of Valsavaranche, there is

one plate (P 5 , Fig. 122, Plate LXI) which contains the image

of the triangulation station Punta Ruja (signal).

The horizontal angle cu, between the optical axis of the camera

for this plate and the horizontal direction to Ruja (signal), is =

5 49' 2 7 ". 75 .

The horizontal distance: Reale Accampamento Ruja = Z> =

5804.2 m. and the elevation of Ruja = 31 73. 5 m. are taken from

the triangulation data.

By careful measurement y is found to be =41 .45 mm.

It is desired to find the focal length, /, for this perspective,

which may be obtained, approximately, by reading the gradua-

tion on the objective tube = 244. 50 mm,

Computation of L :

Elevation of station mark at Ruja = 3 I 73-5 na-

Elevation of the camera horizon (OO') = 2i9i.8 m.

True difference in elevation = 981.7 m.

Correction for refraction and curvature = 2.3 m.

Apparent difference in elevation = 979.4 =L

92 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

Computation of /:

log D = log 5804.2 m. =3.7637424

log ;y = log 0.04145 m. =8.6175245

log cos w = log cos 5 49' 27^.75 =9.9977522

colog L = colog 979.4 = 7.0090399

log/ = 2.3880590

/ = 244.38 mm.

Scale reading = 2 44. 50 mm.

Difference = 0.12 mm.

Had we measured the abscissa x instead of the ordinate y

the focal length for the negative could have been computed by

the formula

/ = x cot co.

Example No. III. The measured value for x may have been

found to be =24.90 mm.

Computation of /:

log * = log 24.90 =1.3961993

log cot w = log cot 5 49' 27^.75=0.9913737

log / = 2.3875730

/ = 244. 1 03 mm.

2. ORIENTATION OF THE PICTURE TRACES.

There exists a close connection between the phototopographic

stations and the new triangulation of Italy. A generous dis-

position of the trigonometric points had been made with the

special purpose in view that they were to serve as the foundation

for the subsequent topographical survey. These points have

PHOTOTOPOGRAPHIC SURVEYING METHODS. 93

been carefully selected, their positions have been precisely com-

puted, and their locations have been permanently marked in the

field, irrespective of the character of the surrounding topography

or of the order of triangulation to which the point may belong.

This large number of triangulation points not only facilitates

the application of the phototopographic surveying method and

assures the accurate determination of the panorama stations

(in the horizontal and vertical sense), but it also greatly simplifies

the subsequent iconometric plotting, as the greater part of the

perspective contains one, two, or more pictured triangulation

points, notwithstanding the instrument commands a field of view

of but 42 horizontally. Two adjoining plates have a common

margin of an angular width of 3, reducing the effective field of

view of one plate to 36 (Fig. 124, Plate LXII).

Thus the picture traces are easily oriented for the icono-

metric work, the salient topographic features (deduced from

the perspectives) may be frequently checked, and such nega-

tives (containing the images of triangulation points) may also

serve to verify the focal length / of the panorama pictures, check

the position of the principal point P, and they give the means

for testing the location of the horizon line OO' on the pictures.

The perspective MN, Fig. 125, Plate LXII, may contain

the images of two trigonometrical points S and 5'.

In the preceding pages it has been shown that the horizontal

distances, d and d', from the camera station V to the pictured

points S and S' may be found from the relations

In the triangle VS\Si' we know the lengths of two sides, d and

d* , and also the value of the included angle, SiVSi', which may

be either measured directly at the camera station or, when the

latter is also a triangulation station, the value for the angle may

be taken from the triangulation records.

94 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

The other two angles, 7- and d, of this triangle may now be

found as follows :

r-d d'-d V

tan L - = , , cot - ,

2 d'+d 2

= I 8o, hence

If we replace i(r + ^) by M and J(r~^) by AT, we find after

adding the equations

and by subtracting the equations

The principal ray VP should be vertical to the horizon line OO f

and both triangles VPSi and VPSi' should be right-angle triangles.

Hence the focal length / should be

and /=d'-sin.

