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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

. (page 13 of 33)
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 13 of 33)
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A protractor may be constructed to measure these angles
directly. It consists of a transparent plate on which lines are
drawn parallel to the principal line containing points of the
same azimuth and curves containing points of the same altitude.

The azimuthal lines are found by plotting the angles in S
and drawing parallels to the principal line SS' through the points
of intersection with the horizon line.

If we take the horizon and principal lines as axes of coor-
dinates and denote the altitude of a point pictured as a by h>
the equation of the curve of altitude h may be written



This also is the equation of a hyperbola of which the prin-
cipal and horizon lines are the transverse and conjugate axes
and of which the principal point is the center.

One of the hyperbola's branches represents the points above
the horizon 'and the other branch those of equal altitude below
the horizon.



156 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

The asymptotes are lines intersecting each other at the prin-
cipal point and making angles equal to h with the horizon line.
This hyperbola is the intersection by the picture plane of the
cone of visual rays forming the angle h with the horizon.

These hyperbolic curves of equal altitude may be obtained by
computation, using the preceding formula and substituting
different values for h, or they may be obtained graphically by
plotting a series of points for each curve, reversing the construc-
tion given above for finding the altitude of the pictured point a,
Fig. 73, Plate XL. The angular distance between the lines
representing points of equal azimuths or those of the same
altitude depends on the degree of precision aimed at.

The complete protractor is shown in Fig. 74, Plate XL. It
may be made in the same manner as mentioned for the per-
spectometer by drawing it on paper on a large scale, reducing
it by photography, and finally making a transparency by bleach-
ing the negative in bichloride of mercury.

D. Method of V. Legros for Locating the Horizon Line of a
Vertically Exposed Plate.

Commandant Legros recommends the use of these hyper-
bolic curves for the location of the horizon line of a vertically
exposed plate.

When the camera with the photographic plate adjusted in
vertical plane is rotated horizontally, the plate remaining ver-
tical, any point a, Fig. 74, Plate XL, will describe a hyperbola
aa' in the picture plane (on the ground -glass plate). The
nearer a approaches the horizon line the smaller the curvature
of its hyperbolic trace on the ground-glass plate will become,
and that point, a, which traverses the ground-glass plate in
a straight line, HH', will have the same elevation as the second
nodal point of the camera-lens its angle of elevation will be= -fO,
or HH' will be the horizon line of the plate.

To locate the horizon line experimentally in this way the



GRAPHICAL ICONOMETRICAL PLOTTING METHODS. 157

ground glass is best provided with a series of equidistant hori-
zontal and vertical lines, after the manner of Dr. Le Bon's ground-
glass plates.

E. Pro). S. Finsterwalder 's Method jor the Iconometric Plotting
oj Horizontal Contours.

Prof. Finsterwalder's method for plotting horizontal contours
is well adapted for the development of the terrene forms of a
moderately rolling country and it is based upon the following
consideration :

The pictured outline of a terrene form may be regarded as
the trace of the terrene surface in a plane (picture plane) ver-
tical to the plotting or ground plane.

The camera stations should be specially selected with refer-
ence to the use of this method with a view toward obtaining
pictures with a sufficient number of such outlines, or silhouettes,
of the terrene forms to enable the iconometric draughtsman to
give a good definition of the relief of the terrene to be plotted.

These terrene-form silhouettes may be regarded as falling
within vertical planes and the rays drawn from the point of
view to the pictured points of the silhouette will form a cone,
with apex in the second nodal point of the lens (or point of view),
its base being formed by the pictured outline (silhouette) of the
terrene. A horizontal plane containing a contour A will inter-
sect such a cone of rays in a curve B, the latter touching A in
one point.

If we designate by h the difference in elevation between
the station (whence the picture was obtained) and the hori-
zontal contour A, by ft the vertical angle of each radial or visual
ray drawn to each point of the silhouette, then the curve B may
be plotted on the working-sheet by laying off, upon a few rays,
from the plotted station to points of the pictured outline the
corresponding distances

h cot 0,



158 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

and the points thus located on the radials drawn from the sta-
tion point, if connected by a continuous line, will represent the
curve B, plotted in horizontal plan.

