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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on the strip) will fall upon their corresponding radials; a line

drawn along the edge of the paper strip while in this position

will represent the oriented picture trace, as indicated by the

line hihi.

If we now draw a perpendicular line (Si PI) to h\h\ from the

plotted station S\, the point PI will be the horizontal projection

of the principal point P and SiPi=f will be the distance line

for the perspective MN.

Should the positions of the points A , B, C . . . with refer-

ence to the station 5 be not known, it will become necessary

to observe the horizontal angles A SB, BSC, CSD . . . instru-

mentally from the station S and plot them in their proper order

upon a sheet of paper (AiSiBi, BiSiCi . . . ) and adjust the

paper strip hh upon these radials in the same manner as just

described.

B. Determination of the Position of the Horizon Line on the

Perspective.

When the elevations AA f , BB', CC' . . . (Fig. 13, Plate VII)

of the points A, B, C . . . above the horizon plane SOO' of the

DETERMINATION OF PRINCIPAL AND HORIZON LINES. 53

station S are known, the position of the horizon line OO r on

the perspective MN may be found by computing the ordinates

aa', W t cd . . . from the equations:

aa':AA'=Sa':SA',

bl/:BB'=Sb':SB',

whence

Sa'xAA

SVXBB'

5BT-*

The distances Sa', 5^, Sc f . . . are taken from the plotting-

sheet. The horizontal distances SA', SB', SC r . . . and the

differences in elevations A A', BE', CC' ... are known.

For example, the difference in elevation between A and

A' = 100 m., the distance of A' from the station S = 1000 m.,

and the distance Sa', measured on the plotting-sheet, =0.05 m.,

then we will have

0.05X100

aa r = y = ^ =0.005 m.

1000

The horizon line OO f on the negative will be 5 mm. ver-

tically below the pictured point a (measured in a direction parallel

with the pictured plumb line w). A line OO' drawn through a'

at right angles with the pictured plumb line w will locate the

horizon line. The computed ordinates bt/=yi, cc f =y 2 ...of

the other pictured points &, c . . . will serve to check the position

of the horizon line OO' ; it should be tangent to the arcs described

with aa', bV> ccf . . . about a, b, c . . . respectively as centers.

54 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

III. Graphic Method for Determining the Positions of the Prin-

cipal and Horizon Lines on the Perspectives.

The following method for orienting the picture trace, pub-

lished by Prof. F. Schiffner, in 1887, and mentioned by Prof.

Steiner, leads to the same result graphically as the preceding

one does arithmetically.

The horizontal projections AI, BI, C\, and Si of three points

A, B, C, and station S, Fig. 14, Plate VII, may be given. From

Si, as center, radials are drawn through A i, BI, and C\. Through

a point a on the radial Si<4i a parallel to SiCi is drawn and

the distance a'b' taken from the negative MN, not shown

in the figure is laid off from a = abi upon this parallel line,

while the distance b'c' is laid off upon the same line from bi

= bi'ci'.

Parallels to the radial 5i^4i are then drawn through the

points bi and c\ and produced to intersect with the radials

SiBi and SiCi. The line h'h' connecting these two points

of intersection will be parallel with the direction of the picture

trace.

The same distances a'b' and b'c f taken from the negative

are laid off upon this line h'h' from a 2 = a 2 b 2 and from b 2 = b 2 c 2

respectively. The parallels to the radial SiAi, drawn through

these points b 2 and c 2 , are brought to intersections with the radials

SiBi and SiCi, when the line hh, passing through these inter-

sections b' and c', will represent the picture trace, oriented with

reference to Si, AI, BI, and Ci.

The distance Si PI of Si from hh represents the distance line

(focal length) of the picture MN, and the point PI will be the

horizontal projection of the principal point of the perspective.

After having transferred PI (with reference to a', b', and c')

to the perspective MN by means of a strip of paper, a parallel

to the pictured plumb line vu drawn through the point P L will

locate the principal line upon the negative.

55

HI. The " Five-point Problem " (by Prof. F. Steiner), or Locating

the Plotted Position of the Camera Station by Means of the

Perspective when Five Triangulation Points are Pictured

on the Same Photographic Perspective.

In the methods considered until now it had been assumed

that the position of the camera station Si on the plotting-sheet

was known with reference to the plotted triangulation points

AI, BI, Ci . . . .

In case the panorama pictures have been taken from a camera

station Si of unknown position and a series of known points

are pictured upon the panorama views, both the position of the

camera station may be found (with reference to the positions

of the surrounding points of known positions) and the picture

trace may be oriented by means of Prof. F. Steiner's " five-point

problem," if one of the panorama views contains the pictures

of five or more points of known positions.

A. Determination oj the Principal Point and Distance Line.

A panorama view MN may contain the images a, b, c, d, e

of the triangulation points A, B, C, D, E, already plotted on

the working-plan, and also the picture of a suspended plumb

line or other vertical (or horizontal) line sufficiently long to be

used for drawing parallel lines to the principal (or horizon)

line.

The points a, b, c, d, and e of the negative MN are projected

upon the straight edge of a strip of paper = a\, bi, c\, di, and e\.

Radials are now drawn from one AI, Fig. 15, Plate VIII

of the five plotted points as center to the other four points, BI,

Ci, DI, and EI. The paper strip is then placed over the radials

AiBi, AiDi, and AiEi, that bi falls upon AiBi, d\ upon AiDi,

and e\ upon AiEi, when the strip will have the position ai,

b\, Cij di, e\. The line drawn through AI and ai (the latter

having been transferred to the sheet by means of the paper strip)

56 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

will be tangent in A to the ellipse E\ (which passes through A\>

BI, DI, and EI and through the plotted station Si).