To ascertain whether the pictured intersection of the cross-

wires P coincides with the principal point of view P upon the

perspective, the measured lengths of the abscissae oc and oc f

(Fig. 125, Plate LXII) should be the same as the computed

values

# = /cot 7,

oc f =fcotd.

The angles of orientation a> and a/ are:

PHOTOTOPOGRAPSIC SURVEYING METHODS.

95

3. DETERMINATION OF THE ELEVATIONS OF PICTURED TERRENE POINTS.

The vertical angles of elevation a and a' of the two refer-

ence points S and S f may be computed from the equations:

tana =-=:,

tana' =

These angles are either taken from the triangulation records

or they may be observed directly from the camera station, and

to check the position of the horizon line OO f the ordinates

y and /, measured on the perspective, are compared with those

computed by means of the equations

COS (Jt)

tana,

cos a>'

tan a'.

Example No. IV. In the panorama obtained Sept. 19, 1884,

from Punta Percia (this peak is on the divide separating the

valleys of the Rhemes and the Valsavaranche) two trigonomet-

rical stations, Punta Rouletta and Gran Punta di Nomenon, of

the new Italian geodetic survey appear upon the same plate (see

Fig. 126, Plate LXIII).

The following values are given for the computation:

Elevation of Punta Rouletta

Elevation of Punta di Nomenon

= 3384. i om. f Taken from the

j catalogue of tri-

= 3488.42 m. [ angulation points

[Elevation of P.

Elevation of horizon of camera station (P. Percia) = 3202. 3 m. \ Percia + height

[ of instrument

Measured upon

= 3250 m. the projection of

the iconometric

= 9720 m. working - sheet,

scale

Distance : Percia.- Rouletta= D

Distance : Percia-Nomenon= D f

96 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

The horizontal angle V (Fig. 126, Plate LXIII) at Punta

Percia, included by the horizontal directions to Rouletta (signal)

and Nomenon (signal), =28 02' 30".

A careful measurement of the coordinates of the pictured

points P. Rouletta and P. Nomenon, on the negative with

a millimeter scale provided with a microscope and vernier, en-

abling the computer to read to 0.05 mm. (the vernier is divided

to read to 1/20 of the graduation unit), produced the following

values :

The coordinates of Punta Rouletta,

# = 46.05 mm.; ^ = 13. 7 5 mm.

The coordinates of Punta di Nomenon,

^ = 75.40 mm.; / = 7.3omm.

It is desired to find:

(1) The focal distance for this negative = /, the preliminary

value, read off on the scale attached to the objective cylinder, is

found to be 244.50 mm.

(2) The correct position of the principal point (P), which

will be fixed by the determination of the abscissae x and oc*.

(3) The position of the line of horizon OO', which will be

located by ascertaining the values for y and /.

Computation to determine the apparent differences in elevation

between the camera horizon (P. Percia) and the two pictured points

P. Rouletta and Punta di Nomenon.

Altitude of P. Rouletta =3384.10 m.

Altitude of camera horizon =3202 .30 m.

True difference in elevation = 181 .80 m.

Correction for curvature and refraction = 0.71 m.

Apparent difference in elevation =L = 181 .09 m.

Altitude of Punta di Nomenon =3488.42 m.

Altitude of camera horizon =3202 .30 m.

PHOTOTOPOGRAPHIC SURVEYING METHODS. 97

True difference in elevation = 286 . 1 2 m.

Correction for curvature and refraction = - 6 . 35 m.

Apparent difference in elevation = Z/ = 279.77

Computation of d= y

log D= 3.5 1 18834

logy =8. 1383027

colog 1=7.7421055

log d= 9.3922916

d= 246.77 mm.

Computation of d'=

log D'= 3.9876663

log /= 7.8633229

colog L'= 7.5531989

log d'= 9.4041881

d'= 253.62 mm.

d+d'= 500.39

d'-d= 6.85

Computation of the angles 7- and d:

r -d d'-d V

~

2 d f + d 2

28 02' 30"; - = i4oi'i5";

>

57' 3o"; ^ = 75 58' 45"=^-

log (d'-d)= log 0.00685 =7.8356906

log cot - = logcoti4 01' 15'' =0.6025567

colog (d'+d} = colog 0.50039 =0.3006914

log tan ^^ = 8 - 7389387

r =7 08' i6".i=N.