The direction of the silhouetted outline is now plotted on
the plan, and where it bisects this curve B will be a point of
the contour A. As we, naturally, would draw not only one
curve B but a series of them corresponding to several horizontal
planes, passing through a series of contours A of various ele-
vations, the construction may be simplified, inasmuch as the
curves B being the lines of intersection of the same cone of
rays with a series of parallel planes containing the horizontal
contours will all be similar in shape, their corresponding points
having the same relative positions with reference to the plotted
station, and the value h cot ft need only be determined for one
point of the remaining curves B if one curve B had been drawn ;
the others will be parallel to it.



CHAPTER VII.

CAMERA-LENSES,
i

THE general theory and laws of optics as applied to lenses
are the same whether the latter are to be mounted in telescopes
or in photographic cameras. The camera may even be regarded
as an incomplete telescope, lacking only a suitable eyepiece to
convert it into a telescope.

Still, photographic lenses are to fulfill requirements differing
widely from those of telescopes, the main difference being in
the field commanded by either. As only the central part of a
telescopic lens is utilized for observing, comprising a field of
but a few degrees, spherical and chromatic aberration do not
affect the latter. Phototopographic lenses, however, should
command as wide an angle as possible (over 60) and still pro-
duce geometrically true perspectives without distortion, with a
sharp definition, a uniformly bright illumination for the entire
plane surface of the sensitive plate, and with a great depth of
focus.

Rapidity in the action of the camera-lens being desirable,
but not of essential importance for surveying purposes, the
quality of the lens will in a great measure determine the value
of the photogrammeter or photographic surveying camera.

A. The Refractive Index.

With reference to Fig. 75, Plate XLI, we designate by AB
the refractive surface, by 57 the incident ray, by IP the refracted
ray, by CC\ the perpendicular to the refractive surface in the
point where the incident ray SI enters the second medium, by a

159



l6o PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

the angle included between the perpendicular CC\ and the inci-
dent ray SI, and by /? the angle of refraction.

75 being equal to IP = r, the ratio of the sines of the angles a
and /? may be expressed by the ratio of the lines DS and EP,
or, in other words,

For all angles a, larger and smaller than the one indicated
in Fig. 75, Plate XLI, the ratio between DS and EP will be
the same for the same two substances.

This constant ratio is termed the " refractive index " of the
two substances that are separated by the refractive surface AB.
The incident ray, the refracted ray, and the emergent ray (coming
from the same source) are all in one plane.

When speaking of the refractive index of any one medium
in optics, it is always to be understood that the incident ray
has passed through air (or space). Thus we have, approxi-
mately, if the refractive index for air or space be assumed as
unity, the refractive index for

Water, about 1.3

For crown glass, about 1.5

For flint glass. 1.6 to 1.9

For diamond, about 2.4, etc.

This means, for instance, that for any angle a (for any
incident ray SI) the vertical DS is 1.5 times as long as PE if
the ray passes through air and is refracted by crown glass.

B. Refraction of Light-rays.

The preceding consideration enables us to find the means
for changing the course of light-rays by refracting them to any
amount desirable.

With reference to Fig. 76, Plate XLI, we have AB and
A \Bi = refracting surfaces of a piece of plate glass.

The incident ray 57 arrives at the surface AB under an angle
a with the perpendicular 1C (perpendicular to AB in 7). Glass
being denser than air the ray will be refracted toward the per-



THE OPTICAL LENS. l6l

pendicular 1C, continuing in a straight line IE as long as it passes
through this second medium (glass) ; arriving at E it passes from
the denser medium into air and at E it will be refracted away
from the perpendicular EC\ (under an angle a) and continue
in the direction EP, parallel to the incident ray SI.

By changing the direction or position of one or of both sur-
faces of the denser medium (glass) the final direction of EP
may be given any course, since the equation

sin a

- ~ = w = refractive index

sin ft

must always be fulfilled.

It becomes plain that the change in the direction of EP from Sf
will increase directly with the angle included between the two
refractive surfaces AB and AiBi.

In Fig. 77, Plate XLI, this change in direction is shown for
three different glass prisms shaped in such a way that their
refractive angles not only decrease from A toward B, but have
been given such values that the three rays emanating from a
certain luminous point 5, after refraction, converge to one
point P.