The paper strip is now placed over the radials AiBi, AiCi,

and AiDi, that bj falls upon A\B\, c\ upon AiCi, and d\ upon

AiD\, when the strip will have the position indicated by a 2 ,

b 2 , 2, d 2y 2, and the line Aia 2 will be the tangent in A\ to the

ellipse E 2 (passing through the points AI, BI, Ci, DI, and the

plotted station point Si).

The plotted position of the station point Si with reference

to the five plotted points AI, BI, C\, DI, and EI will be at the

fourth point of intersection Si of the two ellipses EI and E 2 .

After drawing the radials SiAi, Si-Bi, SiCi, SiDi, and SiEi

the paper strip is placed over these radials in such manner that a\

falls upon SiAi, bi upon SiBi, . . . and e\ upon Si-Ei, in the

position indicated by a, b, c, d, e=HH, when HH will be the

plotted picture trace.

The perpendicular upon HH passing through Si = Si PI

represents the distance line and PI is the principal point of the

negative projected into the horizontal plan, which, in order

to locate the principal line, may now be transferred to the per-

spective by means of the paper strip in the manner already

described.

B. Simplified Construction for Locating the Plotted Position o)

the Camera Station by Means oj the " Five- point Problem."

The method just described being rather complicated, Prof.

SchifTner recommends the following construction, Fig. 16, Plate IX,

in which the drawing of the two ellipses EI and E 2 is avoided:

The plotted positions of the same five points A, B, C, D^

and E, together with a negative containing the images a, b, c, d,

and e, of these points may be given. It is desired to find the

fourth point of intersection Si of the two ellipses EI arid E 2

without actually drawing their perimeters.

57

The two tangents b s Bi and M*i to the ellipses EI and E 2

in BI are located in precisely the same manner as the two tan-

gents aiAi and a 2 Ai for the point A\ were found in Fig. 15,

Plate VIII. The intersections R\ and R 2 of the tangent pairs

aiAi, bzBi, and a 2 Ai, bBi, Fig. 16, Plate IX (belonging respec-

tively to the ellipses EI and E 2 ), are situated on a line QX, form-

ing one side of the polar triangle QXT, common lo both ellipses.

This line RiR 2 = QX intersects the diagonal AiD\ in X and

the quadrilateral side BiDi in the point Q. The lines drawn

through Q from A\ and through X from B\ will intersect each

other in the fourth point of intersection Si of the two ellipses.

This method may also appear rather complicated in view

of the many lines that have to be drawn before the picture trace

H H and the position of the camera station may be plotted.

C. Application of the " Five- point Problem "to the Special Case,

where the Five Points range themselves into a Triangle on the

Working- sheet.

The application of the five-point problem becomes very

much simplified when the five points A, B, C, D, and E form

a triangle of which two sides A\C\ and C\E\ 9 Fig. 17, Plate X,

contain three points each.

If we place the strip of paper upon the radials, drawn from AI,

that e\ falls upon A\E\,d\ upon AiDi, and c\ upon A\C\> it will

have the position indicated by a 2 , b 2 , c 2t d 2 , e 2 , and the first ellipse

will resolve into the lines C\E\ and A\a 2 . If we now place the

paper strip #1, bi, c\, d\, e\ upon the radials drawn from E\ to AI,

to BI and to Ci, that #1 falls upon E\Ai, bi upon EiBi, and c\

upon EiCi, it will assume the position a\ t bi, c\, d\, e\, and the

second ellipse will resolve into the lines A\C\ and EI^I.

The intersection Si, of the two lines Aia 2 and 1^1, locates

the position of the plotted station point Si with reference to the

five given points A\, B\, Ci, D\, E\. By placing the paper

strip upon the radials SiA 1} SiBi, SiCi, SiA, and SiEi in

58 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

such manner that ai falls upon Si^A, upon SiJBi, c\ upon

SiC\ . . . , its edge HH will locate the picture trace, PI ' will be

the horizontal projection of the principal point P, and S\Pi will

be the distance line.

D. To find the Elevation of the Camera Horizon for a Station

that has been located by means of the "Five- point Problem."

To ascertain the elevation of the station S, plotted after one

of the preceding methods, it will be necessary to know the ele-

vations of at least two of the five given points. The elevation

of the station horizon SOO', Fig. 13, Plate VII, above the datum

or ground plane S'OiOi', may be designated by X, H and HI

may be the elevations of A and B respectively, both supposed

to be known. The ordinates of the pictured points a and b

are aa'=y and bb'=y.

From the relation S'ai' -.S'A^ = aa f :AA'

or Sa':SA'=y:(H-X)

we find y = ^7 (HX),

Sb'

and yi = 7 (Hi - X).

As the difference between y and y\ may be found by direct

measurements made on the negative, y yi=m will be known

and the value for X may be computed from the equation

since the measures for Sa', SA', 56', and SB' may be obtained

from the plotting-sheet, measured in the scale of the map.

THE " THREE-POINT PROBLEM." 59

The above equation may be written in the general form:

H-X H-X i Sa' i SV

m = ? where * = S^ and 7 = 5^ ;

hence

m-n-p H-p HI n

-P

By substitution of this value in the equations

n

the numerical values for the ordinates y and yi (governing the

position of the horizon line) may be found.

V. The " Three-point Problem."

If the triangulation points are not sufficiently close together

that five or more points may be pictured on one photographic

perspective, and if stations are occupied with the camera that

are not directly connected with the trigonometric system, it will

become necessary to employ other means than those hereto-

fore considered for locating the position of such detached camera

stations with reference to the triangulation system.