Hence

50

98 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

(a) Computation of the Focal Length (=/).

/ = d - sin 7- / = d f sin d

log ^ = 9. 3922916 log d' = 9. 4041881

log sin f g. 9921180 log sin = 9. 9802269

lg / = 9 - 3 8 44096 log / = 9 . 38441 50

l=o. 242331 m. /=o. 242334 m.

Mean value for / = 242. 33 2 mm.

{/?) Computation of the Abscisses (x and x') for Platting Lines of Hori-

zontal Directions to Pictured Points of the Terrene and for Checking

pictured in MN as a, we first locate the orthogonal projection a a

of the pictured point a in the ground plane (plotting-plane).

We project the image point a, Fig. 37, Plate XXI, upon //' or

upon PP =o:, and describe a circle about P with P a=P (a)

to locate the position (a) of the projected point on the principal

line (/)(/')> revolved into the ground plane. (The positions

of the pictured points a in Figs. 36 and 37 do not correspond;

both should be on the same side of //' in the picture planes,

The perpendicular to 5'P , Fig. 37, Plate XXI, in a and the

vertical that passes through a intersect each other in a . The

point a , Fig. 36, Plate XX, is located on the plotting-sheet as the

intersection of (a )o (perpendicular to 5'P through ( ) -)

and (a)a Q (parallel with 5'P or with (/)(/') through (a) -).

S'a , Fig. 36, Plate XX, will be the horizontal projection

8o PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

(in the plan) of the line of direction (or radial) from S' to the

point A to be plotted.

C. Determination of the Altitudes oj Points pictured on an

Inclined Plate.

It is desired to find the elevation H of the point A, pictured

in MN as a, above the ground plane GG. With reference to

Fig. 36, Plate XX, the elevation aa =aa , in Fig. 37, Plate XXI,

corresponds to (a)ao.

If D= horizontal distance of the plotted point A from the

station S f (taken from the plotting-sheet), h=aao = aao = (a)aQ,

H = elevation of A above GG, and d=S'a (Fig. 36), taken from

the plotting-sheet, then the elevation H of the point A may be

found, either graphically from a diagram, Fig. 39, Plate XXIII,

>r it may be computed from the relation

D. Applications of Prof. Guido Hauck's Method.

The constructions described for locating the horizontal direc-

tions to points photographed on inclined plates may be greatly

simplified by applying Prof. Hauck's method, utilizing the prop-

erties of the " kernel points" of two photographs obtained from

different stations, but covering the same terrene.

In Fig. 38, Plate XXII, S and S' may represent the two camera

stations, S and So are the foot points of S and S' respectively,

MN and M'N' may represent the inclined picture planes, both

containing the images a and a', respectively, of a point A and the

pictures s' and 5 of the stations S' and S. The orthogonal pro-

jections of the pictured points a and a' in the ground plane are

o and (/ ^o is the orthogonal projection of A into the ground

plane GG. We had seen that 2, s', and n are "kernel points "

PHOTOGRAPHS ON INCLINED PLATES. 8l

for the picture plane MN and 2', s, and T^ are the "kernel

points " for M'N'.

The line connecting a and s* in MN and the line a's in M'N'

intersect each other in the same point Q of the line of inter-

section of the two pictures planes (MN and M'N'), and they

also intersect the ground lines gg' in n and K' respectively.

All lines in MN connecting s' with pictured points and those

in M'N' connecting s with the images in M'N' of the same points

will intersect each other in points Q of the line of intersection

("perspective axis ") of the picture planes.

The points 2 and 2 r (the intersections of the verticals passing

through the camera stations S and 5' with the inclined picture-

planes MN and M'N') are the vanishing points for the pictures

of all verticals shown in the negatives. Whenever the pictures

contain images of vertical lines, the intersections of their pictures-

would locate I and 2 r on MN and M'N' respectively ; still, when

the picture plane is inclined in such a way that the principal

line of the same would coincide with that of the vertically ex-

posed plate (if the former were revolved about a line as axis

passing through the second nodal point and being parallel with

the horizon line OO', or HH f ), the kernel point 2 may more

readily be located upon //', as previously shown for 5 in Fig. 37,

Plate XXI.

