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ground and on the photograph, which is then also fixed in position and orientation. Where the tilt is known, approximately three points common to map and photograph will furnish a solution. Two further points of importance are that (l) straight lines upon a plane remain straight lines upon any perspective view of that plane; (2) at any point on an inclined perspective view the scale is not the same in directions parallel to and perpendicular to the axis of tilt.

In plotting detail from vertical photographs certain errors, due to instrumental imperfections, may arise but are seldom of sensible magnitude. These are due to (i) a change in the relative position of plate lens and area photographed, due to the shutter moving so comparatively slowly that the movement of the aeroplane becomes noticeable on the plate; (2) distortion due to optical imperfec- tions of the lens.

Construction of the Map. In order to explain how a map is built up, wholly or partly, from air-photographs it is advisable to take some illustrative cases. The simplest case is where it is desired to produce a map of an area in which a sufficient trigonometrical control already exists and of which there is available a complete collection of cadastral plans, which are, however, much out of date. Ground features are low and gently undulating, extreme difference of altitudes being two or three hundred feet only.

In such a case topography can be brought up to date from air- photographs taken with the axis as nearly vertical as possible, and fitted upon the cadastral framework by one or other of the methods described below. When this is complete the map is contoured in the field, names are added and the topography examined for omis- sions or mistakes.

The most difficult case arises when the area to be mapped is rugged and mountainous, and the inhabitants hostile: the positions and heights of a few peaks in it, visible from accessible ground, have been fixed trigonometrically, but no reliable map exists.

As a preliminary measure oblique photographs are taken from a variety of points of view. The positions of the camera in space are calculated, and from measurements on each photograph a number of rays are drawn to noteworthy points in the valleys and on the hills. Positions and heights are thus determined for a subsidiary control. With the axis vertical a series of photographs of valleys and of watersheds are taken, pasted together, and fitted to the control. When the map has been thus built up, form-lines are added from the oblique photographs and upon the fixed heights. In the majority of surveys difficulties will be of an order intermediate between those of the foregoing two cases.

Applications of Air-photography. In any particular survey air- photographs may be used then for any or all of the following processes, viz.: (i) Air-photo control, (2) Air-photo-topography, (3) Air-photo contouring.

In taking vertical photographs for air-photo control, exposures are so regulated as to ensure a substantial overlap, generally amount- ing to 50% at least. Each successive photograph may therefore be fitted to its predecessor, and lines or traverses of photographs may be mounted and scaled between fixed points. Any two or more traverses of different and independent lines may be made to inter- sect over some topographical object, the position of which may be determined as the simple, or weighted, mean of the individual posi- tions from each traverse. Traverses may also be made to converge and end upon some prearranged and hitherto unfixed object. This method has given fairly accurate results in flat country on the scale of 1/40,000, and is dependent upon the ability of the pilot to main- tain an even keel and a constant height.

In broken and hilly country no method can be regarded as trust- worthy which does not take into account differences of altitude. We must then lie content to limit the use of each photograph to the measurement of horizontal and vertical angles and to fix the posi- tions of hew points by intersection from two or more photographs. Where this principle is decided upon there remains no advantage to be derived from the vertical photograph, and oblique photographs are used in preference as covering larger areas and allowing greater refinement in the measurement of vertical angles.

The first stage of this photo-topography from the air is to inter- polate the position of exposure in space from three or more points, the positions of which on the earth s surface are known, and which appear on the photograph. If we consider the pyramids whose apices are the lens and the bases of which are the triangles formed by the three fixed points respectively on the ground and on the photo- graphic plate (see fig. 2), we see that (a) the angles at the apex are a function of the lengths ab, ac, be (which can be measured upon the plate), and of the focal length, (b) the inclination of the ground pyramid to its base is determined by the direction and magnitude of tilt ; at the present time there are no means of measuring accurately the tilt of the plate at the moment of exposure, and calculation follows by successive approximations from a preliminary estima- tion ; (c) the position of O in space can be calculated and plotted in its correct projection on the plane of A, D, C; (d) angles may be measured upon the plate and rays drawn to additional points from O.

From the nature of the case air-photo control must be limited to the provision of a few supplementary points.

Where the area to be mapped contains a sufficiently close control the filling in of topographical detail is more easily done from vertical than from oblique photographs, providing that the area in question

is not markedly hilly. The scale on which photographs are taken may be larger or smaller than that of the map, but it must be suffi- ciently large to allow of clear identification of detail.

The area to be mapped is photographed from a prearranged height in strips allowing for an overlap in all directions. Much depends upon the training of the pilot in maintaining his height and his over- lap. It is usual to arrange for a mechanical control of exposures regulated according to the ground speed of the aeroplane.

Photo Pyramid

Ground Pyramid

Intersected Point

FIG. 2.

The plotting of detail from these photographs would be simple if the axis of the camera could be maintained in a vertical position. It would then be necessary only to bring the photograph to the scale of the map. No means of ensuring this verticality has, as yet, been evolved. It often becomes necessary, therefore, to fit photographs individually upon the control points. This can be done graphically, or optically by the camera lucida, or by the enlarging camera.

The graphic method depends upon the principle that as straight lines on one plane remain straight lines on any perspective of that plane the position of a point which lies upon the intersection of two lines common to the ground and to the photograph may be readily determined. Within narrow limits the proportional compass set to the difference of scale between map and photograph at this point may be used to fix additional points. It is more accurate, however, to maintain the straight line principle and to cover the map and photograph with a "grid" of corresponding lines, as in fig. 3.

PHOTO

MAP

FIG. 3.

The photograph is mounted on a sheet of paper a, b, c, d and A, B, C, D are four points the positions of which are known and are also identifiable on the photograph. Subsidiary common points at oO are established by drawing the diagonals, and four subsidiary quadrilaterals may then be formed by drawing lines through oO from vV and wW the intersections of the prolonged sides of the quadrilaterals. The same principle may be applied to any poly- gons formed by joining up any number of points (more than four) which are fixed on the ground and identifiable on the photograph. Detail may be sketched in by eye.

A useful method of plotting, known as the four-point method, is as follows (fig. 4) : It can be proved that the cross ratios of four points which lie upon a straight line are the same upon any perspec- tive view of that line, hence we can readily plot the position of a fifth point (S) if we know the positions of four points A,B,C,D.

Let A, B, C, D be four known points on the photograph and o, b, c, d their positions on the map, and let S be a point on the photo- graph the position of which on the map is to be found.

Join A B, AC, A S, and A D, a b, a c, and a d.