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Geometrical Considerations Deform Ation of the Reconstructed Object 819

separation, corresponding, distance, plane, axes and base

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GEOMETRICAL CONSIDERATIONS ; DEFORM ATION OF THE RECONSTRUCTED OBJECT 819. Corresponding Points -Variations of Their Separation. The pairs of points appearing one in each image of a stereoscopic pair and corresponding with a single point in the object are termed homologous (corresponding) points.

Consider (Fig. 207) two photographs T and T' taken from two points S and 5', the optical axes SP, S'P' being parallel and perpendicular to the base SS', the images T and if' are in the same plane, which we will assume vertical, the optical axes being then horizontal. The two images rr' of a point R at infinity appear at a distance rr' apart equal to the distance PP' of the two principal points and the distance SS' of the viewpoints.

The two corresponding points aa' of a point A at finite distance are farther apart. Draw a line through S parallel to S' a,' cutting the picture T in a" ; the length aa" represents the variation. of separation of corresponding points in pass ing from a point at infinity to the point A in question.

Denote the length of the base SS' by h; let F be the common principal distance of the two perspectives (equal to the focal length if the camera is focussed for infinity) ; d is the distance from the point A to the base SS', and e the increase of separation of the corresponding points. From the similar triangles SAS' and aSa" we get the relation e bFld; we thus see that the increase in separation of the correspond ing points in passing from a point at infinity to a point at finite distance (the increase being measured on the negatives in the same position as when recording the images) is proportional to the distance apart of the viewpoints, to the principal distance, and inversely proportional to the distance d 6o) of the points considered. It will be noticed that the value (h e) of the separation of corresponding points is constant for all points in the plane AC, which passes through the point A and is perpendicular to the optical axes ; its value will be smaller for more distant planes and greater for planes nearer the camera. Conversely, in a stereoscopic pair taken

under normal conditions, all corresponding points having the same separation correspond with points in the object situated in the same front plane.

On transposition, the differences of separation between corresponding points on the positives T, and retain the same values, but in the inverse sense ; the nearer the object point was to the observer, the less the separation of the corresponding points, which causes the variation of convergence of the optical axes in the same sense as when the object itself is examined.

82o. Parallax. The angle subtended by the base at the point A, SAS' (Fig. 207), is termed the parallax of the point A. Notice that there is no relation between the parallax and the separation (or variation of separation) of cor responding points. We have seen that this separation is constant for all points of the space object in the same front plane, whilst the paral lax is constant for all points of the space-object situated on the circumference of a circle drawn through the points S. A, and S', and thus for all points on the surface generated by the rotation of the circle about the base SS'.

Considerations of parallax are of little im portance in stereoscopy, and the subject is only mentioned here owing to the confusion which sometimes arises between it and the idea of separation of corresponding points in the minds of many students of the subject.

821. Deformations of the Reconstructed Object Due to the Circumstances in which the Photo graphs are Taken. In the various cases hitherto considered we have always assumed the images of the two perspectives to be in the same plane, and the optical axes perpendicular to this plane, and therefore parallel. Under these conditions the reconstructed object is similar to the object itself if viewed correctly in the stereoscope.

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