The horizontal direction SoA (S 'A Q ) intersects the ground

line gg' of MN (M'N') in a (respectively in a '), Fig. 38, Plate

XXII, In order to locate the position of A with reference

to a on MN (to a' on M'N') we connect a and 2 (also a' with

2 f )j which line locates a (and a ') upon the ground line gg f of

the picture plane MN (and M'N' respectively).

The interesction A of the lines Soa and S'a ' will now give

the plotted position in the ground plane GG of the point A.

CHAPTER VI.

PHOTOTOPOGRAPHIC SURVEYING METHODS.

FROM' the preceding chapters we find that in order to utilize

for iconometric purposes the data contained in a photographic

perspective we, should know:

First. The three constants or elements of the perspective,

which are the focal length, together with the horizon and

the principal line, or the focal length and the principal

point, together with either the horizon line or the principal

line of the perspective.

Second. The position of the picture plane with reference to

fixed points of the terrene, which means the elements for

the orientation of the picture trace in the plotting-plane.

To plot the position of any geodetic point in both the horizontal

and in the vertical sense, we should know, or be able to ascertain,

First. The horizontal angles included between the principal

plane and the lines of direction from two or more stations

to the geodetic point.

Second. The angle of elevation (or depression) which is the

vertical angle included between the horizon plane and the

line of direction to the geodetic point.

If the constants or the elements of the perspective are known,

the geodetic elements (the horizontal and vertical a'ngles) needed

for plotting the position of the geodetic point may be ascertained

either graphically or arithmetically.

Phototopographic methods being generally applied with a view

toward obtaining a graphic record of the measurements in the

82

PHOTOTOPOGRAPHIC SURVEYING METHODS. 83

form of cartographic representation of the terrene, we shall give

in these pages principally graphic solutions of the more important

problems met with in phototopography.

I. Analytical or Arithmetical Phototopographic Methods.

A. Method oj Pro}. Jordan.

In Chapter I, section IH, mention has been made of Prof.

Jordan's map of the oasis "Dachel " and village "Gassr-Dachel,"

based on Remele's photographs. Care was exercised to expose

the plates in vertical plane, and horizontal directions to at least

three points of each photograph were measured instrumentally

to obtain the required data for the orientation of the pictures.

Vertical angles to at least two such points for every picture were

also observed to give the means for locating the horizon lines

of the pictures, thus enabling the draughtsman to deduce the

elevations of other points pictured on the photographs. With

reference to Fig. 40, Plate XXIII, we have:

OO' = horizon line of photographic perspective MN\

//' = principal line ;

P = principal point ;

5 = second nodal point (focus) of camera lens;

SP = } = focal length of picture MN = principal ray;

a, b, and c = images of three points A, B, and C;

i, 2 > and a 3 = horizontal angles a'SP, b'SP, and c'SP\

SN= direction of the meridian passing through the

station 5;

<j>\ fa, and (f>3 = azimuthal angles NSa', NSb', and NSc' re-

spectively;

HI, H 2 , and H 3 = elevations of the points A, B, and C above the

plane of reference or ground plane.

The photographic plate MN having been exposed in vertical

plane, it will be evident that for the three points a, b, and c

respectively the abscissae #1, x 2 , and x 3 should be

84 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

Xi=j tan i,

x 2 = t tan a 2

xz = j tan 0:3,

f sin (0:20:1)

or x 2 #i=/(tano:2 tanai)=; - -

1 coscti cosa 2

, , /<t x , sin -

and * 3 -* 2 = /(tan a 3 -tan ,)/.

The values for (x 2 Xi) and (#3 #2) may be scaled off directly

on the negative, MN, and the values for (a. 2 ai) and (0:3 0:2)

may be taken from the field records of the observed horizontal

angles, when the value for ^ may be computed by means

of the formula

%2Xi cos 0:3 sin (a 2 -fli)

#3^2 cos ai sin (0:3 a 2 )'

cos 0:3

If we substitute tan r for - , and as

cos a i

i +tan

we may write

tan (45 + r)

_ cos o; 3 cos ai- cos o: 3

cos i

cos 5I5L cos 21=2!