To connect detached camera stations with the triangulation

by observations made at the camera station, at least three tri-

angulation points should be visible from such station. When

the camera party is in advance of the triangulation party many

camera stations will be located by the triangulation party by

observing upon a signal left at the camera station, if such signal

be visible from two or more triangulation stations (the camera

station will be a " concluded point " of the triangulation system).

60 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

The determination of the position of a detached camera

station by observing upon three fixed and known points (pro-

vided with signals) is generally known as the " three-point prob-

lem " (station-plotting, station-pointing, etc.), or " Pothenot's

method," although Snellius was probably the first to use this

method (in his trigonometric surveys in the Netherlands in the

second decade of the seventeenth century).

A. Mechanical Solution oj the " Three- point Problem"

(using a Three-arm Protractor or Station- pointer).

The simplest solution of the three-point problem is purely

mechanical in application. The two observed angles M and N

are laid off upon a three-arm protractor (" station-pointer ")

or upon a sheet of tracing-paper, and the three arms or lines S\Ai,

SiB ly and SiCi, Fig. 18, Plate X, are placed over the three fixed

and plotted points AI, BI, and C\ in such manner that the three

lines of direction 5i^4i, S\B\-> S\C\ pass through their respective

points AI, BI, and C\, the point S being transferred to the

working-sheet while holding the two horizontal angles M and N

in unchanged position.

B. Graphic Solution of the " Three- point Problem."

i. USING THE SO-CALLED "TWO-CIRCLE METHOD."

Theoretically the best graphic method is probably that by

which the position of the fourth, or station, point is located at

the intersection of two circles, one passing through A i and BI

and having over AiB, as chord, the angles of circumference

=AiSiBi=M, Fig. 1 8, Plate X, the second circle passing through

BI and Ci and having over the chord B\Ci the angles of cir-

cumference equal to BiSiCi=N.

6i

From the plotted triangle side AiB\ we lay off at A\ and B\

the angles BiAiCi and AiBiCiy each equal to

iSo-2(A l S l B l ) o* n u

= 90 4iSi>i=9o M,

and about the point Ci, thus obtained, we describe a circle

,4i#iSi with the radius = CiAi=CiBi. The observed angle

AiSiBi=M will then be an angle of circumference over AiBi,

and the point Si will be located somewhere on the arc over A\B\.

By means of the angle BiS\C=N another circle BiCSi is

described over the triangle side B^C, in a similar manner, about

the point C% as center, having C<iB\=C<2.C as radius. The

observed second angle B\S\C=N will be an angle of circum-

ference over the chord B\C and the point Si will be on the arc

over BiCy hence its true position is at the (second) point of inter-

section Si of the two circles.

2. USING THE METHOD OF BOHNENBERGER AND BESSEL.

The following method, by Bohnenberger and Bessel, is readily

applied and simple in construction. If we describe a circle

through two of the given points, through A\ and B\, Fig. 19,

Plate XI, and through the station Si, the angles designated by M

and those designated by AT" in the figure will be respectively

equal, being angles of circumference over the same arcs 4iZ>i

and DiCi respectively.

Hence if we lay off the observed horizontal angle N on AiCi

at A i, and the other observed horizontal angle M on the line

^4iCi at Ci, the point of intersection D\ of their convergent

sides CiZ>i and A^D\ will fall upon the line connecting the third

plotted triangulation point BI with the station point Si.

After having thus determined the direction of the line BiDi

or BiSi the position of the point sought may be found as follows :

At any point x on the produced line BiDi the observed angles

M and N are laid off to either side of #iA, in the sense in which

62 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

they were observed at S. Lines AiSi and CiSi, drawn through

A i and Ci, parallel to xy and ^2 respectively, will locate the

plotted position of the station point 5*1 (upon D\B\) with refer-

ence to the three plotted points A\, B\, and C\.

This solution is recommended only when B\D\ is sufficiently

long (in Fig. 19, Plate XI, it evidently is too short) to assure a

correct prolongation toward S\.

The picture trace HH, containing the horizontal projections

of the pictured points a, b, c, may now be oriented in the known

manner by adjusting the paper strip, having the three points a\,

bi, and c\ marked on its edge, over the radials S\A\, S\B\> and

Sid to bring a\ on SiAi, bi on SiBi, and c\ on S\C\.

VI. The Orientation of Picture Traces, Based on Instrumental

Measurements Made in the Field.

When no points of the area to be mapped phototopographic-

ally are known, the elements (horizon line, principal point, and

distance line) of the photographic perspectives can no longer

be determined from the photographs alone. Instrumental obser-

vations will have to be made at the camera stations in the

field to supply the data needed for their determination. This

method, among others, having been adopted by Capt. Deville,

will be described in the chapter giving the description of the

Canadian surveying-camera.

VII. Relations between Two Perspectives of the Same Object,

Viewed from Different Stations.

'(Prof. Guido Hauck's Method.)

A more .general application of photogrammetric methods

dates j from the publication of Prof. G. Hauck's investigations

and results regarding the relationship between trilinear systems

of different planes (Guido Hauck, " Theorie der trilinearen

TWO PERSPECTIVES OF THE SAME OBJECT. 6j

Verwandtschaft ebencr Systeme," Journal fuer reine und ange-

wandte Mathematik, L. Kronecker und Weierstrass, Bd. 95,

1883). In this publication Prof. Hauck discusses the relation-

ship between the projections of the same object upon three differ-

ent planes. The practical value of his theoretical deductions

was fully established and tested practically by the students of

the Royal Technical High School of Berlin who attended ProL

Hauck's lectures and exercises connected with the course in

descriptive geometry in 1882.

