Application to Perspective Drawing

Vn° must be situated at the intersection of TP and TR, but these two vanishing traces do not intersect within the limits of the paper. A line may be drawn through nP to meet TP and TR at their intersection in the following manner. Draw any triangle as nPvw with its apex at nP and its base, vw, stretching between TR and TP. Draw any other triangles as xyz with its sides re spectively parallel to nPvw and its base vz stretching between TR and TP. A line from nP through x will meet TR and TP at their intersection.

Lines Parallel to the Picture Plane.

It has been stated that the perspectives of vertical lines on a vertical plane are always vertical and not convergent. The reason for this follows directly from Rule D. If a vertical line is drawn through the observer's eye it cannot pierce a vertical picture plane within finite limits. The perspective of the vanishing point of such a system will theoretically be vertically over SP?) at an infinite distance. As the perspectives of all vertical lines must meet at the perspective of their vanishing point they cannot meet within a finite distance and must be drawn parallel and vertical. The parallel vertical lines drawn in perspective are themselves, how ever, subject to the laws of appearance, as are all lines in space, and, though actually drawn vertical, will appear to the observer to converge in just the right degree as they recede in the picture from his eye. This is true not only of vertical lines but of all lines which are parallel to the picture plane; i.e., all lines whose projections in the diagram are parallel to HPP. From this truth three additional and very useful axioms may be deduced.

Axiom 6. Lines in space which are parallel to the picture plane will have their perspectives drawn parallel to one another and not convergent.

Axiom 7. Any line parallel to the picture plane must have its perspective drawn parallel to the vanishing trace of every plane in which it lies.

Axiom 8. If the line in which two planes intersect is parallel to the picture plane, the vanishing traces of the two planes must be parallel to one another.

These axioms are exemplified in the vertical lines in the draw ing and also in the lines kf and lh whose projections in the diagram show them to be parallel to the picture plane. kf being the intersection of M and N, TM and TN should be parallel to one another (Axiom 8) as they are found to be, and kf should be drawn parallel to TM and TN (Axiom 7). Similarly lk is drawn parallel to TO and TP, and these two vanishing traces are parallel to one another.

Vanishing Point Diagram.—The more or less geometrical figure composed of the vanishing traces of the planes, the vanish ing points of the lines, VH, HPP, and the projections of the station point form the vanishing point diagram. This vanishing point diagram is quite independent of the particular location of the revolved plan, provided the directions of the lines in the revolved plan are not changed. It would serve to determine the perspective of the house were it placed to the right or the left of its present position, or it might be brought forward or moved backward in relation to the picture plane. It might also rest on a ground plane higher or lower than the one used, provided always that the lines in the object do not change their direction. The vanishing point diagram would therefore serve for a number of similar objects placed at different positions but always having the same angular relation to the co-ordinate planes. As an illustration of this a second perspective projection has been drawn in Plate III. on a ground plane represented by VH2 some distance below VH1. The same vanishing point diagram serves for both perspectives. Should, however, the angular relation between the diagram and the picture plane be changed it would be necessary to construct a new vanishing point diagram.

Parallel Perspective.—Sometimes instead of the diagram being turned with its principal horizontal lines oblique to the picture, it is placed with one of its principal systems of horizontal lines parallel and the other perpendicular to the picture plane, as in fig. 21. The result is often referred to as a parallel perspec tive and is frequently used in showing interior views. The picture plane may be taken coincident with the nearest wall of the room. The horizontal system perpendicular to the picture plane will have its vanishing point coincident with SPy (Rule D). The hori zontal system parallel to the picture plane will, in perspective, show as true horizontal lines (Axioms 6 and 7). This is simply a special case under the general method already explained, and involves no new principles. The vertical and horizontal edges of the room, ab, bc, cd and da being lines in the picture plane will be measure lines and true lengths can be laid off on them directly, as dm the height of the door and dp the projection of the lower step from the wall ; or any line as ef can be extended forward into the picture plane where it will show its true height sn, above the ground plane.