All the planes of one system appear to approach one another as they recede from the eye, and to meet at infinity in a single straight line called the vanishing trace of the system. Thus, the upper and lower faces of a cube seen in space, will appear to converge toward a straight line at infinity.
18. If the eye is so placed as to look directly along one of the planes of a system, that plane will be seen edgewise, and will appear as a single straight line exactly covering the vanishing trace of the system to which it belongs. The plane of any system that passes through the observer's eye is called 'the visual plane of that system.
19. From § 18, it follows that the vanishing trace of a system of planes that vanishes upward, will be found above the level of the eye, while the vanishing trace of a system of planes vanishing downward, will be found below the level of the eye. The vanish ing trace of a system of vertical planes will be a vertical line ; and of a system of horizontal planes, a horizontal line, exactly on a level with the observer's eye.
20. The vanishing trace of the system of horizontal planes is called the horizon.
The visual plane of the horizontal system is called the plane of the horizon. The plane of the horizon is a most important one in the construction of a perspective projection.
21. From the foregoing discussion the truth of the following statements will be evident. They may be called the Five Axioms of Perspective.
(a) All the lines of one system appear to converge_ and to Meet at an infinite distance from the observer's eye, in a single point called the vanishing point of the system.
(b) All the planes of one system appear, to converge as they recede from the eye, and to meet at an infinite distance from the observer, in a single straight line called the vanishing trace of the system.
(c) Any line lying in a plane will have its vanishing point somewhere in the vanishing trace of the plane in which it lies.
(d) The vanishing trace of any plane must pass through the vanishing points of all lines that lie in it. Thus, since the van ishing trace of a plane is a straight line (§ 18), the vanishing points of any two lines lying in a plane will determine the vanishing trace of the system to which the plane belongs.
(e) As the intersection of two planes is a line lying in both, the vanishing point of this intersection must lie in the vanishing traces of both planes, and hence, at the point where the vanishing traces of the two planes cross. In othei words, the vanishing point
of the intersection of two planes must lie at the intersection of the vanishing traces of the two planes.
22. The five axioms in the last paragraph are the statements of purely imaginary conditions which appear to exist, but in reality do not. Thus, parallel lines appear to converge and to meet at a point at infinity, but in reality they are exactly the same distance apart throughout their length. Parallel planes appear to converge as they recede, but this is a purely apparent condition, and not a reality; the real distance between the planes does not change.
23. The perspective projection represents by real conditions the purely imaginary conditions that appear to exist in space.
Thus, the apparent convergence of lines in space is represented by a real convergence in the perspective projection. Again, the vanishing point of a system .of lines is a purely imaginary point which does not exist. But this imaginary point is represented in perspective projection by a real point on the picture plane.
From § 14, the vanishing point of. any system of lines lies upon the visual element of that system. This visual element may be considered to be die visual ray which projects the vanish ing point to the observer's eye. Hence, from § 7, the intersection of this visual element with the picture plane will be the perspective of the vanishing point of the system to which it belongs. This is illustrated in Fig. 6. The object in space is shown on the right of the figure. If the observer wishes to find the vanishing point of the oblique line ab in the object in space, he imagines a line parallel to ab to enter his eye, and looks along this line (§ 13). Where this line along which he is looking pierces the picture plane, will be the perspective of the vanishing point. Further more, the perspective of the line ab has been found by drawing the visual rays from a and b respectively, and finding where these rays pierce the picture plane (§ 7). • These points are respectively, e and b', and the straight lin? drawn between 'e and bP is the perspective of the line ab. The perspective of the line which is parallel to ab, has been found in a similar way, and it will be noticed that its perspective projection (al actually converges towards arbP in such a manner that if these two lines are pro duced they will actually meet at the perspective of the vanish ing point of their system.