HPP represents the horizontal projection of the picture plane, and VII represents the vertical projection of the plane of the horizon.

As horizontal projections are never compared with vertical projections (§ 38), HPP may be drawn as far from, or as near, VH as desired, without in any way affecting the resulting perspective drawing. HPP and VH were coincident in Fig. 9, and the distance between them in Fig. 10 simply shows the distance that the two planes have been slid apart, as illustrated in Fig. 9a. As already stated, this distance is immaterial, and may be made whatever is most convenient, according to the nature of the problem.

If HPP should be placed nearer the top of the sheet, aH and SPH, both being horizontal projections, would follow it, the relation between these horizontal projections always being preserved.

On the other hand, SPV, aV, aP, VH, and VH1, all being projections on the vertical plane, must preserve their relation with one another, and will in no way be affected if the group of projections on the horizontal coordinate is moved nearer or farther away. It must be borne in mind, however, that, in all cases, the vertical and horizontal projections of corresponding points must be kept vertically in line. Thus, aH must always be vertically in line with aV. The vertical distance between these two projections does not matter, provided the distance from aH to HPP, or the distance from aV to VH, is not changed. This point cannot be too strongly emphasized. 41. Suppose it is desired to determine from Fig. 10 how far the station point lies in front of the picture plane. This is a horizontal distance, and therefore will be shown by the distance between the horizontal projection of the station point and the horizontal projection of the picture plane, or, in other words, by the distance between SPH and HPP.

42. The point a is a certain distance above or below the plane of the horizon. This is a vertical distance, and will be shown by the distance between the vertical projection of the point a and the vertical projection of the plane of the horizon ; in other words, by the distance between aV and VH. It will be seen that in Fig. 10 the point a lies below the plane of the horizon.

43. If it be desired to find how far in front or behind the picture plane the point a lies, this is a horizontal distance, and will be shown by the distance between the horizontal projection of the picture plane and horizontal projection of the point a, that is, by the distance between HPP and aH. In Fig. 10 the point a lies behind the picture plane.

44. The distance between the plane of the ground and the plane of the horizon is a vertical distance, and will be shown by the distance between the vertical projection of the plane of the horizon and the vertical projection of the plane of the ground ; i.e., the distance between VH and VH 1. The distance between the observer's eye and the plane of the ground is also a vertical distance, and will be shown by the distance between SPV and VH 1 ; but as SPV must always be found in VII, the distance of the observer's eye above the plane of the ground will always be shown by the distance between VH and VH1,

45. To find the perspective of the point a, Fig. 10, draw the visual ray through the point, and find where this visual ray pierces the picture plane (§ 24f). The horizontal projection of the visual ray is shown by the line RH drawn through the horizontal projection SPH of the observer's eye and the horizontal projection aH of the point a. The vertical projection of the visual ray is shown by the line RV drawn through the vertical projection SPV of the observer's eye and the vertical projection aV of the point a. This visual ray pierces the picture plane at the point aP on RV vertically in line with the point where

RH crosses HPP (§§ 35 and 36). aP is the perspective of the point a.

Note. - To find where any line, represented by its horizontal and vertical projections, pierces the picture plane, is one of the most used and most important problems in perspective projection. The point where any line pierces the picture plane will always be found on the vertical projection of the line, vertically above or below the point where the horizontal projection of the line crosses HPP (§§ 35 and 36).

## Notation

46. In order to avoid confusion between the vertical, horizontal, and perspective projections of the points and lines in the drawing, it becomes necessary to adopt some systematic method of lettering the different points and lines. The following method will be found convenient, and has been adopted in these notes.

If the student will letter each point or line as it is found, in accordance with this notation, he will be able to read his drawings at a glance, and any desired projection of a point or line may be recognized instantly.

The picture plane (or vertical coordinate) is indicated by the capital letters PP.

The plane of the horizon (or horizontal coordinate) is indicated by the capital letter H.

A point in space is indicated by a small letter.

The same small letter with an index V, H, or P, indicates its vertical, horizontal, or perspective projection, respectively.

A line in space is indicated by a capital letter, usually one of the first letters in the alphabet.

The same capital letter with an index V, H, orP, indicates its vertical, horizontal, or perspective projection, respectively.

All lines which belong to the same system may be designated by the same letter, the different lines being distinguished by the subordinate l, 2, 3, etc., placed after the letter.

The trace of a plane upon the picture plane is indicated by a capital letter (usually one of the last letters in the alphabet) with a capital V placed before it.

The same letter preceded by a capital H indicates the trace of the plane upon the horizontal coordinate.

The perspective of the vanishing trace of a system of planes is indicated by a capital letter preceded by a capital T.

The perspective of the vanishing point of a system of lines is indicated by a small v with an index corresponding to the letter of the lines which belong to the system.

PP = vertical coordinate, or picture plane.

HPP= horizontal trace of the vertical coordinate, or picture plane.

H= horizontal coordinate, or plane of the horizon.

VH = vertical trace of the horizontal coordinate, or plane of the horizon.

H 1 = plane of the ground.

VH1 = vertical trace of the plane of the ground.

a = point in space.

aV = vertical projection of the point.

aH = horizontal projection of the point.

aP = perspective projection of the point.

A = line in space.

AV = vertical projection of the line.

AP = perspective projection of the line.

VS = trace of the plane S upon PP (vertical trace).

HS = trace of the plane S upon H (horizontal trace).

TS = perspective of the vanishing trace of the plane S. (See Note 1 below.) vA = perspective of the vanishing point of a system of lines, the elements of which are lettered A1, A2, A3, A4, etc. (See Note 2 below.)

Note 1.- A plane in space may also be designated by the letters of any two lines which lie in it. Thus, the plane AB would be a plane determined by the two lines A and B. TAB would indicate the perspective of the vanishing trace of the plane.

Note 2.-A straight line may be designated by the letters of any two points which lie in it. Thus, the line ab would be a straight line determined by the two points a and b. vab would indicate the perspective of the vanishing point of the line. It is some-times convenient to use this notation in place of the general one.