A point P, where several converging rays (originally ema-
nating from a point 5 in space) intersect one another, is termed
an " image point."

C. The Optical Lens.

If the directions not only of three, Fig. 77, Plate XLI, but
of an infinite number of light-rays emanating from a luminous
point S are to be so changed that all will converge into a point P,
we will have to superimpose an infinite number of prisms one
upon the other. The heights of these prisms will have to be
made infinitesimally small and the refractive angles of two neigh-
boring ones will differ by an infinitesimal small amount.

This means that the broken lines AB and ABi, Fig. 77,



162 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

Plate XLI, will become curves, and a piece of glass with its
two faces shaped in such a manner that all light pencils ema-
nating from the same point S will converge to meet in its image
point P is termed an optical lens.

Evidently such a lens is a body formed by rotating the fig-
ure ABB i, composed of an infinite number of prism sections,
about the line BB i as axis. This axis of rotation is termed
the optical axis of the lens, and the latter may be considered
as composed of concentric zones or rings with spherically shaped
outer surfaces. The question now arises what form should
be given the figure ABBi to obtain a lens that will produce optical
images of luminous points.

Opticians can produce in the manufacture of lenses only
spherical surfaces with any degree of precision; therefore all
optical lenses are inclosed by spherical surfaces. Still, spherical
lenses produce well-defined and sharp images of luminous points
only within certain limits, limits between which the spherical
surface approaches very closely that ideal shape which is best
adapted for the purpose in view, but which cannot be manu-
factured owing to mechanical difficulties encountered in the
grinding or cutting process of the lens. In our superficial treat-
ment of the laws of optics considered inasmuch as they apply
to phototopography only we shall assume that the spherical
lenses are optically perfect and of a small thickness.

The deduction of the optical laws governing the action of
lenses of various shapes would require complicated computa-
tions; still, at least a general consideration of certain optical
laws and facts should not be omitted in this treatise on photo-
topography, in order to better elucidate the formation of the
optical images and to determine such elements of the photo-
graphic lens as will be needed in iconometric plotting.

Generally speaking, we meet with so-called simple lenses
and with combinations or sets of lenses in photography. The
symmetrical combinations are preferable for topographic sur-
veying purposes, as they command a wider field or larger view



OPTICAL DISTORTION. 163

angle and as they are less affected with distortion and aberration
than is generally the case with the simple or single lenses.



D. Optical Distortion.

So-called " spherical aberration " is more commonly pro-
duced by those light rays which pass through the marginal zone
of the lens, as this part of the lens is less perfect than the cen-
tral part. Spherical aberration may be reduced by decreasing the
effective diameter of the lens which is generally done by insert-
ing a so-called " diaphragm " between the lenses forming the
combination, or by a reduction of the curvature of the faces of
the lens.

Lenses corrected for spherical aberration are known as
aplanatic lenses.

In so-called " chromatic aberration " the different color
rays which compose the white light are unevenly refracted, and
colored, ill-defined images are the result.

Lenses corrected for chromatic aberration are termed achro-
matic lenses.

Probably the greatest source of error introduced into photog-
raphy is due to distortion of the image when using an inferior
lens. It is caused primarily by the greater refraction in the
direction toward the optical axis of those light-rays which pass
through the marginal or border zones of the lens. When the
image on the ground glass of a test-screen of the form shown
in Fig. 78, Plate XLII, assumes the form indicated in Fig. yg r
Plate XLII, so-called " pin-cushion distortion " has taken place,
whereas an image of the form shown in Fig. 80, Plate XLII,
is produced by " barrel-shape distortion." A lens affected by
either is unfit for phototopographic purposes.

Commandant Moessard has invented an ingenious little con-
trivance the tourniquet by means of which the field or
angle that is affected by distortion of the image may readily
be determined experimentally.