0:1+0:3 (OL\ a3\

hence tan =cot (45+7*) cot 1 ).

PHOTOTOPOGRAPHIC SURVEYING METHODS. 8$

From this equation i +0:3 may be computed.

By inspection we find from Fig. 40, Plate XXIII,

3 2 = 03 - 02 = 2,

2~ 0:1 =02 01 = l.

By adding these two equations we obtain

i -0:3 = 02 03-

Knowing (i +ct 3 ) and (i 3 ) we can readily find i and 3 ;

also ct2 =

or =0:3 2-

We had found

. sin (2 i) t sm *i

X 2 Xi=f- - /- ;

7 cos 0:1 cos a 2 cos i cos az

(x2%i) cos 0:1 cos o: 2

hence * = : ,

sm 1

, sin (0:3-0:2) 9 sin 2

and ^a~^

whence /=

COS 3 COS 2 COS O: 3 COS 2

(^3^2) cos 0:3 cos a 2

sm

Thus two elements of the perspective MN, the focal length /,

and the principal line //' (given by the abscissae Xi, x 2 , and #3),

may be found.

With the aid of the observed vertical angles /?, the third ele-

ment, the horizon line OO', may now be located on the photo-

graph.

86 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

The vertical angle fo = cSc' having been observed at S to

the point C we find

CC ' = y 3 = Sc' tan #, - - - tan /? 3 ,

and for the point a the vertical distance to the horizon line would

be

aai = y l = Sa f tan ft = ^^- tan ft.

The horizon line OO f will be the common tangent to two

circles, one described with the radius = - tan ft about a

cosai

and the other with a radius = - tan 3 3 about c.

cos a: 3

At least two vertical angles having been observed for each

exposed plate, the horizon line OO' may thus be located and

marked upon the negative, when the principal point P may

also be located on OO' by means of the principal line //', the

latter being tangent to the three circles described about a, b t

and c with the radii x\, x 2 , and #3 respectively.

B. Method oj Dr. Le Bon.

Dr. G. Le Bon (who used his instrument chiefly for the plot-

ting of ancient buildings and monuments in India) provided

the ground-glass plate of his camera with a net of squares, each

square having i cm. sides, one set of the latter being drawn

parallel with the horizon, while the second set of lines is paral-

lel with the principal line of the perspective. The lines repre-

senting the horizon and principal lines are again subdivided into

millimeters.

This arrangement enables the operator to obtain the measure-

ments of objects directly by inspection of the image on the gradu-

ated ground-glass plate.

PHOTOTOPOGRAPHIC SURVEYING METHODS. 87

To determine the dimensions of the front of a building Dr. Le

Bon measures a certain length directly upon the same and then

takes a picture by exposing a photographic plate in vertical

plane and parallel with the base of the front (facade) of the

building.

For example, to find

First. The distance D of an object, the height H of which

is not known, Fig. 41, Plate XXIII:

Two stations S and S' are occupied on a base line B (which

is measured directly in the field) laid off in a direction perpen-

dicular to the base of the object.

If the height of the image measured on the ground glass

at the first station is h, at the second station h', and if the focal

length for both exposures be the same and=/, then

and for the second station 5'

h and h' being known they may be measured directly on the

negative or on the ground-glass plate we find, after dividing

the second equation by the first,

D+B h

B h h-h'

or

whence

D~W h'

Bh'

88 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

Second. The height H of an object is to be found when

the fractional length H' has been obtained by direct

measurement (Fig. 42, Plate XXIV).

On the image of the object on the graduated ground-glass

plate the lengths for the heights h and hf may be read off directly,

and as H r is also known, we find H from the equation

H'

-u. h ,.

C. Method of L. P. Paganini (Italian Method).

This method has been extensively used for the new topo-

graphic survey of the kingdom of Italy and for the Colonial

possessions in East Africa (" Eritrea").