In the discussion of the relation between three perspectives

of the same object Prof. Hauck refers to some properties of

decided value in iconometric plotting. The principal law (as

deduced by Prof. Hauck) with reference to phototopographj

may be stated as follows:

If an object be projected from three different points as cen-

ters upon three different planes that may have any position in

space, one of these projections (perspectives) can be evolved

from the other two by means of graphic construction. Or,

expressed in terms more suited to our case, if an object has

been photographed on three plates exposed from different

stations, any one of these photographic perspectives may be

evolved graphically from the remaining two. A topographic

map (the orthogonal projection of the terrene) may be regarded

as a central projection or perspective in a horizontal plane,

having its center of projection (point of view) at infinite dis-

tance, and we may state Prof. Hauck's law as follows :

From two photographs MN and M'N' of the same terrene,

taken from different stations 5 and S', the orthogonal horizontal

projection of the terrene may be obtained graphically by means

of rays emanating from the so-called " kernel points " (" Kern-

punkte ") as centers.

The line of intersection of the two photographs (the two

planes of projection or perspective planes) MN and M'N 9 ,

Fig. 20, Plate XI, will be the " perspective axis."

To better illustrate the connection between two different

64 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

photographs we will first refer to the simple case of two vertical

perspective planes or photographs MN and M'N', Fig. 20,

Plate XI.

A. " Kernel Points " and " Kernel Planes."

Let S and S' represent the two camera stations (centers of

projection or points of view for the two vertical photographic

perspectives MN and M'N'), let s f be the picture in MN of S',

and s be the picture of S in M'N', then these two pictured sta-

tions s and s' will be so-called " kernel points " (" Kernpunkte ").

The two picture planes MN and M'N' intersect each other

in the line IQ, the so-called " perspective axis."

Planes passing through the line SS' (base line) will contain

the " kernel points " 5 and s'; they are termed " kernel planes "

(" Kernebenen ").

A kernel plane M 2 N 2 , laid through any point A, pictured

as a and a', will intersect the first picture plane MN in the line as'

and the second picture plane M'N' in the line sa'. These lines

of intersection (as' and a's) will intersect the " perspective axis "

IQ in the same point Q] they will contain the pictures a and a'

of the point A, and they will pass through the picture s and s'

of the two camera stations S and S'.

The lines S'A, SA, SS', as', and a's fall within the " kernel

plane " M 2 N 2 . All lines as' for all points pictured in MN

will pass through the pictured station point s' (image of the second

camera station S'), and all lines as for the picture plane M'N'

will pass through the pictured point s of the camera station S.

Furthermore, all pairs of lines (as' and a's) joining the per-

spectives (a and a') of identical points (A) with their corre-

sponding pictured station points (" kernel points " s' and s)

will intersect the " perspective axis " (IQ) of the two pcture

planes (MN and M'N') in identical points (Q).

From two photographs of the same object which also con-

tain the pictures of the two reciprocal stations peculiar advan-

TWO PERSPECTIVES OF THE SAME OBJECT. 65

tages may be gained for the iconometric plotting, inasmuch as

such pictured stations s f and s will be " kernel points."

The perspective axis of the picture planes may also play an

important part in iconometric plotting, not only for pictures

exposed in vertical planes, but even more so for inclined picture

planes.

If two photographs MN and M'N' are given (in Fig. 21,

Plate XII, their traces are represented by the lines HH and H'H')

representing the same object, viewed from the two stations S

and S' without containing the pictures of the stations, the posi-

tions of the pictures s and s' of the corresponding camera sta-

tions S and S' may be located upon the picture planes (out-

side of the actual field of the photograph) by construction.

The horizontal projections Si and Si' of the " kernel points "

s and s* are identical with the points of intersection of the plotted

base line SS' and the picture traces HH and H'H', Fig. 21,

Plate XII. Hence, if we revolve the picture planes MN and

M'N' about their ground lines, until they coincide with the

ground plane, the line IQ, common to both picture planes (the

" perspective axis "), will be represented by the two lines i(I),

and the " kernel points " 5 and s f of the revolved planes MN

and MN will fall upon the lines si(Si) and V(Si') respectively

(these lines are perpendiculars upon the picture traces in the

horizontal projections si and Si' of the kernel points 5 and s f )..

To find the lengths si(Si) and s^ (Si')' (the ordinates of

the " kernel points " in the picture planes above the ground

lines) we erect perpendiculars to the base line in S and S' with

lengths equal to the elevations of the camera stations above

the ground plane =S(S) and S'(S') respectively.

The line (S)(S') the vertical plane passing through the base

line S has been revolved about the horizontal projection of the

base line into the ground plane to coincide with the latter

will intersect the lines $i(Si") and Si'(Sn') they are perpen-

diculars to SS' in the " kernel points " 5 and s and the lengths

$i(Si") and Ji'(Sn') will be equal to the ordinates of the " ker-

66 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

nel points " J and s f above the ground lines of MN and

M'N'.

The " kernel points " s and s' , Fig. 20, Plate XI, may be

located after this manner in the picture planes of any two photo-

graphs, provided such picture planes are not parallel (or even

nearly so) with the base line SS'.

B. Use of the "Perspective Axis " (Line of Intersection) of two

Picture Planes that show identical Objects viewed from

different Stations.

If a series of characteristic points of the terrene, pictured

in a vertical plane 'MN, Fig. 22, Plate XII, are connected with

the " kernel point " s by straight lines, these will, when pro-

drawn along the edge of the paper strip while in this position

will represent the oriented picture trace, as indicated by the

line hihi.

