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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duced, intersect the perspective axis 18, and if the images of

the corresponding identical points in the vertical picture plane

M'N' are joined with the " kernel point " s f , and if these lines

are likewise produced to intersect the " perspective axis " IQ,

the points of intersection of IQ with the first group of lines (belong-

ing to MN) will be identical with the points of intersection of

IQ with the second group of lines (belonging to M'N').

If we now provide the " perspective axis " with a scale of

equal parts (having the zero or origin of graduation in the ground

plane), lines drawn through the " kernel points " and through

corresponding images of identical points in both picture planes

will intersect identical points of this scale.

The length O'O, Fig. 22, Plate XII, intercepted on the scale

of the " perspective axis " by the two horizon lines of the picture

planes MN and M'N' represents the difference in elevation

of the two camera stations 5 and S'. The scale IQ may be

drawn on both pictures to show on both lines i(I), Fig. 21, Plate

XII, after the pictures have been separated. Frequently the

picture itself will not be sufficiently extended to contain the

line IQ, in which case such a scale may still be used by placing

it upon a line XX", in MAT, and upon zz", in M'N', some dis-

TWO PERSPECTIVES OF THE SAME OBJECT. 67

tance from but parallel with the perspective axis 1Q, Fig. 22,

Plate XII, provided the following relation remain satisfied:

sQ:soc'=s'Q:s'z'.

For any other point B, photographed as b and 6' in the pic-

ture planes M N and M'N' respectively, the following propor-

tional equation should be fulfilled:

The triangles sxoxf, s!Q and Vz z', ^^Q being, respectively,

similar, XQOC* must be equal to ZQZ' (as fiQ is common to both

triangles spQ and s'^Q), which means the spaces on the scales

XX" and zz" are to be identical in numerical value. The two

scales (or either of them) may, if more convenient, be placed

beyond 5 or s f y f. i. at #", in which case

s{! : 5/0 = sfi : sx = s f ^ : s f z .

It should be noted that the scale is now to be read from f toward

/ . It may be stated generally that the scales should be placed

parallel with the " perspective axis " IQ and at distances from the

" kernel points " directly proportional to the distances of the latter

from the " perspective axis " of the picture planes, their correct

position being found from the horizontal projection or from

the ground plane. To avoid obscurity and obliteration of details

in the field of the photograph it will generally be more expe-

dient to draw these scales outside of the picture proper.

To find the proper position of the second scale on the second

picture, after the position of the scale on the first picture has

been decided upon, we again refer to Fig. 21, Plate XII, where

HH and H'H' are the two picture traces, 5 and S' are the

horizontal projections of the camera stations, P and P f are the

traces of the principal lines // and /'/' (Fig. 22), or the horizontal

68 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

projections of the principal points, and, finally, h the selected

position for the first scale.

To find the corresponding position h f of the second scale

we draw a line hh parallel to SS f through h, when

s f i\s'h=s l r i:s l 'h'\

hence

= distance of the second scale from

the " kernel point " s f in the second picture.

The conditions and relations described in the foregoing

paragraphs may often prove of value in iconometric plotting;

f. i., if we consider the case of a straight line L, Fig. 23, Plate XIII,

the image of which appears in picture MN as /, but in the second

picture M'N' only a short piece /' is seen. It may be desirable

to locate in the picture plane MN the reciprocal position of a

point x, shown on the line / in M'N, but falling outside of the

picture limit of M'N' on the prolongation of /'.

To find the position of x f we proceed as follows:

The pictured point x of the line /, pictured in MN, is con

nected with the kernel point (V) and the line (s')x is produced

to its intersection (x) with li. After transferring the point (x)

to the line il of the second picture plane M'N', to ((x)), and

connecting the latter with the " kernel point " (s), the intersec-

tion of ((x))(s) with /' produced will represent the point sought,

x*, on the prolongation of the line /'.

VIII. To Plot a Figure, Situated in a Horizontal Plane, on the

Ground Plan by Means of its Perspective.

Excepting the shore lines of lakes and coasts and the out-

lines of marshes, figures in horizontal planes are not frequently

met with in topographic surveys, and the simplest way to map

these would be to expose photographic plates in a horizontal

TO PLOT A FIGURE BY MEANS OF ITS PERSPECTIVE. 69

position from a captive balloon at points of known positions

and at identical or known elevations.

The mapping of such figures, when photographed on ver-

tically exposed plates, from stations above the figure's plane

is also an easy matter. It may even be done with but a single

perspective view of such figure (obtained on a vertically exposed

plate from a station of known position), provided we also know

the difference in elevation between the camera station and the

horizontal plane containing the figure, and provided we know

the positions of the principal point and horizon line together

with the length of the distance line (focal length) of the photo-

graphic perspective.

We have, with reference to Fig. 24, Plate XIII, HH= horizon

plane of the camera station S, OO' = horizon line of the photo-

graphic perspective MN, GG= ground plane or horizontal plane

coinciding with the surface plane of the lake A BCD, SSo = h

= difference in elevation between the camera station 5 and the

water level of the lake.

With a given perspective abed of the lake A BCD in the ver-

tical picture plane MN y known focal length, given position

of the principal point P and known difference in elevation, h,

between the water surface of the lake and the camera station,

the projection of the lake-outline (AiBiC\D\) in horizontal plan

may be drawn.

The ground line OoCV (line of intersection of ground plane

GG with the vertical picture plane MN) is drawn through P

(horizontal projection of P) parallel with the horizon line OO r ,

PP Q being equal to h (measured in the plotting -scale). If we

now project the pictured points a, b, c, d upon OoOo'=0o> b ,

c Q , d , the radials from the foot S of the station S drawn through

the points a ^o? CQ> do, will pass through the corresponding

points of the lake shore-line AI, BI, Ci, DI that are to be plotted.