COMPUTATION OF THE ELEMENTS OF THE PHOTOGRAPHIC

PERSPECTIVE WHICH ARE NEEDED FOR THE

ICONOMETRIC PLOTTING.

i. DETERMINATION OF THE FOCAL LENGTH or THE PHOTOGRAPHIC

PERSPECTIVE.

(a) When the Reference Point is Bisected by the Principal Line of the

Perspective.

A triangulation point S, Fig. 121, Plate LX, may be visible

from the camera station V. The camera is directed toward 5

in such manner that the image s of the distant peak S is bisected

by the vertical thread //'.

V camera station or the point of view of the perspective

MN-

PI = principal point of the photograph;

VPi = focal length or distance line of the perspective, denoted

by/;

PHOTOTOPOGRAPHIC SURVEYING METHODS. 89

5' = orthogonal projection of 5 in the horizontal plane which

passes through F;

F5i' = horizontal distance from V to 5, designated by D\

55' = apparent difference in elevation between V and 5, des-

ignated by L.

After having carefully measured the ordinate Ps=y on the

negative, we can determine the focal length from the equation

Example No. I. The station V may be occupied over the

centre of the triangulation point Reale Accampamento and the

bisected point 5 be the signal upon Cian del Lei. The camera

having been leveled and adjusted over Reale Accampamento is

turned in azimuth until the signal Cian del Lei is bisected by the

vertical thread //' and the first plate is then exposed (Fig. 122,

Plate LXI).

The focal distance, read off on the scale a of the lens tube

= 244.5 mm -> an d the values for D and L, taken from the records

of the new trigonometrical survey of Italy, are

D= distance from Reale Accampamento (signal) to

Punta Cian del Lei (signal) =3270.7 m.

Elevation of station mark at Punta Cian del Lei =2811.72 m.

Elevation of camera horizon at Reale Accampamento = 2 191.80 m.

Difference of true elevation = 619.92 m.

The ordinate y, carefully measured on the negative (from the

principal point P to the image s of the point Cian del Lei) gave

46.25 mm.

Computation of L :

True difference in elevation =619.92 m.

Correction for curvature and refraction = 0.72 m.

Z, = apparent difference in elevation = 619. 20 m.

90 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

Computation of /:

logZ> = log 3270.7 =3.5146407

log y= log 0.04625=8.6651117

colog L = colog 619.20 =7.2081691

Scale-reading for / was = 244. 50

Difference = 0.20 mm.

(b) The Image of the Reference Point Falls to either Side of the Principal

Line of the Photographic Perspective.

If the image 5 of the reference point S be to either side of

the vertical thread //' of the perspective MN, Fig. 123, Plate

LX, the principal ray VP making an angle e with the horizontal

direction VS' to the reference point s, then the value VP = f may

be found as follows :

d= horizontal distance Vs f (Fig. 123);

y and x coordinates of the image s\

D = horizontal distance between the camera station V and the

reference point 5, and

L= apparent difference in elevation between V (or s f ) and s.

From the similar triangles VSS' and Vss' we find

_

D d'

hence

PHOTOTOPOGRAPHIC SURVEYING METHODS. 91

From the horizontal triangle s'P V we find

d= :

hence /= cos e.

Example No. II. In the panorama (series of ten photographic

perspectives) obtained September 21, 1884, vertically above the

trigonometrical point (of the new Italian geodetic triangulation

system), near Reale Accampamento of Valsavaranche, there is

one plate (P 5 , Fig. 122, Plate LXI) which contains the image

of the triangulation station Punta Ruja (signal).

The horizontal angle cu, between the optical axis of the camera

for this plate and the horizontal direction to Ruja (signal), is =

5 49' 2 7 ". 75 .

The horizontal distance: Reale Accampamento Ruja = Z> =

5804.2 m. and the elevation of Ruja = 31 73. 5 m. are taken from

the triangulation data.

By careful measurement y is found to be =41 .45 mm.

It is desired to find the focal length, /, for this perspective,

which may be obtained, approximately, by reading the gradua-

tion on the objective tube = 244. 50 mm,

Computation of L :

Elevation of station mark at Ruja = 3 I 73-5 na-

Elevation of the camera horizon (OO') = 2i9i.8 m.