If we now draw a perpendicular line (Si PI) to h\h\ from the

plotted station S\, the point PI will be the horizontal projection

of the principal point P and SiPi=f will be the distance line

for the perspective MN.

Should the positions of the points A , B, C . . . with refer-

ence to the station 5 be not known, it will become necessary

to observe the horizontal angles A SB, BSC, CSD . . . instru-

mentally from the station S and plot them in their proper order

upon a sheet of paper (AiSiBi, BiSiCi . . . ) and adjust the

paper strip hh upon these radials in the same manner as just

described.

B. Determination of the Position of the Horizon Line on the

Perspective.

When the elevations AA f , BB', CC' . . . (Fig. 13, Plate VII)

of the points A, B, C . . . above the horizon plane SOO' of the

DETERMINATION OF PRINCIPAL AND HORIZON LINES. 53

station S are known, the position of the horizon line OO r on

the perspective MN may be found by computing the ordinates

aa', W t cd . . . from the equations:

aa':AA'=Sa':SA',

bl/:BB'=Sb':SB',

whence

Sa'xAA

SVXBB'

5BT-*

The distances Sa', 5^, Sc f . . . are taken from the plotting-

sheet. The horizontal distances SA', SB', SC r . . . and the

differences in elevations A A', BE', CC' ... are known.

For example, the difference in elevation between A and

A' = 100 m., the distance of A' from the station S = 1000 m.,

and the distance Sa', measured on the plotting-sheet, =0.05 m.,

then we will have

0.05X100

aa r = y = ^ =0.005 m.

1000

The horizon line OO f on the negative will be 5 mm. ver-

tically below the pictured point a (measured in a direction parallel

with the pictured plumb line w). A line OO' drawn through a'

at right angles with the pictured plumb line w will locate the

horizon line. The computed ordinates bt/=yi, cc f =y 2 ...of

the other pictured points &, c . . . will serve to check the position

of the horizon line OO' ; it should be tangent to the arcs described

with aa', bV> ccf . . . about a, b, c . . . respectively as centers.

54 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

III. Graphic Method for Determining the Positions of the Prin-

cipal and Horizon Lines on the Perspectives.

The following method for orienting the picture trace, pub-

lished by Prof. F. Schiffner, in 1887, and mentioned by Prof.

Steiner, leads to the same result graphically as the preceding

one does arithmetically.

The horizontal projections AI, BI, C\, and Si of three points

A, B, C, and station S, Fig. 14, Plate VII, may be given. From

Si, as center, radials are drawn through A i, BI, and C\. Through

a point a on the radial Si<4i a parallel to SiCi is drawn and

the distance a'b' taken from the negative MN, not shown

in the figure is laid off from a = abi upon this parallel line,

while the distance b'c' is laid off upon the same line from bi

= bi'ci'.

Parallels to the radial 5i^4i are then drawn through the

points bi and c\ and produced to intersect with the radials

SiBi and SiCi. The line h'h' connecting these two points

of intersection will be parallel with the direction of the picture

trace.

The same distances a'b' and b'c f taken from the negative

are laid off upon this line h'h' from a 2 = a 2 b 2 and from b 2 = b 2 c 2

respectively. The parallels to the radial SiAi, drawn through

these points b 2 and c 2 , are brought to intersections with the radials

SiBi and SiCi, when the line hh, passing through these inter-

sections b' and c', will represent the picture trace, oriented with

reference to Si, AI, BI, and Ci.

The distance Si PI of Si from hh represents the distance line

(focal length) of the picture MN, and the point PI will be the

horizontal projection of the principal point of the perspective.

After having transferred PI (with reference to a', b', and c')

to the perspective MN by means of a strip of paper, a parallel

to the pictured plumb line vu drawn through the point P L will

locate the principal line upon the negative.

55

HI. The " Five-point Problem " (by Prof. F. Steiner), or Locating

the Plotted Position of the Camera Station by Means of the

Perspective when Five Triangulation Points are Pictured

on the Same Photographic Perspective.

In the methods considered until now it had been assumed

that the position of the camera station Si on the plotting-sheet

was known with reference to the plotted triangulation points

AI, BI, Ci . . . .

In case the panorama pictures have been taken from a camera

station Si of unknown position and a series of known points

are pictured upon the panorama views, both the position of the

camera station may be found (with reference to the positions

of the surrounding points of known positions) and the picture

trace may be oriented by means of Prof. F. Steiner's " five-point

problem," if one of the panorama views contains the pictures

of five or more points of known positions.

A. Determination oj the Principal Point and Distance Line.

A panorama view MN may contain the images a, b, c, d, e

of the triangulation points A, B, C, D, E, already plotted on

the working-plan, and also the picture of a suspended plumb

line or other vertical (or horizontal) line sufficiently long to be

used for drawing parallel lines to the principal (or horizon)

line.

The points a, b, c, d, and e of the negative MN are projected

upon the straight edge of a strip of paper = a\, bi, c\, di, and e\.

Radials are now drawn from one AI, Fig. 15, Plate VIII

of the five plotted points as center to the other four points, BI,

Ci, DI, and EI. The paper strip is then placed over the radials

AiBi, AiDi, and AiEi, that bi falls upon AiBi, d\ upon AiDi,

and e\ upon AiEi, when the strip will have the position ai,

b\, Cij di, e\. The line drawn through AI and ai (the latter

having been transferred to the sheet by means of the paper strip)

56 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

will be tangent in A to the ellipse E\ (which passes through A\>

BI, DI, and EI and through the plotted station Si).