Referring to the vertical plane passing through the camera

station S and through the pictured point a (it intersects the ground

plane in SoA or in SoAi) we find from the similar triangles

70 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

SSoA i and aa^A i the horizontal distance S$A i from the camera

station to the point sought, either graphically or arithmetically.

Imagining the vertical plane SSoAi to be revolved a,bout

SoAi until it coincides with the ground plane GG, the points S

and a will assume the positions (S) and (a), Fig. 24, Plate XIII,

and the line (5) (a) will pass through A\, hence AI may be

located in the ground plane as the intersection of (5) (a) with

Sodo. The same may be done for the other points BI, C\, and

DI by revolving the vertical planes SSoBi, SSoC\, and SSoDi

about Sob , S c , and S d Q into the ground plane GG to locate

the positions of BI, Ci, and D\.

To avoid the drawing of so many auxiliary lines on the work-

ing- or plotting-sheet, these constructions are preferably made

on a separate sheet of paper, and the following method may be

adopted :

The vertical planes passing through SQ^O* Sob , SQC Q , and

So^o m ay be supposed to be revolved about SSo, as common

axis of rotation, until they all coincide with the principal plane

SSoPo, Fig. 25, Plate XIV, the surface of the paper representing

the principal plane, when HH = trsice of the horizon plane

in the principal plane, MN = trace of the picture plane in

the principal plane, GG = trace of the ground plane in the

principal plane, SSo = ^= difference in elevation between the

station 5 and the ground plane GG, measured in the plotting-

scale, SP = SoPo = true length of the focal distance of the pho-

tograph MN.

The radials 5o#o> Se^o? So c o> and Sodo are laid off upon the

line GG from SO = SO(<IQ), So(bo), >>o(o)> and 5o(^o)> and the

verticals (a )(a), (^o)(^)> ( c o)( c )> an d (do)(d) are, made equal to

the ordinates aa , bb Q , cc , and dd , respectively, measured on

the picture.

Radials drawn through (a), (6), (c), and (d) from S will

cut off on the line GG the horizontal distances S (A), So(B),

5o(C), and -So(D). These distances, laid off on the radials

S a , Sob , S c , and S d 0) on the plotting-sheet will locate, in

TO PLOT A FIGURE BY MEANS OF ITS PERSPECTIVE. 71

the scale of the map, the plotted positions of the characteristic

points AI, BI, Ci, and DI of the lake, with reference to the ground

line O Oo', which is identical on the plotting-sheet with the

picture trace.

We may reach the same results by utilizing the orthogonal

projections of the points a, b, c, and d and those of AI, BI, Ci,

and DI into the principal plane instead of revolving the ver-

tical planes separately into the principal plane, as done above.

With reference to Fig. 26, Plate XIV, we would then have:

PP = principal plane, MN = picture plane, HH= horizon

plane, containing the camera station 5, GG= ground plane or

surface plane of the lake A BCD.

If we draw the radials SQCLQ, S Q b Q , SQC O , and S d from 5

(the orthogonal projection of 5 in GG) through the orthogonal

projections of the pictured points a, b, c, d on the ground line

O O ', the points sought will fall upon those radials. After

projecting the points a, b, c, and d, in the picture plane MN,

upon the principal line (=a, /?, ?-, and d) the radials Sa, S/?,

Sf, and Sd (drawn in the principal plane PP) will locate the

points a , A)> ro and d , respectively, upon the line SoPo (in

the ground plane), and these represent the orthogonal projec-

tions of the points A , B, C, and D in GG upon SoPo. Hence

the points A, B, C, and D may be found by erecting perpen-

diculars upon SoPo in a , /? , Toy an d <^o> respectively, and their

points of intersection with the radials Soa , Sob , SQC O , and Sodo,

respectively, will be the positions of the plotted points A, B, C,

and D.

Also this construction is preferably made upon a separate

sheet of paper, Fig. 27, Plate XV, where the radials Soa , Sob ,

SQCQ, and Sodo are drawn through their corresponding points

on the plotted picture trace or ground line O O ', but the rest

of the construction is made on the separate sheet of paper, con-

sidering the surface of the latter to coincide with the principal

plane (Fig. 28, Plate XV, where the designations are the same

as in Fig. 25, Plate XIV).

72 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

The points d, /?, a, and 7- on the line PPo (principal line)

represent the projections into the principal plane of the pictured

points a, b, c, and d, their positions being found by transferring

the ordinates dd , bb Q , aa , and CC Q of the pictured points d, k,

a, and c to PPo from P , Pod=ddo, Po^=bb 0) P a = aa 0) and

The radials from S through d, /?, a, and 7- locate the points

<^o> A)> <*o> and 7-0 on the line GG or on 5oPo> and by transferring

the distances Sod Q) SoAb S'o^oj and Soro> Fig. 28, Plate XV, to

the principal line SoPo> Fig. 27, Plate XV, and drawing lines

through d 0) /? , o> and fo parallel with O O ', their intersec-

tions with the corresponding radials 5o^o> Sob Q , Soa , and SQC&

will locate the plotted positions D\, BI, A\, and Ci of the points

D, B, A, and C of the shore line of the lake.

IX. To Draw the Horizontal Projection of a Plane Figure ABCD

on the Ground Plan by Means of the So-called " Method of

Squares," if its Perspective in Vertical Plane, abed, and the

Elements of the Perspective are given.