True difference in elevation = 981.7 m.

Correction for refraction and curvature = 2.3 m.

Apparent difference in elevation = 979.4 =L

92 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

Computation of /:

log D = log 5804.2 m. =3.7637424

log ;y = log 0.04145 m. =8.6175245

log cos w = log cos 5 49' 27^.75 =9.9977522

colog L = colog 979.4 = 7.0090399

log/ = 2.3880590

/ = 244.38 mm.

Scale reading = 2 44. 50 mm.

Difference = 0.12 mm.

Had we measured the abscissa x instead of the ordinate y

the focal length for the negative could have been computed by

the formula

/ = x cot co.

Example No. III. The measured value for x may have been

found to be =24.90 mm.

Computation of /:

log * = log 24.90 =1.3961993

log cot w = log cot 5 49' 27^.75=0.9913737

log / = 2.3875730

/ = 244. 1 03 mm.

2. ORIENTATION OF THE PICTURE TRACES.

There exists a close connection between the phototopographic

stations and the new triangulation of Italy. A generous dis-

position of the trigonometric points had been made with the

special purpose in view that they were to serve as the foundation

for the subsequent topographical survey. These points have

PHOTOTOPOGRAPHIC SURVEYING METHODS. 93

been carefully selected, their positions have been precisely com-

puted, and their locations have been permanently marked in the

field, irrespective of the character of the surrounding topography

or of the order of triangulation to which the point may belong.

This large number of triangulation points not only facilitates

the application of the phototopographic surveying method and

assures the accurate determination of the panorama stations

(in the horizontal and vertical sense), but it also greatly simplifies

the subsequent iconometric plotting, as the greater part of the

perspective contains one, two, or more pictured triangulation

points, notwithstanding the instrument commands a field of view

of but 42 horizontally. Two adjoining plates have a common

margin of an angular width of 3, reducing the effective field of

view of one plate to 36 (Fig. 124, Plate LXII).

Thus the picture traces are easily oriented for the icono-

metric work, the salient topographic features (deduced from

the perspectives) may be frequently checked, and such nega-

tives (containing the images of triangulation points) may also

serve to verify the focal length / of the panorama pictures, check

the position of the principal point P, and they give the means

for testing the location of the horizon line OO' on the pictures.

The perspective MN, Fig. 125, Plate LXII, may contain

the images of two trigonometrical points S and 5'.

In the preceding pages it has been shown that the horizontal

distances, d and d', from the camera station V to the pictured

points S and S' may be found from the relations

In the triangle VS\Si' we know the lengths of two sides, d and

d* , and also the value of the included angle, SiVSi', which may

be either measured directly at the camera station or, when the

latter is also a triangulation station, the value for the angle may

be taken from the triangulation records.

94 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

The other two angles, 7- and d, of this triangle may now be

found as follows :

r-d d'-d V

tan L - = , , cot - ,

2 d'+d 2

= I 8o, hence

If we replace i(r + ^) by M and J(r~^) by AT, we find after

adding the equations

and by subtracting the equations

The principal ray VP should be vertical to the horizon line OO f

and both triangles VPSi and VPSi' should be right-angle triangles.

Hence the focal length / should be

and /=d'-sin.

To ascertain whether the pictured intersection of the cross-

wires P coincides with the principal point of view P upon the

perspective, the measured lengths of the abscissae oc and oc f

(Fig. 125, Plate LXII) should be the same as the computed

values

# = /cot 7,

oc f =fcotd.

The angles of orientation a> and a/ are:

PHOTOTOPOGRAPSIC SURVEYING METHODS.

95

3. DETERMINATION OF THE ELEVATIONS OF PICTURED TERRENE POINTS.

The vertical angles of elevation a and a' of the two refer-

ence points S and S f may be computed from the equations:

tana =-=:,

tana' =

These angles are either taken from the triangulation records

or they may be observed directly from the camera station, and

to check the position of the horizon line OO f the ordinates

y and /, measured on the perspective, are compared with those

computed by means of the equations

COS (Jt)

tana,

cos a>'

tan a'.

