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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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Online LibraryJohn Adolphus FlemerAn elementary treatise on phototopographic methods and instruments, including a concise review of executed phototopographic surveys and of publicatins on this subject → online text (page 7 of 33)
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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 .



Online LibraryJohn Adolphus FlemerAn elementary treatise on phototopographic methods and instruments, including a concise review of executed phototopographic surveys and of publicatins on this subject → online text (page 7 of 33)