As must be evident from the foregoing discussion, linear per spective is essentially a science of straight lines. When curves appear in the plan or elevation their perspectives are usually con structed by reference to straight lines. If the curve is of regular form it can be enclosed in a polygon, usually a rectangle, the per spective of the rectangle found and the curve then constructed within the perspec tive rectangle. Fig. 22 shows a curve so enclosed, the perspective of which has been found. The diagonals of the perspec tive rectangle locate its centre. Lines drawn through the centre respectively par allel to the adjacent sides give by their intersections with the sides, points at which the curve is tangent. The sides of the rectangle give the directions of the curve at the points of tangency. Fig. 23 is an other illustration of curves of regular form constructed in per spective. If the curve is not regular in form the enclosing rec tangle can be subdivided into smaller rectangles, as shown in fig. 24, as a further aid to locating the curve.
Although methods can be devised for constructing the exact per spective of any curved object, they are complex and seldom used in practice. A few important points are usually located and the outline sketched in freehand, if in a position corresponding to that in the picture.
Apparent Distortion.—It is evident from the statement in connection with Plate I., fig. 3 that an object in space is exactly represented by its perspective prospective projection, or in other words, no distortion or exaggeration can exist in correctly con structed perspective projection. Notwithstanding this fact, very disagreeable effects and very apparent distortions are often noticed in perspective projections, the accuracy of which cannot be ques tioned. A few examples will suffice to show what is meant. Plate II., fig. I, is a true perspective. It is supposed to represent a number of perfect spheres of equal size. The view as seen does not convey this impression. A sphere in space always appears as a perfect circle and never as the oval-shaped areas seen near the edges of the photograph. Again, Plate II., fig. 2, is a correct per spective of five circular cylinders all having the same diameter. The ones farthest to the right or left should appear smaller than the nearest one at the centre. In the photograph just the opposite is true and the farther from the eye the larger they appear. The explanation of these seeming anomalies is as follows : Before any perspective projection is constructed the position of the ob server's eye is definitely fixed, and, in order that the perspective shall represent the view in space, the observer must close one eye and place the other exactly in the predetermined position. It is seldom that an observer looks at a drawing with one eye only, or places either eye even approxi mately in the correct relation to the drawing. This limitation
of a perspective view and the failure to understand it is the cause of all apparent distortion.
In Plate II., figs. i and 2, the station point has purposely been chosen so close to the paper that it is impossible for the observer to see the view from the correct position. Should either of these views be enlarged so that the distance from the paper to the station point became considerably greater, and should the observer examine the enlarged view with one eye only, placed exactly at the station point, the elliptical projections of the outside spheres would be foreshortened by the obliquity of his line of vision and would appear as perfect circles, representing to him perfect spheres. The cylinders would also appear in their proper relations.
When the eye is not in its proper position all parts of the draw ing show more or less distortion. This is most noticeable in reg ular curved forms or in the human figure, especially when these are located near the edges of the drawing. The disagreeable effects are much more pronounced when the station point is taken too near the picture plane. Thus the apparent distortion seen in a cor rectly constructed perspective is due, not to inaccuracies in the perspective theory, but to an unwise choice in the station point, a faulty arrangement of the view, and the failure of the observer to recognize the limitations of a perspective projection. In making a perspective, the station point should always be chosen in such a position that the observer will naturally place his eyes approxi mately at the chosen point when viewing the drawing. An arbi trary rule sometimes given is to assume the station point directly in front of the centre of the drawing, at the apex of an equilateral triangle the base of which just covers the width of the view. The station point may be chosen farther away than this without much danger but not nearer, and never nearer than eight or ten inches.
Curved forms of regular shape and human figures or animals should be kept as near the centre of the drawing as possible. Plate II., fig. 3, shows a view of the same cylin ders and spheres seen in the two previous figures but with the station point much far ther from the picture plane. The result is a vast improvement in the view obtained, though the spheres on the outskirts of the picture still appear slightly elliptical. After all precautions have been taken, if disa greeable effects still persist in the drawing it is customary to introduce certain so-called corrections such as making the perspective of the spheres in Plate II., fig. 1, all perfect circles. These are really not corrections but actual transgressions of the rules of perspective which alter the view so that it will not be exactly correct at any point, yet it may not be noticeably dis agreeable from any position likely to be taken by the observer.