Example No. IV. In the panorama obtained Sept. 19, 1884,

from Punta Percia (this peak is on the divide separating the

valleys of the Rhemes and the Valsavaranche) two trigonomet-

rical stations, Punta Rouletta and Gran Punta di Nomenon, of

the new Italian geodetic survey appear upon the same plate (see

Fig. 126, Plate LXIII).

The following values are given for the computation:

Elevation of Punta Rouletta

Elevation of Punta di Nomenon

= 3384. i om. f Taken from the

j catalogue of tri-

= 3488.42 m. [ angulation points

[Elevation of P.

Elevation of horizon of camera station (P. Percia) = 3202. 3 m. \ Percia + height

[ of instrument

Measured upon

= 3250 m. the projection of

the iconometric

= 9720 m. working - sheet,

scale

Distance : Percia.- Rouletta= D

Distance : Percia-Nomenon= D f

96 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

The horizontal angle V (Fig. 126, Plate LXIII) at Punta

Percia, included by the horizontal directions to Rouletta (signal)

and Nomenon (signal), =28 02' 30".

A careful measurement of the coordinates of the pictured

points P. Rouletta and P. Nomenon, on the negative with

a millimeter scale provided with a microscope and vernier, en-

abling the computer to read to 0.05 mm. (the vernier is divided

to read to 1/20 of the graduation unit), produced the following

values :

The coordinates of Punta Rouletta,

# = 46.05 mm.; ^ = 13. 7 5 mm.

The coordinates of Punta di Nomenon,

^ = 75.40 mm.; / = 7.3omm.

It is desired to find:

(1) The focal distance for this negative = /, the preliminary

value, read off on the scale attached to the objective cylinder, is

found to be 244.50 mm.

(2) The correct position of the principal point (P), which

will be fixed by the determination of the abscissae x and oc*.

(3) The position of the line of horizon OO', which will be

located by ascertaining the values for y and /.

Computation to determine the apparent differences in elevation

between the camera horizon (P. Percia) and the two pictured points

P. Rouletta and Punta di Nomenon.

Altitude of P. Rouletta =3384.10 m.

Altitude of camera horizon =3202 .30 m.

True difference in elevation = 181 .80 m.

Correction for curvature and refraction = 0.71 m.

Apparent difference in elevation =L = 181 .09 m.

Altitude of Punta di Nomenon =3488.42 m.

Altitude of camera horizon =3202 .30 m.

PHOTOTOPOGRAPHIC SURVEYING METHODS. 97

True difference in elevation = 286 . 1 2 m.

Correction for curvature and refraction = - 6 . 35 m.

Apparent difference in elevation = Z/ = 279.77

Computation of d= y

log D= 3.5 1 18834

logy =8. 1383027

colog 1=7.7421055

log d= 9.3922916

d= 246.77 mm.

Computation of d'=

log D'= 3.9876663

log /= 7.8633229

colog L'= 7.5531989

log d'= 9.4041881

d'= 253.62 mm.

d+d'= 500.39

d'-d= 6.85

Computation of the angles 7- and d:

r -d d'-d V

~

2 d f + d 2

28 02' 30"; - = i4oi'i5";

>

57' 3o"; ^ = 75 58' 45"=^-

log (d'-d)= log 0.00685 =7.8356906

log cot - = logcoti4 01' 15'' =0.6025567

colog (d'+d} = colog 0.50039 =0.3006914

log tan ^^ = 8 - 7389387

r =7 08' i6".i=N.

Hence

50

98 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

(a) Computation of the Focal Length (=/).

/ = d - sin 7- / = d f sin d

log ^ = 9. 3922916 log d' = 9. 4041881

log sin f g. 9921180 log sin = 9. 9802269

lg / = 9 - 3 8 44096 log / = 9 . 38441 50

l=o. 242331 m. /=o. 242334 m.

Mean value for / = 242. 33 2 mm.

{/?) Computation of the Abscisses (x and x') for Platting Lines of Hori-

zontal Directions to Pictured Points of the Terrene and for Checking