164 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

Astigmatic distortion in an image is produced when well-
defined images of the lateral points of an image may be obtained
for two different positions of the ground -glass plate, and yet
neither of these two images of the same .points will represent
the true shape of the original. Using a test-screen of the shape
shown in Fig. 81, Plate XLII, radial distortion will be shown,
Fig. 82, Plate XLII, for one position of the f ocusing-glass ;
the distortion will be in directions radiating from the center
of the ground-glass plate. In the other position of the focusing-
glass tangential distortion will be observed, Fig. 83, Plate
XLII; the distortions will appear in directions at right angles
to the directions radiating from the center of the ground-glass
plate. Both radial and tangential distortions increase from the
center toward the extraaxial zones of the lens.

Lenses corrected for astigmatic distortion are termed anastig-
matic lenses.

The distortion shown in Fig. 84, Plate XLIII, of the image
of the test-screen, Fig. 81, Plate XLII, is due to imperfect regis-
tering of two lenses composing a double lens; the component
lenses are not " centered."

The Zeiss anastigmatic lens has a perfectly flat field. That
is to say, if the ground glass has been focused for the sharp
definition of a central point, extraaxial points will also be well
defined on the focusing-plate.

Nearly all the older lens types were characterized by more
or less curvature of the field, which means the focal length when
focusing for a central point would be longer than when focusing
for sharp definitions of a marginal point shown on the image
plate.



NODAL POINTS AND NODAL PLANES OF A LENS. 165



E. Nodal Points and Nodal Planes of a Lens.

Formerly the thickness of a lens was disregarded when inves-
tigating its action upon light- rays passing through it and it was
generally assumed that the central rays those passing through
the so-called " optical center " of a lens suffered no change
of direction.

Lenses are generally regarded as being bounded by two
spherical surfaces. If both sides are convex (the lens is thicker
in the center than at the edge) it is termed a biconvex or
positive lens, Fig. 85, Plate XLIII.

If the spherical surfaces are concave on both sides of the
lens (its center is thinner than its edge) it will be a biconcave
or a negative lens, Fig. 89, Plate XLV.

Fig. 86, Plate XLIII, represents the cross-section of a con-
cave-convex or a periscopic convex lens, the convex surface
having a shorter radius than the concave surface, the lens
being thicker at its center than at its margin. When the con-
cave surface has the shorter radius the lens would be called
convex-concave. The principal 'elements of a lens (Figs. 85 and
86, Plate XLIII) are:

First. The geometrical centers; they are the centers C
and Ci of the spherical surfaces forming the faces of the
lens.

The line passing through C and Ci is termed the principal
axis of the lens.

Second. The vertices A and B of the lens are the inter-
sections of the principal axis with the two spherical lens
surfaces.

Third. The thickness AB of the lens is the distance between
the vertices of the lens.

A lens is centered " when the plane PP, passing through the
circumference of the lens passing through the circular line of



1 66 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

intersection of the two lens surfaces is intersected by the prin-
cipal axis at right angles.

A lens-combination is centered when the planes PP\ of the
individual lenses are parallel and if they are intersected by the
principal axis at right angles.

The foci of the separate lenses should also fall upon the
principal axis or the images of the discs shown on the test-screen,
Fig.. 81, Plate XLII, will show so-called " flare spots," some-
what like those represented in Fig. 84, Plate XLIII.

A large flare spot, or halo, in the center of an image or
picture may be produced by halation, caused by light- rays that
have passed through the diaphragm aperture being reflected from
the lens surfaces.

There exist certain relationships between the curvature of a
lens, the distance of a luminous point from the lens, and the
distance of its image from the lens which we will now briefly
consider.

An incident ray SI, Fig. 85, Plate XLIII, will be refracted
at / toward the radius R ( = Co/), glass being a denser medium
than air; it will continue through the lens in the direction /,
and the emergent ray EP will be parallel to the incident ray SI.
The radii CI and C\E are also parallel. The point C , where
the refracted part IE of the light-ray intersects the optical axis
of the lens, is known as the optical center of the lens and
the following relation exists between its distances from the geo-
metrical centers and the radii of the two lens surfaces :

CC R

R\



The triangles ICoC and EC$C\ are similar.

Every lens has two nodal points N and NI, Figs. 85 and
86, Plate XLIII, on the optical axis of the lens. The rays reach-
ing the first nodal point N from luminous points S in space
are parallel with the rays connecting the second nodal point NI
with the corresponding images P of the luminous points.



