Elements of perspective

Chapter I

PerspECTivk is the art of representing objects as they appear and not as they really are.'

A perspective drawing is one in which we have the relative heights and distances of different parts according to the angles and distances at which they stand in reference to the observer. If the student understands the principles by which a square or cube is correctly placed in perspective, he can, with little difficulty, make a representation of more complicated objects in their true proportions.

EI.BMENTS AND Rules of Perspective.

Certain lines are supposed to exist by which we determine the direction of the lines in the object to be drawn ; each one of these lines occupies a certain position with regard to the observer, and also with regard to the picture plane, and they are governed by rules, deduced from the sciences of Optics and Geometry. Before giving any of these rules, it will be necessary to define terms with which the student must become familiar — picture plane, ground plane, horizon, base, etc. The most important of these is the term picture plaiie or plane of delineation (page 12). The position of this plane with regard to the object to be represented determines whether the drawing is to be made in parallel, angular, or oblique perspective (page 11).

The picture plane or plane of delineation is an imaginary plane parallel to the observer and perpendicular to 'the ground plane. The rays proceeding from an object (visual rays) pass through this plane.

The visual angle is the angle formed by drawing lines from the extremities of any line to the eye ; the angle varies with the size and distance of an object.

The visual angles are formed by lines extending .from a and b to the eye.

If these rays proceed from a rectangular surface they form a pyramid of rays (Fig. 3), the vertex in the eye of the observer. If the rays proceed from a round surface, they form a cone of rays (Fig- 4).

Elements and Rules of Perspective.

Fig' 3-

The base from which the rays proceed is termed the ^e/d of view. The distance cd, in Fig. 4, represents the cross diameter of the field of view. Lines drawn from these points to the eye form an angle of 60° (one-sixth of a circle). When looking at a point one can not see more than is included in an angle of 60°, and not more than one-half of this (30°) distinctly.

Chapter II

The apparent size of an object varies. according to the distance at which it is situated from the eye of the observer. The size of an object appears to diminish as it recedes from the eye; an object at a distance of thirty feet from the observer appears one-half as large as the same object at a distance of fifteen feet from the observer. At a distance of one hundred feet, it will appear one- fourth as large as it will at a distance of twenty-five feet from the observer.

a c

1 1 1 1 1 1 1 —

- ft.

n.

- b d

Figure 5 represents the Hue 8 feet long at a distance of 15 feet, and the same line again at a distance of 30 feet. At a distance of 30 feet, it appears one-half the length of the line at a distance of 15 feet.

If an object is at R. A. to the Hne of direction, (a Hne that extends from the eye of the observer to the object,) no matter how far removed from the eye, the apparent shape remains the same, the size only varies. A cube viewed in this position would be represented on a plane surface by a square, and a rectangular box by a

Elements and Rules of Perspective.

rectangle, whether viewed at a distance of 5 feet or at a distance of 50 feet.

Fig. 6 represents the appearance of a cube when the line of direction is perpendicular to the point a.

Fig. 7 represents the appearance of a box when the line of direction is perpendicular to the point b.

Tig. 6.

If an object is viewed at an angle, the apparent shape then varies. If we take a square figure and view it at a distance of 20 feet, the line of direction at right angles with the plane of the figure, the apparent shape is a square^ and will be a square as long as it is viewed in that position. But let it occupy any position save that in which it is at right angles to the line of direction, and it will no longer appear of a square form. It will, if transferred to a flat surface, be represented by a figure that is not square.

As long as the square is parallel to the observer the opposite parallel lines do not appear to meet, but if we turn the square so that the right side is at a greater distance from the observer than the left side, then the upper and lower lines of the square, though parallel in reality, appear to approach. If a picture were made of it on paper, in this position, we should have the two vertical lines of the square, those that are not at an angle with, but parallel to the obser^^er, represented by vertical lines. The line representing the right side, of course, is shorter than the line representing the left side, because it is farther from the observer ; but those lines that are at an angle with the plane of delineation or picture plane (this is an imaginary plane between the observer and the object) are not represented as they are, but are represented in the drawing by two

Elements and Rules of Perspective.

Fig. 8,

Fig. 9.

lines that will, if suflficiently prolonged, meet in a point.

Fig. 8 represents a square adcd in perspective, the side dc at a greater distance from the observer than the side ad, and the lines ad and dc (parallel in reality) tending toward the point P.

If the square is de/ow the level of the eye, the sides that recede from the observer tend to a point above the square.

