(c) Any line lying in a plane will have its vanishing point somewhere in the vanishing trace of the plane in which if lies.

(d) The vanishing trace of any plane mast pass through the vanishing points of all lines that lie in it. Thus, since the vanishing 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 other 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 the visual ray which projects the vanishing 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. Furthermore, 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, aP and bP, and the straight line drawn between aP and bP is the perspective of the line ab. The perspective of the line albl which is parallel to ah, has been found in a similar way, and it will be noticed that its perspective projection (a1P b1P) actually converges towards aPbP in such a manner that if these two lines are produced they will actually meet at the perspective of the vanishing point of their system.

Note.- It is evident that the perspective of a straight line will always be a straight line, the extreme points of which are the perspectives of the extremities of the given line.

24. Thus, the five axioms of perspective may be applied to Perspective Projection as follows : -

(a) Parallel lines do converge and meet at the vanishing point of their system.

(b) Parallel planes do converge and meet at the vanishing trace of their system.

(c) The vanishing point of any line lying in a plane will be found in the vanishing trace of the plane.

Therefore, the vanishing points of all horizontal lines will be found in the horizon (§ 20).

(d) The vanishing trace of any plane will be determined by the vanishing points of any two lines that lie in it, and must contain the vanishing points of all lines that lie in it.

(e) The vanishing point of the intersection of two planes will be found at the intersections of the vanishing traces of the two planes.

To the five axioms of perspective projection already stated may be added the following three truths concerning the construction of the perspective projection: -

(f) The perspective of any point in space is where the visual ray through the point pierces that picture plane (§ 7).

(g) The perspective of the vanishing point of any system of lines is when; the visual element of that system pierces the picture plane.

Rule for finding the perspective of the vanishing point of any system of lines:-Draw an element of the system through the observer's eye, and find where it pierces the picture plane.

(h) Any point, line, or surface which lies in the picture plane will be its own perspective, and show in its true size and shape.

25. Knowing how to find the perspective of any point, and how to find the vanishing point of any system of lines, any problem in perspective may be solved. Therefore, it may be said that the whole process of making a perspective projection reduces itself to the problem of finding where a line pierces a plane.

Before proceeding farther, the student should review the first twenty-five paragraphs by answering carefully the following questions : -

(1) What does a perspective projection represent?

(2) What is a visual ray?

(3) How is a perspective projection formed?

(4) How does a perspective projection differ from an orthographic projection ?

What is the plane called on which the perspective pro jection is made?

(6) What is meant by the term Station Point?

(7) What is the most important phenomenon of perspective

(8) What is meant by a system of lines?

(9) What is meant by a system of planes?

(10) What is a visual element?

(11) Define vanishing point.

(12) Define vanishing trace.

(13) Describe the position of the vanishing point of any sys tern of lines.

(14) Give the five axioms of perspective.

(15) Do parallel lines in space really converge?

(16) Do the perspective projections of parallel lines really converge?

(17) Where will the perspective projections of parallel lines meet?

(18) How is the perspective of any point found?

(19) How is the perspective of the vanishing point of any system of lines found ?

(20) What will be the perspective of a straight line?

(21) What is meant by the horizon ?

(22) What is meant by the plane of the horizon ?