The paper strip is now placed over the radials AiBi, AiCi,

and AiDi, that bj falls upon A\B\, c\ upon AiCi, and d\ upon

AiD\, when the strip will have the position indicated by a 2 ,

b 2 , 2, d 2y 2, and the line Aia 2 will be the tangent in A\ to the

ellipse E 2 (passing through the points AI, BI, Ci, DI, and the

plotted station point Si).

The plotted position of the station point Si with reference

to the five plotted points AI, BI, C\, DI, and EI will be at the

fourth point of intersection Si of the two ellipses EI and E 2 .

After drawing the radials SiAi, Si-Bi, SiCi, SiDi, and SiEi

the paper strip is placed over these radials in such manner that a\

falls upon SiAi, bi upon SiBi, . . . and e\ upon Si-Ei, in the

position indicated by a, b, c, d, e=HH, when HH will be the

plotted picture trace.

The perpendicular upon HH passing through Si = Si PI

represents the distance line and PI is the principal point of the

negative projected into the horizontal plan, which, in order

to locate the principal line, may now be transferred to the per-

spective by means of the paper strip in the manner already

described.

B. Simplified Construction for Locating the Plotted Position o)

the Camera Station by Means oj the " Five- point Problem."

The method just described being rather complicated, Prof.

SchifTner recommends the following construction, Fig. 16, Plate IX,

in which the drawing of the two ellipses EI and E 2 is avoided:

The plotted positions of the same five points A, B, C, D^

and E, together with a negative containing the images a, b, c, d,

and e, of these points may be given. It is desired to find the

fourth point of intersection Si of the two ellipses EI arid E 2

without actually drawing their perimeters.

57

The two tangents b s Bi and M*i to the ellipses EI and E 2

in BI are located in precisely the same manner as the two tan-

gents aiAi and a 2 Ai for the point A\ were found in Fig. 15,

Plate VIII. The intersections R\ and R 2 of the tangent pairs

aiAi, bzBi, and a 2 Ai, bBi, Fig. 16, Plate IX (belonging respec-

tively to the ellipses EI and E 2 ), are situated on a line QX, form-

ing one side of the polar triangle QXT, common lo both ellipses.

This line RiR 2 = QX intersects the diagonal AiD\ in X and

the quadrilateral side BiDi in the point Q. The lines drawn

through Q from A\ and through X from B\ will intersect each

other in the fourth point of intersection Si of the two ellipses.

This method may also appear rather complicated in view

of the many lines that have to be drawn before the picture trace

H H and the position of the camera station may be plotted.

C. Application of the " Five- point Problem "to the Special Case,

where the Five Points range themselves into a Triangle on the

Working- sheet.

The application of the five-point problem becomes very

much simplified when the five points A, B, C, D, and E form

a triangle of which two sides A\C\ and C\E\ 9 Fig. 17, Plate X,

contain three points each.

If we place the strip of paper upon the radials, drawn from AI,

that e\ falls upon A\E\,d\ upon AiDi, and c\ upon A\C\> it will

have the position indicated by a 2 , b 2 , c 2t d 2 , e 2 , and the first ellipse

will resolve into the lines C\E\ and A\a 2 . If we now place the

paper strip #1, bi, c\, d\, e\ upon the radials drawn from E\ to AI,

to BI and to Ci, that #1 falls upon E\Ai, bi upon EiBi, and c\

upon EiCi, it will assume the position a\ t bi, c\, d\, e\, and the

second ellipse will resolve into the lines A\C\ and EI^I.

The intersection Si, of the two lines Aia 2 and 1^1, locates

the position of the plotted station point Si with reference to the

five given points A\, B\, Ci, D\, E\. By placing the paper

strip upon the radials SiA 1} SiBi, SiCi, SiA, and SiEi in

58 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

such manner that ai falls upon Si^A, upon SiJBi, c\ upon

SiC\ . . . , its edge HH will locate the picture trace, PI ' will be

the horizontal projection of the principal point P, and S\Pi will

be the distance line.

D. To find the Elevation of the Camera Horizon for a Station

that has been located by means of the "Five- point Problem."

To ascertain the elevation of the station S, plotted after one

of the preceding methods, it will be necessary to know the ele-

vations of at least two of the five given points. The elevation

of the station horizon SOO', Fig. 13, Plate VII, above the datum

or ground plane S'OiOi', may be designated by X, H and HI

may be the elevations of A and B respectively, both supposed

to be known. The ordinates of the pictured points a and b

are aa'=y and bb'=y.

From the relation S'ai' -.S'A^ = aa f :AA'

or Sa':SA'=y:(H-X)

we find y = ^7 (HX),

Sb'

and yi = 7 (Hi - X).

As the difference between y and y\ may be found by direct

measurements made on the negative, y yi=m will be known

and the value for X may be computed from the equation

since the measures for Sa', SA', 56', and SB' may be obtained

from the plotting-sheet, measured in the scale of the map.

THE " THREE-POINT PROBLEM." 59

The above equation may be written in the general form:

H-X H-X i Sa' i SV

m = ? where * = S^ and 7 = 5^ ;

hence

m-n-p H-p HI n

-P

By substitution of this value in the equations

n

the numerical values for the ordinates y and yi (governing the

position of the horizon line) may be found.

V. The " Three-point Problem."

If the triangulation points are not sufficiently close together

that five or more points may be pictured on one photographic

perspective, and if stations are occupied with the camera that

are not directly connected with the trigonometric system, it will

become necessary to employ other means than those hereto-

fore considered for locating the position of such detached camera

stations with reference to the triangulation system.