Fig, 9 represents a square below the level of the eye, which is here denoted by the horizontal line.

In this the lines dc and ab at an angle to the observer, recede toward and appear to meet in the point P.

If the square is above the

level of the eye, the lines that

recede from the observer tend

to a point below the square.

Fig. lo represents a square in perspective above the level of the eye. The lines ad and dc, that are at an angle with the picture plane, recede and meet in the point P.

In Fig. II a very large square is represented, resting on the plane on which the observer stands, and extending above his head. The lines dc and ad that are at angles to the plane meet, if produced, in the point P. The line dc above

Fig. lo.

Fig, J J.

Elements and Rules of Perspective.

the level of the eye tends down to the point P, and the line ab below the level of the eye tends up to the same point.

The point to which these receding lines recede, tend, or vanish, is called a vanishing point ; it is on a level line, vSituated opposite the eye of the observer. In this case the vanishing point is on the right of the squares.

If these squares were so placed that the left side was farther from the observer than the right side, then the receding lines would vanish to a point on the left of the squares.

In Figs. 12 and 13 the vanishing points for the squares are on the left.

In representing a solid, such as a cube, if the front face is parallel to the obser^T-er, the receding lines vanish in one point (Fig. 14). If the cube is at an angle to the obsen^er, then instead of one vanishing point there are two, — one on each side of the object (Fig. 15).

Fig. 14 represents a view of a cube parallel to the observer.

Pig. X5.

Fig. 15 represents a view of a cube at an angle to the observer.

The illustrations thus far show the direction of the lines when lO

E1.KMKNTS AND RUI.KS OF Perspective.

the objects are at right angles with the plane on which the observer is standing. We will now notice the direction of the lines in a square \ym.% fiat upon the same surface upon which the observer is standing.

Place the square at a distance of eighteen or twenty feet from the observer, so that a line drawn from the point where the observer is stationed, through the center of the square, will divide the front line into two equal parts ; the front side of the square will, of course, appear longer than the far side, because it is nearer the observer. The line receding from the nearest right-hand corner, because it forms an angle with the picture plane, appears to tend to a point on the left of it, and the line extending from the nearest left-hand corner tends to a point on the right of it. These two receding lines approach nearer and nearer as they recede from the observer, and would, if produced, meet in a point opposite the ej^e of the observer (Fig. 16). This point is called the point of sight, and it locates the horizon, a horizontal line (page 8) to which all lines in a perspective drawing vanish, and on which are situated all vanishing points in parallel and angular perspective. All lines that are at right angles to the observer vanish in the point of sight.

All lines that are at an angle (greater or less than a right

angle) to the observer, vanish in the — horizon, but not in the point of sight.

Fig. 16 represents a square lying flat on the surface upon which the observer is standing. The point m is opposite the observer.

P.S. Horizon a ni b

ttg, z6.

Chapter III

p.s.

Hor.

Objects rest upon what is termed the ground plane in many different positions, which may be reduced to three general ones.

I. The objects may be so placed that their surfaces are at right angles and parallel with the picture plane. Objects viewed in this position are said to be in /«ra//^/ perspective (Fig. 17).

abed represents the picture plane; abef represents the ground plane.

The face hiln and the face parallel to it, are parallel to the picture plane; the four remaining surfaces are at right angles to it.

Fig. X7.

2. The objects may be so placed that some of the surfaces are at an angle with the spectator, and others parallel with the ground plane. Obj ects viewed in this position are said to be in «;2- ^2^/<2r perspective (Fig. 18).

abed represents the picture plane ; abe represents the ground plane. When the cube is in this position, the faces ijk and ghn are parallel to the ground plane, and the four remaining sides are at angles with the picture plane.

3. The objects may be so placed that the surfaces are not par-

(II)

Elements and Rules of Perspective.

P.S allel with either the picture plane or the ground plane. Objects viewed in this position are said to be in oblique perspective (Fig. 19).

abed represents the picture plane ; abef represents the ground plane.

The cube in this illustration is viewed in oblique perspective, and the faces are at angles with both the picture plane and the ground plane. In parallel and angular perspective, the lines vanish in the horizon ; but in oblique perspective, the lines vanish on lines perpendicular to the horizon.

Oblique perspective is necessary in making drawings of roofs of houses, steeples of churches, gables, roads that are not level, etc. ; in fact, in all views where there are oblique lines sloping from or toward the observer.

Before proceeding further, it is necessary to define the following terms :

(Note. — if these definitions are thoroughly understood, and committed to memory now, the after-work can be accomplished with very little diflSculty.)