If we imagine the figure covered with a net of squares in

such manner that one set of sides is parallel with, while the other

is perpendicular to, the ground line, such net may be used to

draw the outline of the figure upon the ground plan. It will

only remain necessary to cover the pictured figure abed with

the perspective of the net that has been selected for the ground

plan. The lines representing the squares in perspective must

have the proper relation with reference to both, the principal

ray and the horizon line, to conform with the net in the

ground plan.

The simplest disposition of the lines forming this auxiliary

net is the one mentioned above, with one set of sides parallel

with, and the other perpendicular to, the horizon line; still, any

other disposition of the net lines or sides may be made : they

THE "METHOD OF SQUARES." 73

may form equal-sized squares or not and their directions may

include any angle.

In Fig. 29, Plate XVI, in illustration of this method, the

lines of the perspective, corresponding to those sides of the

rectangular figures that had been drawn at right angles to the

ground line O O ', will vanish in the principal point P, while

those drawn parallel with the ground line OoOo' will be parallel

with the horizon line OO'.

Selecting the lines of this rectangular system so that one

line of each system passes through each one of the characteristic

points fl, b y c, and d of the pictured lake, the perspective of this

net will appear as shown by the fine lines in Fig. 29, Plate XVI,

where O O ' represents the ground line of the picture plane

MN.

If we again plot in the principal plane SS Q P Q P, Fig. 30,

Plate XVI, and retain the same designations as in Fig. 25, Plate

XIV, the points d , A)> <*o> and J-Q will represent, in the ground

plane GG, the intersections of the horizontal projection of the

principal ray SP=SoPo with those net lines that had been drawn

parallel with the ground line through D, B, A, and C.

After plotting the picture trace OoO</, of the perspective

MN, Fig. 29, Plate XVI, in the ground plan by means of the

radials SQ^O* ^o^o> etc., Fig. 31, Plate XVII, the distances Sod 0l

SoPo, etc., taken from Fig. 30, Plate XVI, and laid off upon

S P , Fig. 31, Plate XVII, will locate the intersections of S P Q

with those net lines (parallel with OoOo') in the ground plan

that correspond to the lines dd, bf), etc., of the perspective MN,

Fig. 29, Plate XVI.

If we now transfer the points a '> PQ> b ', d Q ', and CQ from

Fig. 29, Plate XVI, to the edge of a paper strip and place the

latter upon the picture trace O O ', Fig. 31, Plate XVII, that

the points P of both will coincide, then the lines a Q 'Ai, b Q 'Bi,

etc., drawn parallel with S P Q will represent those net lines

that are perpendicular to the ground line O O(/, and the plotted

positions AI, BI, Ci, and DI of the points A, B, C, and D are

74 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

treated on the ground plan as the intersections of corresponding

net lines of both systems as indicated in Fig. 31, Plate XVII.

The points A\, BI, Ci, and D\ will, of course, also be bisected

by the radials S a Q , S &o, S c Q , and S d , which fact may make

some other disposition of the net lines more desirable for a figure

of a different shape.

When the figure is bounded by a sinuous, perimeter, the

squares of the net should be selected sufficiently small to enable

the draughtsman to .draw the perimeter sections falling within

each square sufficiently accurate to. obtain a correct reduction

representing the general course of the figure's outline.

X. The " Vanishing Scale."

We had seen Fig. 31, Plate XVII that the radials drawn

from the so-called foot s of the station S represent directions

to the points AI, BI, C\, and D\ in the ground plane. If we

now could determine from the perspective the distances (S ^i,

SoBi, etc.) from the foot So of the station to the points to be

plotted their location in the ground plane would become an easy

matter.

The distances S Ai, S i, etc., may be determined from

the perspective by means of the so-called vanishing scale, which

may be constructed as follows, with reference to Fig. 32,

Plate XVII, where MN = trace of picture plane, HH = trace

of horizon plane, and GG = trace of ground plane, all in the

principal plane, and where SSo = elevation of the station 5

above the ground plane GG, or above the foot So of the station.

A scale of equal parts is laid off upon GG to both sides of P Q

and radials are then drawn from S through the graduation-

marks. Their intersections with MN form the so-called van-

ishing scale which may serve to locate the distances from the

foot So of the station S to points that are to be plotted in the

ground plane from the picture.

75

The picture trace O O ', Fig. 33, Plate XVIII, may have

been plotted and the radials SQOQ, Sob Q , etc., may have been

drawn on the working- sheet. It is desired to locate the posi-

tion A i of a point A in the ground plane that is pictured as a

in MN, Fig. 34, Plate XVIII, by means of the vanishing scale.

Take the ordinate aa Q from the photographic perspective MN

(the vertical distance of a above the ground line O Oo') and

lay it off upon the vanishing scale PoP, Fig. 32, Plate XVII,

from P > equal to PoX.

The line ax in the picture plane MN, Fig. 34, Plate XVIII,

drawn parallel with the horizon line OO' and passing through a,

is the perspective of the line AIOC, drawn parallel with the ground

line and passing through AI, Fig. 33, Plate XVIII, in the ground

plane. Hence, if we lay off SoX, Fig. 32, Plate XVII, upon

S Po, from 5 , Fig. 33, Plate XVIII, the point AI in the ground

plane will be situated upon the line XA\, drawn parallel with

the ground line OoO ' through X. The plotted position AI

of the point A will be at the intersection of the radial S 0o with

this line XA\.

CHAPTER V.

PHOTOGRAPHS ON INCLINED PLATES.