To connect detached camera stations with the triangulation

by observations made at the camera station, at least three tri-

angulation points should be visible from such station. When

the camera party is in advance of the triangulation party many

camera stations will be located by the triangulation party by

observing upon a signal left at the camera station, if such signal

be visible from two or more triangulation stations (the camera

station will be a " concluded point " of the triangulation system).

60 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

The determination of the position of a detached camera

station by observing upon three fixed and known points (pro-

vided with signals) is generally known as the " three-point prob-

lem " (station-plotting, station-pointing, etc.), or " Pothenot's

method," although Snellius was probably the first to use this

method (in his trigonometric surveys in the Netherlands in the

second decade of the seventeenth century).

A. Mechanical Solution oj the " Three- point Problem"

(using a Three-arm Protractor or Station- pointer).

The simplest solution of the three-point problem is purely

mechanical in application. The two observed angles M and N

are laid off upon a three-arm protractor (" station-pointer ")

or upon a sheet of tracing-paper, and the three arms or lines S\Ai,

SiB ly and SiCi, Fig. 18, Plate X, are placed over the three fixed

and plotted points AI, BI, and C\ in such manner that the three

lines of direction 5i^4i, S\B\-> S\C\ pass through their respective

points AI, BI, and C\, the point S being transferred to the

working-sheet while holding the two horizontal angles M and N

in unchanged position.

B. Graphic Solution of the " Three- point Problem."

i. USING THE SO-CALLED "TWO-CIRCLE METHOD."

Theoretically the best graphic method is probably that by

which the position of the fourth, or station, point is located at

the intersection of two circles, one passing through A i and BI

and having over AiB, as chord, the angles of circumference

=AiSiBi=M, Fig. 1 8, Plate X, the second circle passing through

BI and Ci and having over the chord B\Ci the angles of cir-

cumference equal to BiSiCi=N.

6i

From the plotted triangle side AiB\ we lay off at A\ and B\

the angles BiAiCi and AiBiCiy each equal to

iSo-2(A l S l B l ) o* n u

= 90 4iSi>i=9o M,

and about the point Ci, thus obtained, we describe a circle

,4i#iSi with the radius = CiAi=CiBi. The observed angle

AiSiBi=M will then be an angle of circumference over AiBi,

and the point Si will be located somewhere on the arc over A\B\.

By means of the angle BiS\C=N another circle BiCSi is

described over the triangle side B^C, in a similar manner, about

the point C% as center, having C<iB\=C<2.C as radius. The

observed second angle B\S\C=N will be an angle of circum-

ference over the chord B\C and the point Si will be on the arc

over BiCy hence its true position is at the (second) point of inter-

section Si of the two circles.

2. USING THE METHOD OF BOHNENBERGER AND BESSEL.

The following method, by Bohnenberger and Bessel, is readily

applied and simple in construction. If we describe a circle

through two of the given points, through A\ and B\, Fig. 19,

Plate XI, and through the station Si, the angles designated by M

and those designated by AT" in the figure will be respectively

equal, being angles of circumference over the same arcs 4iZ>i

and DiCi respectively.

Hence if we lay off the observed horizontal angle N on AiCi

at A i, and the other observed horizontal angle M on the line

^4iCi at Ci, the point of intersection D\ of their convergent

sides CiZ>i and A^D\ will fall upon the line connecting the third

plotted triangulation point BI with the station point Si.

After having thus determined the direction of the line BiDi

or BiSi the position of the point sought may be found as follows :

At any point x on the produced line BiDi the observed angles

M and N are laid off to either side of #iA, in the sense in which

62 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

they were observed at S. Lines AiSi and CiSi, drawn through

A i and Ci, parallel to xy and ^2 respectively, will locate the

plotted position of the station point 5*1 (upon D\B\) with refer-

ence to the three plotted points A\, B\, and C\.

This solution is recommended only when B\D\ is sufficiently

long (in Fig. 19, Plate XI, it evidently is too short) to assure a

correct prolongation toward S\.

The picture trace HH, containing the horizontal projections

of the pictured points a, b, c, may now be oriented in the known

manner by adjusting the paper strip, having the three points a\,

bi, and c\ marked on its edge, over the radials S\A\, S\B\> and

Sid to bring a\ on SiAi, bi on SiBi, and c\ on S\C\.

VI. The Orientation of Picture Traces, Based on Instrumental

Measurements Made in the Field.

When no points of the area to be mapped phototopographic-

ally are known, the elements (horizon line, principal point, and

distance line) of the photographic perspectives can no longer

be determined from the photographs alone. Instrumental obser-

vations will have to be made at the camera stations in the

field to supply the data needed for their determination. This

method, among others, having been adopted by Capt. Deville,

will be described in the chapter giving the description of the

Canadian surveying-camera.

VII. Relations between Two Perspectives of the Same Object,

Viewed from Different Stations.

'(Prof. Guido Hauck's Method.)

A more .general application of photogrammetric methods

dates j from the publication of Prof. G. Hauck's investigations

and results regarding the relationship between trilinear systems

of different planes (Guido Hauck, " Theorie der trilinearen

TWO PERSPECTIVES OF THE SAME OBJECT. 6j

Verwandtschaft ebencr Systeme," Journal fuer reine und ange-

wandte Mathematik, L. Kronecker und Weierstrass, Bd. 95,

1883). In this publication Prof. Hauck discusses the relation-

ship between the projections of the same object upon three differ-

ent planes. The practical value of his theoretical deductions

was fully established and tested practically by the students of

the Royal Technical High School of Berlin who attended ProL

Hauck's lectures and exercises connected with the course in

descriptive geometry in 1882.

