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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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-

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-