Until now we have regarded phototopographic plates exposed

in vertical planes, and although the general use of inclined

plates is not recommended for phototopographic purposes on

account of the complications that will arise in the generally

simple constructions underlying the iconometric plotting from

vertically exposed plates, and because the relations that exist

between the elements of the perspective and the orthogonal

projection into horizontal plan will not be so readily recognized.

Occasions may arise, however, where the selection of the availa-

ble or accessible stations will be so circumscribed as to make

exposures on inclined plates a necessity (to insure a good con-

trol of the inaccessible terrene forms). Photographs may also

have been obtained from balloons or with an ordinary camera

not supplied with devices for adjusting the plate into vertical

plane, or photographs originally taken for illustrative purposes

may perchance find use for iconometric plotting.

With reference to Fig. 35, Plate XIX, we have PP = prin-

cipal plane, HH= horizontal plane passing through the nodal

point of the camera-lens at station S, GG= ground plane, MN =

picture plane, O'P = trace of the picture plane MN in the hori-

zon plane HH, Oo'Po = ground line of the picture plane, So = foot

of the station 5, P'Po= principal line of the picture plane,

P'= principal point of the perspective MN, SS Q = vertical of

the station 5. It pierces the ground plane in the foot of the

station and passes through the picture plane MN above (or

below) the horizon line at s. The point s is the vanishing point

76

PHOTOGRAPHS ON INCLINED PLATES. 77

for the perspectives of all vertical lines that may be pictured

in MN. P'SP = P'sS =a= angle of inclination of the plate MN,

SP = perpendicular through 5 to the horizon line O'P, SA = line

of direction from 5 to a point A, pictured as a in MN.

If we revolve SP in the vertical plane PP about P until

SP falls within the picture plane, the point 5 will fall into (S)

and the line Sa will fall into (5)a.

The vertical plane, passing through SS and containing

the line SA, will intersect the ground plane in SQ^O- If we

revolve the line SoPo within the vertical plane PP about P

until S Po falls into the picture plane MN, the point S will

fall into (So) and the trace So^o will have assumed the position

The intersection A of the trace S a o with the line of direc-

tion Sa would locate the plotted position hi GG of the pictured

point a.

The line sa intersects the ground line in a , and S a Q will

be the radial in the ground plane from the foot SQ of the station S

that passes through the plotted position (in GG) of A . To

find A on So fl o we first locate in the picture plane the inter-

section (4) of the revolved lines (S)a and (So)a Q . This point

(A), revolved within the vertical plane doSoS, will locate A

upon So fl o-

To locate the position of A in GG in the manner just indi-

cated we should know the position of the line O'P, as well as

the points 5 and P. These are known, or may readily be found,

if the position of the principal point P', the length of the dis-

tance line SP', and the value of the angle of inclination a are

known.

When a photographic plate is purposely exposed in aa inclined

position in a surveying camera, it will generally be done in such

manner that the principal line //' still coincides with the inter-

section of the picture plane MN and the principal plane PP,

Fig. 35> Plate XIX.

When the angle of inclination a is an angle of elevation (depres-

78 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

sion) the horizon line O'P will fall below (above) the line repre-

senting the horizon line of the plate when exposed vertically.

The angles of inclination for inclined plates should be

observed directly in the field, and if the constant focal length

of the camera = / is known, the line SP may be found as the

hypothenuse of the right-angle triangle having the angle = a

and adjoining

A. To Plot the Picture Trace of an Inclined Plate.

To plot the picture trace the horizontal angle included between

the optical axis of the inclined camera and the horizontal direc-

tion to some known point should be known or measured.

Should the length S'Si', Fig. 36, Plate XX, and the posi-

tion of the line connecting two camera stations be known and

also the position of a third point A, visible from both stations,

no instrumental measurement of a horizontal angle a need

be made, provided the plates containing the pictures a of the

third point A are oriented in such manner that the picture a

of that third point be bisected by the vertical thread, by the

principal line //' of the perspective.

We have with reference to Fig. 36, Plate XX: S' = plotted

position of the station 5, S'Si f = plotted length and direction of

the base line, SS = elevation of the station S (laid off in the reduced

plotting-scale), Fig. 37, Plate XXI. The horizontal angle a

(at S', Fig. 36, Plate XX), included between the plotted base

line S'Si' and the principal plane (or the horizontal projection

S'Po of the optical axis SP') may have been observed in the field.

We revolve the line S'S about S'P , Fig. 36, Plate XX and

Fig. 37, Plate XXI, into the plotting- plane, when it will assume

the position 5' (5), and erect at (S) a line (S)(P) perpendicular

to S' (S). The angle of inclination of the plate MN = ? is laid

off from (5) upon (S)(P). We make (S)(P') equal to the

constant focal length of the camera = /, when the line (/)(/')>

drawn perpendicular to (S)(P f ) through (P), will represent

PHOTOGRAPHS ON INCLINED PLATES. 79

the principal line //' of the perspective MN, Fig. 37, Plate XXI r

revolved about 5'Po into the plotting-plane.

The point of intersection (s) of (5)5' with (/)(/') represents

the vanishing point for all vertical lines that may be shown in

picture MN.

The intersection P of the perpendicular line (/)(/') with

the horizontal projection of the optical axis S'Po will be the

trace of the inclined principal line //' in the ground plane (draw-

ing plan). The line Pog, perpendicular in PQ to 5'Po, is the

ground line or the trace of the inclined picture plane MN in

the drawing plan GG.

B. Plotting the Lines oj Direction to Points pictured on an

Inclined Photographic Plate.