1 . The ground plane is the plane or surface upon which the object or objects to be drawn are situated.

2. The base or ground line is a horizontal line that marks the nearest limit of. the view to be taken ; in a perspective drawing it is

the lowest line of the picture.

3. The picture plane is an imaginary plane between the observer and the object, and is perpendicular to the ground plane (Fig. 17, page m&, 20, II, and Fig. 20).

c d

1)\

EI.EMENTS AND Rui.ES OF PERSPECTIVE. 1 3

Fig. 20 represents the ground plane abef, and on the base line ah, the picture plane, abed.

4. The li7ie of direction is a horizontal line extending from the e3^e of the observer to the object ; it is perpendicular to the picture plane.

In perspective drawings, the horizon is usually fixed at five feet above the base line ; this distance is generall}^ adopted (being about the distance from the ground plane to the line of direction) for the sake of uniformity.

In landscapes and sketches, this distance between the horizon and base line varies according to the nature of the scene to be represented.

1. In a representation of a low, flat, marshy desert, or of a prairie country, the horizon is quite low in order to give the idea of a level landscape.

2. If the picture is intended to convey the idea of great height, the horizon is placed low. In a sketch of a mountain gorge, cataract, mountain peak, or any scenery where an idea of great height is to be given, it is always low.

3. If an idea of depth, as of a scene viewed from a great height, is desired, the horizon is placed high — as the view of a city from a tower or steeple, or an expanse of country from a high hill.

4. In an ordinary view where no special height, depth, or distance is required, the horizon is placed about one-third the height of the picture above the base line.

5. The horizon is a horizontal line opposite the e^^e of the observer, parallel with the ground line, and distant from it five feet.

6. T'\iQpoi?it of sight is a point on the horizon opposite the eye of the observer ; its position on the horizon varies as the spectator changes his point of observation.

14 KlvEMKNTS AND RULES OF PERSPECTIVE.

7. The prime vertical is a vertical line passing through the point of sight.

8. ^hQ point of station is the point where the observer stands. The proper position from which to view the picture, is distant from the picture three times the height of the highest object in the scene ; or, if the length exceeds the height, then the point of station will be distant from the picture three times the length of the object.

9. Point of distance is a point on the horizon to which all measurement lines vanish. In a drawing, it is as far from the point of sight as the point of .station is from the point of sight (see page 16).

10. Vanishing points are points in which vanishing lines meet.

OBSERVATIONS.

1. Horizontal lines seen obliquely or at angles, if above the eye, appear to incline downward; if below the level of the eye, they appear to incline upward.

2. Straight lines may be drawn by finding the position of the extremities of the lines.

3. Curved lines may be drawn by finding their points of intersection.

4. The center of a perspective square is found by drawing its diagonals.

5. The center of a perspective rectangle is found by drawing its diagonals.

6. The perspective of any surface is determined by the perspective of its boundary lines.

H

Chapter IV

The following frame- work (Fig. 21) is necessary before a correct perspective drawing can be made ; the points are placed and the lines are drawn in the following order :

1. Place the point of sight and mark it — P. S.

2. Through the P. S. draw a horizontal line of indefinite length, and mark it — Horizon.

3. Through the P. S. draw a vertical line, and mark this prime vertical — P. V.

4. Five feet below the P. S. on the prime vertical place a point through which draw a horizontal line of indefinite length, and divide it off into feet according to the scale. Mark this the base or ground line. Measure off on each side of the P. V.

5. Fix the point of distance on the horizon line, three times the height of the object from the

point of sight, (if the length of the object exceeds the height, then take three times the length to avoid a violent fore-shortening) and mark it P. D.

In this case the supposed height of the object is 3^ feet; the point of distance is on the horizon three times 3^ feet from Fig, zi.

the point of sight.

p.

Base or Ground Line.

Scale Vg, in. to 1 ft.

i6

E1.KMKNTS AND RuivKs OF Perspective.

Another form is here given (Fig. 22) which is often used in preference to the one on the preceding page. In this form the

points are placed and the lines drawn as in the form on page 15; but the points of distance are found by describing a half circle, with the point of sight as a center, and the line extending from the P. S. to the point of station as a radius. Where this half circle crosses the horizon, the points of distance are located.

p. D. Horizon

P.

s.

P. D

Baae_ _ / of,

a

Pi

Scale 3^ in. to 1 ft.

Fig. 2Z,

Chapter V

RULKS GOVERNING THE DIRECTION OF I.INES AND SURFACES OF OBJECTS IN PARAI,I,EIv PERSPECTIVE.

