26. Two planes of projection at right angles to one another, one vertical and the other horizontal, are used in making a perspective. In Fig. 7 these two planes are shown in oblique projection. The vertical plane is the picture plane (§ 6 and Fig. 7) on which the perspective projection is made, and corresponds exactly to the vertical plane, or vertical coordinate used in orthographic projections.

The Planes Of Projection 070084

27. The horizontal plane, or plane of the horizon (§ 20 and Fig. 7), always passes through the assumed position of the observer's eye, and corresponds exactly to the horizontal plane or horizontal coordinate used in orthographic projections.

28. All points, lines, surfaces, or solids in space, the perspective projections of which are to be found, are represented by their orthographic projections on these two planes, and their perspectives are determined from these projections.

29. Besides these two principal planes of projection, a third plane is used to represent the plane on which the object is supposed to rest (Fig. 7). This third plane is horizontal, and is called the plane of the ground. Its relation to the plane of the horizon determines the nature of the perspective projection. To illustrate : The observer's eye must elways be in the plane of the horizon (§ 27), while the object, the perspective of which is to be made, is usually supposed to rest upon the plane of the ground. In most cases the plane of the ground will also be the plane on which the observer is supposed to stand, but this will not always be true. The observer may be standing at a much higher level than the plane on which the object rests, or he may be standing below that plane. It is evident, therefore, that if the plane of the ground is chosen far below the plane of the horizon, the observer's eye will be far above the object, and the resulting perspective projection will be a "bird's-eye view." If, on the other hand, the plane of the ground is chosen above the plane of the horizon, the observer's eye will be below the object, and the resulting perspective projection will show the object as though being viewed from below. This has sometimes been called a "worm's-eye view," or a " toad's-eye view."

Usually the plane of the ground is chosen so that the distance between it and the plane of the horizon is about equal (at the scale of the drawing) to the height of a man. This is the position indicated in Fig. 7, and the resulting perspective will show the object as though seen by a man standing on the plane on which the object rests.

30. The intersection of the picture plane and the plane of the horizon corresponds to the giound line used in the study of projections, in Mechanical Drawing. For more advanced work, however, there is some objection to this terra. The intersection of the two coordinate planes has really no connection with the ground, and if the term "ground line" is used, it is apt to result in a confusion between the intersection of the two coordinate planes, and the intersection of the auxiliary plane of the ground, with the picture plane.

31. The intersection of the two coordinate planes is usually lettered VII on the picture plane, and HPP on the plane of the horizon. (See Fig. 7.) That is to say : When the vertical plane is being considered, VH represents the intersection of that plane with the plane of the horizon. It should also be considered as the vertical projection of the plane of the horizon. See Mechanical Drawing Part III, page 5, paragraph in italics. All points, lines, or surfaces lying in the plane of the horizon will have their vertical projection in VH.

32. On the other hand, when the horizontal plane is being considered, HPP represents the intersection of the two planes, and also the horizontal projection of the picture plane. All points, lines, or surfaces in the picture plane will have their horizontal projections in HPP. Thus, instead of considering the intersection of the two coordinate planes a single line, it should be considered the coincidence of two lines, i.e.: First, the vertical projection of the plane of the horizon; second, the horizontal projection of the picture plane.

33. The plane of the ground is always represented by its intersection with the picture plane (see VH1 Fig. 7). Its only use is to determine the relation between the plane of the horizon and the plane on which the object rests (§ 29). The true distance between these two planes is always shown by the distance between VH and VH1 as drawn on the picture plane.

34. To find the perspective of a point determined by its vertical and horizontal projections.

Fig. 8 is an oblique projection showing the two coordinate planes at right angles to each other. The assumed position of the plane of the ground is indicated by its vertical trace (VH.)

Note. - The vertical trace of any plane is the intersection of that plane with the vertical coordinate. The horizontal trace of any plane is the intersection of that plane; with the horizontal coordinate.

The assumed position of the statian point is indicated by its two projections, SPV and SPH. Since the station point lies in the plane of the horizon (§ 27), it is evident that its true position must coincide with SPH, and that (§ 31) its vertical projection must be found in VII, as indicated in the figure. Let the point a represent any point in space. The perspective of the point a will be at a P, where a visual ray through the point a pierces the picture plane (§ 24). We may find a p in the following manner, by using the orthographic projections of the point a p, instead of the point itself. a H represents the horizontal projection, and aV represents the vertical projection of the point a. A line drawn from the vertical projection of the point a to the vertical projection of the station point, will represent the vertical projection of the visual ray, which passes through the point a. In Fig. 8 this vertical projection is represented by the line drawn on the picture plane from aV to SPv.