The inclined picture plane MN is revolved about P g into

the drawing or ground plane, Fig. 37, Plate XXI, when it will

appear as (M)(N), the principal point P falling upon 5'P =

(/)(/') in (P) and (P)P is equal to PP .

the corresponding identical points in the vertical picture plane

M'N' are joined with the " kernel point " s f , and if these lines

are likewise produced to intersect the " perspective axis " IQ,

the points of intersection of IQ with the first group of lines (belong-

ing to MN) will be identical with the points of intersection of

IQ with the second group of lines (belonging to M'N').

If we now provide the " perspective axis " with a scale of

equal parts (having the zero or origin of graduation in the ground

plane), lines drawn through the " kernel points " and through

corresponding images of identical points in both picture planes

will intersect identical points of this scale.

The length O'O, Fig. 22, Plate XII, intercepted on the scale

of the " perspective axis " by the two horizon lines of the picture

planes MN and M'N' represents the difference in elevation

of the two camera stations 5 and S'. The scale IQ may be

drawn on both pictures to show on both lines i(I), Fig. 21, Plate

XII, after the pictures have been separated. Frequently the

picture itself will not be sufficiently extended to contain the

line IQ, in which case such a scale may still be used by placing

it upon a line XX", in MAT, and upon zz", in M'N', some dis-

TWO PERSPECTIVES OF THE SAME OBJECT. 67

tance from but parallel with the perspective axis 1Q, Fig. 22,

Plate XII, provided the following relation remain satisfied:

sQ:soc'=s'Q:s'z'.

For any other point B, photographed as b and 6' in the pic-

ture planes M N and M'N' respectively, the following propor-

tional equation should be fulfilled:

The triangles sxoxf, s!Q and Vz z', ^^Q being, respectively,

similar, XQOC* must be equal to ZQZ' (as fiQ is common to both

triangles spQ and s'^Q), which means the spaces on the scales

XX" and zz" are to be identical in numerical value. The two

scales (or either of them) may, if more convenient, be placed

beyond 5 or s f y f. i. at #", in which case

s{! : 5/0 = sfi : sx = s f ^ : s f z .

It should be noted that the scale is now to be read from f toward

/ . It may be stated generally that the scales should be placed

parallel with the " perspective axis " IQ and at distances from the

" kernel points " directly proportional to the distances of the latter

from the " perspective axis " of the picture planes, their correct

position being found from the horizontal projection or from

the ground plane. To avoid obscurity and obliteration of details

in the field of the photograph it will generally be more expe-

dient to draw these scales outside of the picture proper.

To find the proper position of the second scale on the second

picture, after the position of the scale on the first picture has

been decided upon, we again refer to Fig. 21, Plate XII, where

HH and H'H' are the two picture traces, 5 and S' are the

horizontal projections of the camera stations, P and P f are the

traces of the principal lines // and /'/' (Fig. 22), or the horizontal

68 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

projections of the principal points, and, finally, h the selected

position for the first scale.

To find the corresponding position h f of the second scale

we draw a line hh parallel to SS f through h, when

s f i\s'h=s l r i:s l 'h'\

hence

= distance of the second scale from

the " kernel point " s f in the second picture.

The conditions and relations described in the foregoing

paragraphs may often prove of value in iconometric plotting;

f. i., if we consider the case of a straight line L, Fig. 23, Plate XIII,

the image of which appears in picture MN as /, but in the second

picture M'N' only a short piece /' is seen. It may be desirable

to locate in the picture plane MN the reciprocal position of a

point x, shown on the line / in M'N, but falling outside of the

picture limit of M'N' on the prolongation of /'.

To find the position of x f we proceed as follows:

The pictured point x of the line /, pictured in MN, is con

nected with the kernel point (V) and the line (s')x is produced

to its intersection (x) with li. After transferring the point (x)

to the line il of the second picture plane M'N', to ((x)), and

connecting the latter with the " kernel point " (s), the intersec-

tion of ((x))(s) with /' produced will represent the point sought,

x*, on the prolongation of the line /'.

VIII. To Plot a Figure, Situated in a Horizontal Plane, on the

Ground Plan by Means of its Perspective.

Excepting the shore lines of lakes and coasts and the out-

lines of marshes, figures in horizontal planes are not frequently

met with in topographic surveys, and the simplest way to map

these would be to expose photographic plates in a horizontal

TO PLOT A FIGURE BY MEANS OF ITS PERSPECTIVE. 69

position from a captive balloon at points of known positions

and at identical or known elevations.

The mapping of such figures, when photographed on ver-

tically exposed plates, from stations above the figure's plane

is also an easy matter. It may even be done with but a single

perspective view of such figure (obtained on a vertically exposed

plate from a station of known position), provided we also know

the difference in elevation between the camera station and the

horizontal plane containing the figure, and provided we know

the positions of the principal point and horizon line together

with the length of the distance line (focal length) of the photo-

graphic perspective.

We have, with reference to Fig. 24, Plate XIII, HH= horizon

plane of the camera station S, OO' = horizon line of the photo-

graphic perspective MN, GG= ground plane or horizontal plane

coinciding with the surface plane of the lake A BCD, SSo = h

= difference in elevation between the camera station 5 and the

water level of the lake.

With a given perspective abed of the lake A BCD in the ver-

tical picture plane MN y known focal length, given position

of the principal point P and known difference in elevation, h,

between the water surface of the lake and the camera station,

the projection of the lake-outline (AiBiC\D\) in horizontal plan

may be drawn.

The ground line OoCV (line of intersection of ground plane

GG with the vertical picture plane MN) is drawn through P

(horizontal projection of P) parallel with the horizon line OO r ,

PP Q being equal to h (measured in the plotting -scale). If we

now project the pictured points a, b, c, d upon OoOo'=0o> b ,

c Q , d , the radials from the foot S of the station S drawn through

the points a ^o? CQ> do, will pass through the corresponding

points of the lake shore-line AI, BI, Ci, DI that are to be plotted.