RUI.E I. All measurements are taken on the base line, which is divided into feet according to a given scale.

RuivE II. All lines parallel to the picture plane are represented in their actual position, and do not vanish.

Rui,E III. All lines perpendicular to the picture plane vanish in the point of sight. (P. S.)

RuiyE IV. All measurement lines vanish in the point of distance. (P. D.)

Rui,E V. All surfaces are governed by the lines that bound them, and their perspective is determined by these lines.

(Note. — After sketching an object or picture, the above rules may be applied to test the accuracy of the representation. )

In drawing a group of objects in still life or a landscape sketch, all the lines should be brought to the test of perspective calculations. It is impossible to place the different objects in their proper relation to each other, and in perfect harmony with the picture as to size and proportions, without a knowledge of the principles of perspective. All nature, animate and inanimate, is impressed on the sense of vision in accordance with the laws of perspective; consequently all art that represents in lines or masses must be in conformity to this most important branch of drawing.

i8

ElvKMKNTS AND RuivKS OF P^RSPKCTIVEJ.

Perspective admits of another division than that presented in this volume — aerial perspective. Aerial perspective has reference to atmospheric and other influences, by which objects more or less remote are effected in regard to light, shadow, color, gradation of tints, etc. , according to their distances and relative positions. The principles for guidance in representations in aerial perspective are not reduced to systematic rules. A close observance of nature will, in time, enable the student to represent these effects with approximate accuracy.

Chapter VI

REMARKS ON MODKI. AND OBJECT DRAWING.

Precision is the basis of correct representation from the model. All questions are readily and satisfactorily settled by reference to the rules and principles oi perspective, vAiv^ are founded on the sciences, and are unchanging. (Geometry, Light and Color, and Optics. )

Practice in model drawing is necessary to satisfactory work in sketching from nature. It develops the power of observation, the ability to concentrate the attention, and the habit of noticing details — their relation to each other and to the whole, and the characteristic features of the object under consideration.

The objects used in sketching are geometrical solids, and objects based on these forms, supplemented by objects of beauty and utility. In drawing from the object, all work is free-hand.

In arranging objects for drawing, they should be placed so that the light comes from one side, thus giving the observer a view of the shaded side and the shadow.

The pupil should draw things as he sees them, and not as they really are.

To secure the right proportion in the drawing, the pencil may be held at arm's length, and the subtended amount of any dimension of the object may be marked on the pencil by moving the thumb-nail until the distance between it and the end of the pencil

exactly covers the desired distance in the object.

20 BiyKMKNTS AND Rui.BS OF PKRSPECTIVK.

By always viewing the object from the same point of station, and holding the pencil (when used as a measure) always at the same distance from the eye, and in the same plane, the copy will have the same relative proportions as the object.

SUGGKSTIONS.

1. Place the model so that the light falls on it from one direction only.

2. Call attention to the facts and the representation of the surfaces.

3. Review rapidly the rules and principles of perspective governing the appearance of lines.

4. Note the distribution of light and shade — high light, shade, reflected light, shadow, and reflections, if any occur.

5. Make the drawing, and if you doubt the correctness of the representation, test by perspective. (Rules, page 17.)

6. The student may draw with board or stretcher at any angle that is convenient for him.

7. In making the drawing for illustration, if the object is not at hand, represent the light as falling from the upper left, falling on the object at an, angle of 45°; the shade will then be on the opposite side.

8. The shade on objects bounded by plane surfaces is on the side away from the light.

9. When a plane is in shadow, the deepest shade is on that part nearest the observer.

10. Reflected lights appear on the farther side of the plane.

11. When a plane is in half-light, the highest light on it is on the part nearest the observer.

12. A plane partly illuminated has the deepest shade adjacent to the illuminated part.

13. A shadow cast by an object is darker than the shade on the object casting the shadow.

14. The darkest part of a shadow is near the object casting it.

NoTK. — Test your comprehension of these observations by drawing perspectives of a cube, on three different sheets of paper, showing : {a) A cubical block. {b) A cubical box without the lid. {c) A cubical box, minus the front and left or right side.

15. If the hght falls on a sphere from the left (7), illuminating the upper left-hand surface a little in from the apparent edge, the deepest shadow is on the lower right side a little in from the apparent edge ; near this edge we find a medium tone that is called reflected light.

16. Reflected light on objects having curved surfaces is seen on the shaded side. The shaded surface is partially illuminated by lights cast from adjacent objects, and from the plane on which the object rests.

