1 2. If any object, as, for illustration, a cube, is studied, it will be seen that the lines which form its edges may be separated into groups according to their different directions ; all lines having the same direction forming one group, and apparently converging to a common vanishing point. Each group of parallel lines is called a system, and each line an element of the system. For example, in Fig. 4, A, A1, A2, and A3 belong to one group or system ; B, B1, B2, and B3, to another; and C, C1, C2, and C3, to a third. Each system has its own vanishing point, towards which all the elements of that system appear to converge. This phenomenon is well illustrated in the parallel lines of a railroad track, or by the horizontal lines which form the courses of a stone wall.

13. As all lines which belong to the same system appear to meet at the vanishing point of their system, it follows that if the eye is placed so as to look directly along any line of a system, that line will be seen endwise, and appear as a point exactly covering the vanishing point of the system to which it belongs.

If, for illustration, the eye glances directly along one of the horizontal lines formed by the courses of a stone wall, this line will be seen as a point, and all the other horizontal lines in the wall will apparently converge towards the point. In other words, the line along which the eye is looking appears to cover the vanishing point of the system to which it belongs. Thus, the vanishing point of any system of lines must lie on that element of the system which enters the observer's eye, and must be at an infinite distance from the observer. Therefore, to find the vanishing point of any system of lines, imagine one of its elements to enter the observer's eye. This element is called the visual element of the system, and may often be a purely imaginary line indicating simply the direction in which the vanishing point lies. The vanishing point will always be found on this visual element and at an infinite distance from the observer.

14. To further illustrate this point, suppose an observer to be viewing the objects in space represented in Fig. 5. He desires to find the vanishing point for the system of lines parallel to the oblique line ab which forms one edge of the roof plane abcd. There are two lines in the roof that belong to this system,namely: ab and dc If he imagines an element of the system to enter his eye, and looks directly along this element, he will be look-ing in a direction exactly parallel to the line ab, and he will be looking directly at the vanishing point of the system (§ 13). This visual element along which he is looking is a purely imaginary line parallel to ab and dc. All lines in the object belonging to this system will appear to converge towards a point situated on the line along which he is looking, and at an infinite distance from him.

This phenomenon is of great importance, and is the foundation of most of the operations in making a perspective drawing.

15. The word "vanish" as used in perspective always implies a recession. Thus, a line that vanishes upward, slopes upward as it recedes from the observer; a line that vanishes to the right, slopes to the right as it recedes from the observer.

16. It follows from paragraphs 13 and 14 that any system of lines that vanishes upward, will have its vanishing point above the observer's eye. Similarly, any system vanishing downward, will have its vanishing point below the observer's eye; any system vanishing to the right, will have its vanishing point to the right of the observer's eye ; and any system vanishing to the left, will have its vanishing point to the left of the observer's eye. Any system of horizontal lines will have its vanishing point on a level with the observer's eye, and a system of vertical lines will have its vanishing point vertically in line with the observer's eye.

17. All planes that are parallel to one another are said to belong to the same system, each plane being called an element of the system.

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 vanishing 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.