Referring to the vertical plane passing through the camera

station S and through the pictured point a (it intersects the ground

plane in SoA or in SoAi) we find from the similar triangles

70 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

SSoA i and aa^A i the horizontal distance S$A i from the camera

station to the point sought, either graphically or arithmetically.

Imagining the vertical plane SSoAi to be revolved a,bout

SoAi until it coincides with the ground plane GG, the points S

and a will assume the positions (S) and (a), Fig. 24, Plate XIII,

and the line (5) (a) will pass through A\, hence AI may be

located in the ground plane as the intersection of (5) (a) with

Sodo. The same may be done for the other points BI, C\, and

DI by revolving the vertical planes SSoBi, SSoC\, and SSoDi

about Sob , S c , and S d Q into the ground plane GG to locate

the positions of BI, Ci, and D\.

To avoid the drawing of so many auxiliary lines on the work-

ing- or plotting-sheet, these constructions are preferably made

on a separate sheet of paper, and the following method may be

adopted :

The vertical planes passing through SQ^O* Sob , SQC Q , and

So^o m ay be supposed to be revolved about SSo, as common

axis of rotation, until they all coincide with the principal plane

SSoPo, Fig. 25, Plate XIV, the surface of the paper representing

the principal plane, when HH = trsice of the horizon plane

in the principal plane, MN = trace of the picture plane in

the principal plane, GG = trace of the ground plane in the

principal plane, SSo = ^= difference in elevation between the

station 5 and the ground plane GG, measured in the plotting-

scale, SP = SoPo = true length of the focal distance of the pho-

tograph MN.

The radials 5o#o> Se^o? So c o> and Sodo are laid off upon the

line GG from SO = SO(<IQ), So(bo), >>o(o)> and 5o(^o)> and the

verticals (a )(a), (^o)(^)> ( c o)( c )> an d (do)(d) are, made equal to

the ordinates aa , bb Q , cc , and dd , respectively, measured on

the picture.

Radials drawn through (a), (6), (c), and (d) from S will

cut off on the line GG the horizontal distances S (A), So(B),

5o(C), and -So(D). These distances, laid off on the radials

S a , Sob , S c , and S d 0) on the plotting-sheet will locate, in

TO PLOT A FIGURE BY MEANS OF ITS PERSPECTIVE. 71

the scale of the map, the plotted positions of the characteristic

points AI, BI, Ci, and DI of the lake, with reference to the ground

line O Oo', which is identical on the plotting-sheet with the

picture trace.

We may reach the same results by utilizing the orthogonal

projections of the points a, b, c, and d and those of AI, BI, Ci,

and DI into the principal plane instead of revolving the ver-

tical planes separately into the principal plane, as done above.

With reference to Fig. 26, Plate XIV, we would then have:

PP = principal plane, MN = picture plane, HH= horizon

plane, containing the camera station 5, GG= ground plane or

surface plane of the lake A BCD.

If we draw the radials SQCLQ, S Q b Q , SQC O , and S d from 5

(the orthogonal projection of 5 in GG) through the orthogonal

projections of the pictured points a, b, c, d on the ground line

O O ', the points sought will fall upon those radials. After

projecting the points a, b, c, and d, in the picture plane MN,

upon the principal line (=a, /?, ?-, and d) the radials Sa, S/?,

Sf, and Sd (drawn in the principal plane PP) will locate the

points a , A)> ro and d , respectively, upon the line SoPo (in

the ground plane), and these represent the orthogonal projec-

tions of the points A , B, C, and D in GG upon SoPo. Hence

the points A, B, C, and D may be found by erecting perpen-

diculars upon SoPo in a , /? , Toy an d <^o> respectively, and their

points of intersection with the radials Soa , Sob , SQC O , and Sodo,

respectively, will be the positions of the plotted points A, B, C,

and D.

Also this construction is preferably made upon a separate

sheet of paper, Fig. 27, Plate XV, where the radials Soa , Sob ,

SQCQ, and Sodo are drawn through their corresponding points

on the plotted picture trace or ground line O O ', but the rest

of the construction is made on the separate sheet of paper, con-

sidering the surface of the latter to coincide with the principal

plane (Fig. 28, Plate XV, where the designations are the same

as in Fig. 25, Plate XIV).

72 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

The points d, /?, a, and 7- on the line PPo (principal line)

represent the projections into the principal plane of the pictured

points a, b, c, and d, their positions being found by transferring

the ordinates dd , bb Q , aa , and CC Q of the pictured points d, k,

a, and c to PPo from P , Pod=ddo, Po^=bb 0) P a = aa 0) and

The radials from S through d, /?, a, and 7- locate the points

<^o> A)> <*o> and 7-0 on the line GG or on 5oPo> and by transferring

the distances Sod Q) SoAb S'o^oj and Soro> Fig. 28, Plate XV, to

the principal line SoPo> Fig. 27, Plate XV, and drawing lines

through d 0) /? , o> and fo parallel with O O ', their intersec-

tions with the corresponding radials 5o^o> Sob Q , Soa , and SQC&

will locate the plotted positions D\, BI, A\, and Ci of the points

D, B, A, and C of the shore line of the lake.

IX. To Draw the Horizontal Projection of a Plane Figure ABCD

on the Ground Plan by Means of the So-called " Method of

Squares," if its Perspective in Vertical Plane, abed, and the

Elements of the Perspective are given.