17. In cylinders, if the highest light falls on the left side a little way in from the apparent vertical boundary, the deepest shade comes on the right side a little from the apparent outline. This shade is darkest at the top, slightl}^ modified in tone at the base by the reflected lights. The top of the cylinder is shaded slightly at the left ; the right side is light. The shade and light do not extend to the limit of the outline.

Note. — Drawings may be finished by hatching, stippling, or shading with the stump.

Chapter VII Rkflkctions

Rkflkctions are produced ( i ) by polished surfaces that give back form and color, as mirrors, etc.; (2) by polished siurfaces that give reflections somewhat distorted, as chinaware; (3) by liquids at rest — the smooth surface of the liquid serving as a mirror.

The rule that covers all reflections is, —

The angle of incidence equals the angle of reflection.

To place reflections in perspective, treat the reflections as realities. Such objects as rise or occupy a position perpendicular to the surface of the water, preserve their real proportions and relative positions. Thus posts, perpendicular cliffs, and masts of boats, throw their reflections to their full height, while surfaces that recede from us — foreshortened surfaces — although much higher in reality, and rising far above the objects mentioned, may scarce be seen at all in the reflection. If the point of sight is placed on a level with the water line, then the reflection will be a perfect repetition of the view, but the slightest elevation of the point of sight above the water line affects everything reflected that is not perpendicular to the water's edge. For instance, a roof jutting out over a boathouse on the bank will be much longer in the reflection than the vanishing lines in the original, and from the point of view we may see the top of the roof, while in the reflection we may see the under side of the projecting part of the roof. The reflection of an arch

Elements and RuIvKS of Perspective. 23

or culvert will be exactly like the original in the general face of the arch, but on account of the position of the observer he sees only the under side of the arch in the reflection.

The same rules that apply to linear perspective, apply to the perspective of reflectionc.

Chapter VIII

Problkm. — Place in perspective a 3-foot cubical box, the front line of the cube resting on the base line, and the nearest corner of the cube 3 feet to the left of the prime vertical.

I

Point of Distance ( 9 Ft.)

Scale M in. to 1 ft

1. Locate the point of sight. If the object is on the left of the prime vertical, fix the point of sight on the right side of the paper. If the object is on the right of the prime vertical, then locate the point of sight near the left edge of the paper.

2. Draw the horizon, prime vertical, and base lines, marking the base line off according to the given scale.

3. lyocate the point of distance. (Page 15.)

As the cube is 3 feet on the left of the prime vertical, move 3 feet to the left on the base line, and mark the point a; from a move

3 feet further to the left, and mark the point d (Rule I). From a

EI.KMENTS AND RULES OF PERSPECTIVE.

and b, rule lines to the point of sight (Rule III). We now have the side ab and two sides of the base extending back from the front line ; to determine the length of these receding lines is the next step.

As lines of the same length at different distances appear of different lengths, we know it is impossible to measure three feet on these lines, so we take the measurement on the base line (Rule I), and from the measurement point to the point of distance rule a line. Where this line crosses the line receding from a, the point c is located ; from the point <:, draw a line parallel to ab, mark the point d, and we have the base of the cube.

J^rom a and b, draw vertical lines 3 feet in length, and draw ef (Rule II).

From e andy, draw lines to the point of sight (Rule III), erect vertical lines from c and d, and where these touch the lines receding from e and/" locate the points g and h, and draw the line gh which completes the outline of the cube.

ProbIvEm. — Place in perspective a box 6 feet high, with a base 3 feet square, the nearest corner of the base 7 feet to the right of the prime vertical on the base line.

From the point where the base line is crossed by the prime vertical, move to the right on the base 7

feet, because the nearest corner of the box is 7 feet to the right of the observer, and locate the point a. From this draw the line ab on the base 3 feet long. From a and b, draw lines to the point of sight (Rule III). From the point a, move toward the point of sight 3 feet, and draw a line from the point of measurement to the point of p.

xr c

Base

11 ,,11..

i

J

Klemknts and RuIvBS of Pkrspectivk.

distance. Where this line crosses the line receding from a, we have a point 3 feet from a. Mark this point d, and from it draw a line dc parallel to the front line of the base. This gives the square abed in perspective, the base of the box. From a and b draw vertical lines, each 6 feet high, and draw the horizontal line ef; from e and / draw lines to the point of sight (Rule III). As the points e and f are above the level of the eye, the lines receding from them appear to incline to the horizon.