In the discussion of the relation between three perspectives

of the same object Prof. Hauck refers to some properties of

decided value in iconometric plotting. The principal law (as

deduced by Prof. Hauck) with reference to phototopographj

may be stated as follows:

If an object be projected from three different points as cen-

ters upon three different planes that may have any position in

space, one of these projections (perspectives) can be evolved

from the other two by means of graphic construction. Or,

expressed in terms more suited to our case, if an object has

been photographed on three plates exposed from different

stations, any one of these photographic perspectives may be

evolved graphically from the remaining two. A topographic

map (the orthogonal projection of the terrene) may be regarded

as a central projection or perspective in a horizontal plane,

having its center of projection (point of view) at infinite dis-

tance, and we may state Prof. Hauck's law as follows :

From two photographs MN and M'N' of the same terrene,

taken from different stations 5 and S', the orthogonal horizontal

projection of the terrene may be obtained graphically by means

of rays emanating from the so-called " kernel points " (" Kern-

punkte ") as centers.

The line of intersection of the two photographs (the two

planes of projection or perspective planes) MN and M'N 9 ,

Fig. 20, Plate XI, will be the " perspective axis."

To better illustrate the connection between two different

64 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

photographs we will first refer to the simple case of two vertical

perspective planes or photographs MN and M'N', Fig. 20,

Plate XI.

A. " Kernel Points " and " Kernel Planes."

Let S and S' represent the two camera stations (centers of

projection or points of view for the two vertical photographic

perspectives MN and M'N'), let s f be the picture in MN of S',

and s be the picture of S in M'N', then these two pictured sta-

tions s and s' will be so-called " kernel points " (" Kernpunkte ").

The two picture planes MN and M'N' intersect each other

in the line IQ, the so-called " perspective axis."

Planes passing through the line SS' (base line) will contain

the " kernel points " 5 and s'; they are termed " kernel planes "

(" Kernebenen ").

A kernel plane M 2 N 2 , laid through any point A, pictured

as a and a', will intersect the first picture plane MN in the line as'

and the second picture plane M'N' in the line sa'. These lines

of intersection (as' and a's) will intersect the " perspective axis "

IQ in the same point Q] they will contain the pictures a and a'

of the point A, and they will pass through the picture s and s'

of the two camera stations S and S'.

The lines S'A, SA, SS', as', and a's fall within the " kernel

plane " M 2 N 2 . All lines as' for all points pictured in MN

will pass through the pictured station point s' (image of the second

camera station S'), and all lines as for the picture plane M'N'

will pass through the pictured point s of the camera station S.

Furthermore, all pairs of lines (as' and a's) joining the per-

spectives (a and a') of identical points (A) with their corre-

sponding pictured station points (" kernel points " s' and s)

will intersect the " perspective axis " (IQ) of the two pcture

planes (MN and M'N') in identical points (Q).

From two photographs of the same object which also con-

tain the pictures of the two reciprocal stations peculiar advan-

TWO PERSPECTIVES OF THE SAME OBJECT. 65

tages may be gained for the iconometric plotting, inasmuch as

such pictured stations s f and s will be " kernel points."

The perspective axis of the picture planes may also play an

important part in iconometric plotting, not only for pictures

exposed in vertical planes, but even more so for inclined picture

planes.

If two photographs MN and M'N' are given (in Fig. 21,

Plate XII, their traces are represented by the lines HH and H'H')

representing the same object, viewed from the two stations S

and S' without containing the pictures of the stations, the posi-

tions of the pictures s and s' of the corresponding camera sta-

tions S and S' may be located upon the picture planes (out-

side of the actual field of the photograph) by construction.

The horizontal projections Si and Si' of the " kernel points "

s and s* are identical with the points of intersection of the plotted

base line SS' and the picture traces HH and H'H', Fig. 21,

Plate XII. Hence, if we revolve the picture planes MN and

M'N' about their ground lines, until they coincide with the

ground plane, the line IQ, common to both picture planes (the

" perspective axis "), will be represented by the two lines i(I),

and the " kernel points " 5 and s f of the revolved planes MN

and MN will fall upon the lines si(Si) and V(Si') respectively

(these lines are perpendiculars upon the picture traces in the

horizontal projections si and Si' of the kernel points 5 and s f )..

To find the lengths si(Si) and s^ (Si')' (the ordinates of

the " kernel points " in the picture planes above the ground

lines) we erect perpendiculars to the base line in S and S' with

lengths equal to the elevations of the camera stations above

the ground plane =S(S) and S'(S') respectively.

The line (S)(S') the vertical plane passing through the base

line S has been revolved about the horizontal projection of the

base line into the ground plane to coincide with the latter

will intersect the lines $i(Si") and Si'(Sn') they are perpen-

diculars to SS' in the " kernel points " 5 and s and the lengths

$i(Si") and Ji'(Sn') will be equal to the ordinates of the " ker-

66 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

nel points " J and s f above the ground lines of MN and

M'N'.

The " kernel points " s and s' , Fig. 20, Plate XI, may be

located after this manner in the picture planes of any two photo-

graphs, provided such picture planes are not parallel (or even

nearly so) with the base line SS'.

B. Use of the "Perspective Axis " (Line of Intersection) of two

Picture Planes that show identical Objects viewed from

different Stations.

If a series of characteristic points of the terrene, pictured

in a vertical plane 'MN, Fig. 22, Plate XII, are connected with

the " kernel point " s by straight lines, these will, when pro-