If we imagine the figure covered with a net of squares in

such manner that one set of sides is parallel with, while the other

is perpendicular to, the ground line, such net may be used to

draw the outline of the figure upon the ground plan. It will

only remain necessary to cover the pictured figure abed with

the perspective of the net that has been selected for the ground

plan. The lines representing the squares in perspective must

have the proper relation with reference to both, the principal

ray and the horizon line, to conform with the net in the

ground plan.

The simplest disposition of the lines forming this auxiliary

net is the one mentioned above, with one set of sides parallel

with, and the other perpendicular to, the horizon line; still, any

other disposition of the net lines or sides may be made : they

THE "METHOD OF SQUARES." 73

may form equal-sized squares or not and their directions may

include any angle.

In Fig. 29, Plate XVI, in illustration of this method, the

lines of the perspective, corresponding to those sides of the

rectangular figures that had been drawn at right angles to the

ground line O O ', will vanish in the principal point P, while

those drawn parallel with the ground line OoOo' will be parallel

with the horizon line OO'.

Selecting the lines of this rectangular system so that one

line of each system passes through each one of the characteristic

points fl, b y c, and d of the pictured lake, the perspective of this

net will appear as shown by the fine lines in Fig. 29, Plate XVI,

where O O ' represents the ground line of the picture plane

MN.

If we again plot in the principal plane SS Q P Q P, Fig. 30,

Plate XVI, and retain the same designations as in Fig. 25, Plate

XIV, the points d , A)> <*o> and J-Q will represent, in the ground

plane GG, the intersections of the horizontal projection of the

principal ray SP=SoPo with those net lines that had been drawn

parallel with the ground line through D, B, A, and C.

After plotting the picture trace OoO</, of the perspective

MN, Fig. 29, Plate XVI, in the ground plan by means of the

radials SQ^O* ^o^o> etc., Fig. 31, Plate XVII, the distances Sod 0l

SoPo, etc., taken from Fig. 30, Plate XVI, and laid off upon

S P , Fig. 31, Plate XVII, will locate the intersections of S P Q

with those net lines (parallel with OoOo') in the ground plan

that correspond to the lines dd, bf), etc., of the perspective MN,

Fig. 29, Plate XVI.

If we now transfer the points a '> PQ> b ', d Q ', and CQ from

Fig. 29, Plate XVI, to the edge of a paper strip and place the

latter upon the picture trace O O ', Fig. 31, Plate XVII, that

the points P of both will coincide, then the lines a Q 'Ai, b Q 'Bi,

etc., drawn parallel with S P Q will represent those net lines

that are perpendicular to the ground line O O(/, and the plotted

positions AI, BI, Ci, and DI of the points A, B, C, and D are

74 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

treated on the ground plan as the intersections of corresponding

net lines of both systems as indicated in Fig. 31, Plate XVII.

The points A\, BI, Ci, and D\ will, of course, also be bisected

by the radials S a Q , S &o, S c Q , and S d , which fact may make

some other disposition of the net lines more desirable for a figure

of a different shape.

When the figure is bounded by a sinuous, perimeter, the

squares of the net should be selected sufficiently small to enable

the draughtsman to .draw the perimeter sections falling within

each square sufficiently accurate to. obtain a correct reduction

representing the general course of the figure's outline.

X. The " Vanishing Scale."

We had seen Fig. 31, Plate XVII that the radials drawn

from the so-called foot s of the station S represent directions

to the points AI, BI, C\, and D\ in the ground plane. If we

now could determine from the perspective the distances (S ^i,

SoBi, etc.) from the foot So of the station to the points to be

plotted their location in the ground plane would become an easy

matter.

The distances S Ai, S i, etc., may be determined from

the perspective by means of the so-called vanishing scale, which

may be constructed as follows, with reference to Fig. 32,

Plate XVII, where MN = trace of picture plane, HH = trace

of horizon plane, and GG = trace of ground plane, all in the

principal plane, and where SSo = elevation of the station 5

above the ground plane GG, or above the foot So of the station.

A scale of equal parts is laid off upon GG to both sides of P Q

and radials are then drawn from S through the graduation-

marks. Their intersections with MN form the so-called van-

ishing scale which may serve to locate the distances from the

foot So of the station S to points that are to be plotted in the

ground plane from the picture.

75

The picture trace O O ', Fig. 33, Plate XVIII, may have

been plotted and the radials SQOQ, Sob Q , etc., may have been

drawn on the working- sheet. It is desired to locate the posi-

tion A i of a point A in the ground plane that is pictured as a

in MN, Fig. 34, Plate XVIII, by means of the vanishing scale.

Take the ordinate aa Q from the photographic perspective MN

(the vertical distance of a above the ground line O Oo') and

lay it off upon the vanishing scale PoP, Fig. 32, Plate XVII,

from P > equal to PoX.

The line ax in the picture plane MN, Fig. 34, Plate XVIII,

drawn parallel with the horizon line OO' and passing through a,

is the perspective of the line AIOC, drawn parallel with the ground

line and passing through AI, Fig. 33, Plate XVIII, in the ground

plane. Hence, if we lay off SoX, Fig. 32, Plate XVII, upon

S Po, from 5 , Fig. 33, Plate XVIII, the point AI in the ground

plane will be situated upon the line XA\, drawn parallel with

the ground line OoO ' through X. The plotted position AI

of the point A will be at the intersection of the radial S 0o with

this line XA\.

CHAPTER V.

PHOTOGRAPHS ON INCLINED PLATES.

Until now we have regarded phototopographic plates exposed

in vertical planes, and although the general use of inclined

plates is not recommended for phototopographic purposes on

account of the complications that will arise in the generally

simple constructions underlying the iconometric plotting from

vertically exposed plates, and because the relations that exist

between the elements of the perspective and the orthogonal

projection into horizontal plan will not be so readily recognized.