NODAL POINTS AND JSTODAL PLANES OF A LENS. 167

Hence a negative produced by an optical-lens system may
be regarded as a central projection or as a perspective image
(the center of which coincides with the second nodal point NI),
Fig. 88, Plate XLIV.

The positions of the nodal points will be constant for all
rays that make a small angle with the optical axis of the lens
(for all rays passing through the small aperture of a diaphragm).
The distances of the nodal points from the corresponding ver-
tices of the lens are constants and their values are given by the
equations (Fig. 85, Plate XLIII)

CN



where n is the refractive index from air into glass, or w=f.
As the distances of the nodal points from the optical center
and CoNi) will be small they may sometimes become



inappreciable or=o we may omit the factors and * *



from the equations (when CoN=o and CVVi=o the factors

7^77- and * ~ l will become =i), hence
CCo CiCo



A AT

AN=

n



A close approximation to the distance between the nodal
points will be



1 68 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

The planes and the points of a lens system are numbered
in the sense of the direction of the incident rays. With refer-
ence to Fig. 87, Plate XLIV, the light is supposed to be coming
from the left, hence

N = first nodal point;
F= first principal focus;
FG= first focal plane;
H U NK= first nodal plane, whereas
NI = second nodal point; ,
FI = second principal focus, etc.

Lengths are considered plus, or positive, if they extend in
the sense of the direction of the incident rays, and minus, or
negative, if they extend in the opposite direction. With refer-
ence to Fig. 87, Plate XLIV, where the light comes from the
left, we have

FN=-f and F^i^+fr.

The nodal planes (passing through the nodal points and
intersecting the principal axis at right angles) coincide with the
principal planes if the extreme or outer medium of the optical
lens system is the same, which is the case in photography where
air surrounds the lens.



F. Principal Foci and Focal Planes of a Lens.

The principal foci F and FI of a biconvex or positive
lens, Fig. 87, Plate XLIV, are two points on the optical axis
one on either side of the lens where those incident rays con-
verge which arrive at the refractive lens surface in a course
parallel to the optical axis.

In Fig. 87, Plate XLIV, the ray S'I IV , coming from a lumin-
ous point S' at infinite distance from the lens, traverses a path
parallel in its course to the optical axis FF\ of the lens, and



THE FOCAL LENGTH OF A LENS. 169

after refraction converges to the point FI', while a similar ray
PE", coming from the opposite direction in a course parallel
with the optical axis, converges to F.

The planes FG and FiGi, passing through the foci F and FI
and intersecting the optical axis FFi at right angles, are termed
focal planes.



G. The Focal Length of a Lens.

FN and FiNi, Fig. 87, Plate XLIV, are termed focal
lengths and they are generally designated by the letter /. The
value of the focal length is expressed by the equation






When this value for /i becomes positive it is an indication
that the incident rays, when coming from infinite distance (parallel
with the optical axis), are refracted to converge to the principal
focus of the lens.

A negative value for /i would indicate that the rays entering
the lens in a course parallel to its optical axis diverge from the
principal focus.

For thin lenses (the distance between A and B, Fig. 85,
Plate XLIII, is very small in comparison with the lengths of
the radii R and RI and it may be assumed =o) the formula
for the focal length would read



RRi



170 PHOTOTOPOGRAPHIC METHODS AND INSTRUMENTS.

After substitution of f for w, the approximate value for the
refractive index of glass, the approximate value of the focal
length for thin lenses would be



2R



H. The Biconvex or Positive Lens.

The image of a point at infinite distance from the biconvex
lens is on the opposite side of the lens and falls together with
its principal focus.

When the distant luminous point approaches the lens the
image will recede, at first slowly, but more rapidly the nearer
the luminous point advances toward the lens, and by the time
the original point will have reached a distance from the lens
equal to its double focal length, its image will have moved to a
point beyond the lens, also at a distance of the double focal length.
When the luminous point continues to approach the lens within
the double focal distance range, its image moves faster and
faster beyond the double focal distance on the opposite side of
the lens, and when the luminous point finally falls together
with the first focus (F) the image will disappear at infinite dis-



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 13 of 33)