Occasions may arise, however, where the selection of the availa-

ble or accessible stations will be so circumscribed as to make

exposures on inclined plates a necessity (to insure a good con-

trol of the inaccessible terrene forms). Photographs may also

have been obtained from balloons or with an ordinary camera

not supplied with devices for adjusting the plate into vertical

plane, or photographs originally taken for illustrative purposes

may perchance find use for iconometric plotting.

With reference to Fig. 35, Plate XIX, we have PP = prin-

cipal plane, HH= horizontal plane passing through the nodal

point of the camera-lens at station S, GG= ground plane, MN =

picture plane, O'P = trace of the picture plane MN in the hori-

zon plane HH, Oo'Po = ground line of the picture plane, So = foot

of the station 5, P'Po= principal line of the picture plane,

P'= principal point of the perspective MN, SS Q = vertical of

the station 5. It pierces the ground plane in the foot of the

station and passes through the picture plane MN above (or

below) the horizon line at s. The point s is the vanishing point

76

PHOTOGRAPHS ON INCLINED PLATES. 77

for the perspectives of all vertical lines that may be pictured

in MN. P'SP = P'sS =a= angle of inclination of the plate MN,

SP = perpendicular through 5 to the horizon line O'P, SA = line

of direction from 5 to a point A, pictured as a in MN.

If we revolve SP in the vertical plane PP about P until

SP falls within the picture plane, the point 5 will fall into (S)

and the line Sa will fall into (5)a.

The vertical plane, passing through SS and containing

the line SA, will intersect the ground plane in SQ^O- If we

revolve the line SoPo within the vertical plane PP about P

until S Po falls into the picture plane MN, the point S will

fall into (So) and the trace So^o will have assumed the position

The intersection A of the trace S a o with the line of direc-

tion Sa would locate the plotted position hi GG of the pictured

point a.

The line sa intersects the ground line in a , and S a Q will

be the radial in the ground plane from the foot SQ of the station S

that passes through the plotted position (in GG) of A . To

find A on So fl o we first locate in the picture plane the inter-

section (4) of the revolved lines (S)a and (So)a Q . This point

(A), revolved within the vertical plane doSoS, will locate A

upon So fl o-

To locate the position of A in GG in the manner just indi-

cated we should know the position of the line O'P, as well as

the points 5 and P. These are known, or may readily be found,

if the position of the principal point P', the length of the dis-

tance line SP', and the value of the angle of inclination a are

known.

When a photographic plate is purposely exposed in aa inclined

position in a surveying camera, it will generally be done in such

manner that the principal line //' still coincides with the inter-

section of the picture plane MN and the principal plane PP,

Fig. 35> Plate XIX.

When the angle of inclination a is an angle of elevation (depres-

78 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

sion) the horizon line O'P will fall below (above) the line repre-

senting the horizon line of the plate when exposed vertically.

The angles of inclination for inclined plates should be

observed directly in the field, and if the constant focal length

of the camera = / is known, the line SP may be found as the

hypothenuse of the right-angle triangle having the angle = a

and adjoining

A. To Plot the Picture Trace of an Inclined Plate.

To plot the picture trace the horizontal angle included between

the optical axis of the inclined camera and the horizontal direc-

tion to some known point should be known or measured.

Should the length S'Si', Fig. 36, Plate XX, and the posi-

tion of the line connecting two camera stations be known and

also the position of a third point A, visible from both stations,

no instrumental measurement of a horizontal angle a need

be made, provided the plates containing the pictures a of the

third point A are oriented in such manner that the picture a

of that third point be bisected by the vertical thread, by the

principal line //' of the perspective.

We have with reference to Fig. 36, Plate XX: S' = plotted

position of the station 5, S'Si f = plotted length and direction of

the base line, SS = elevation of the station S (laid off in the reduced

plotting-scale), Fig. 37, Plate XXI. The horizontal angle a

(at S', Fig. 36, Plate XX), included between the plotted base

line S'Si' and the principal plane (or the horizontal projection

S'Po of the optical axis SP') may have been observed in the field.

We revolve the line S'S about S'P , Fig. 36, Plate XX and

Fig. 37, Plate XXI, into the plotting- plane, when it will assume

the position 5' (5), and erect at (S) a line (S)(P) perpendicular

to S' (S). The angle of inclination of the plate MN = ? is laid

off from (5) upon (S)(P). We make (S)(P') equal to the

constant focal length of the camera = /, when the line (/)(/')>

drawn perpendicular to (S)(P f ) through (P), will represent

PHOTOGRAPHS ON INCLINED PLATES. 79

the principal line //' of the perspective MN, Fig. 37, Plate XXI r

revolved about 5'Po into the plotting-plane.

The point of intersection (s) of (5)5' with (/)(/') represents

the vanishing point for all vertical lines that may be shown in

picture MN.

The intersection P of the perpendicular line (/)(/') with

the horizontal projection of the optical axis S'Po will be the

trace of the inclined principal line //' in the ground plane (draw-

ing plan). The line Pog, perpendicular in PQ to 5'Po, is the

ground line or the trace of the inclined picture plane MN in

the drawing plan GG.

B. Plotting the Lines oj Direction to Points pictured on an

Inclined Photographic Plate.

The inclined picture plane MN is revolved about P g into

the drawing or ground plane, Fig. 37, Plate XXI, when it will

appear as (M)(N), the principal point P falling upon 5'P =

(/)(/') in (P) and (P)P is equal to PP .