The Vertical Vanishing Point in Linear Perspective [Reprint 2014 ed.] 9780674183735

Table of contents :
PREFACE
CONTENTS
I. VISIBLE SPACE RELATIONS
II. GEOMETRY OF SPACE RELATIONS
III. PHOTOGRAPHIC PERSPECTIVE

Citation preview

THE VERTICAL VANISHING POINT IN LINEAR PERSPECTIVE

BY

STANLEY BRAMPTON PARKER, A.I.A.

HARVARD UNIVERSITY PRESS CAMBRIDGE, MASSACHUSETTS LONDON: GEOFFREY CUMBERLEGE OXFORD UNIVERSITY PRESS

1947

Copyright, 1947 By the President and Fellows of Harvard College

Printed in the United States of America

To WILLIAM T. ALDRICH

PREFACE THE FOLLOWING PAGES represent a brief of twelve years' research in linear perspective. My interest in the vertical vanishing point was originally instigated in 1935 by William T . Aldrich, F.A.I.A., who was the first to introduce a new type of visual screen which I have referred to as the Aldrich Viewer. F o r him, in the next four years, I reviewed all the books on perspective listed in the four great local libraries—the Boston Athenaeum, the Boston Public Library, the library of the Boston Museum of Fine Arts, and the Harvard College Library—and developed with him a set of illustrations and text which exhaustively described the whole subject of linear perspective. In 1940, I carried on the perspective practice and research alone with particular concentration on stereoscopic drawing. In this work, I discovered that the short cuts developed in stereoscopic drawing, such as the use of verticals as a space measurement and "protractor rabattement," were equally important in all linear perspective drawing.

This research also led to setting aside the

Monge quadrant as an axiom of centrolinear projection and necessitated a revised group of geometric assumptions and definitions. Since M r . Aldrich has not yet published his book and since I have been urged from many sources to make our work known to the public, with his approval I have taken the responsibility of publishing this brief. STANLEY B .

Cambridge, Massachusetts December, 1946

PARKER

CONTENTS PART I · VISIBLE SPACE RELATIONS

Plate I Plate II Plate III Plate I V

The Aldrich Viewer Device to Demonstrate the Relation of the Horizon to Converging Verticals Drawing Made with the Aldrich Viewer (Looking Down) Drawing Made with the Aldrich Viewer (Looking U p )

PART II · GEOMETRY OF SPACE RELATIONS

Plate Plate Plate Plate Plate Plate Plate Plate Plate Plate Plate

V VI VII VIII IX X XI XII XIII XIV XV

Plate X V I Plate X V I I

Fundamentals Rabattement Space Diagram Right Trihedron Repeated Vertical Measurement Sun's Shadows Reflection; Perspective Table Perspective with Low Horizon Perspective with Horizon Near the Top of the Picture Perspective with Horizon Near the Bottom of the Picture Horizon Off the Picture Horizon Oil the Picture

PART III · PHOTOGRAPHIC PERSPECTIVE

Plate X V I I I Plate X I X Plate X X

Camden, Maine. Drawing Made from Photograph Correct and Incorrect Perspective Compared to Correct and Incorrect Photography Memorial Hall, Cambridge. Drawing Made from Photograph

Plates V to X I I were made by Miss Charlotte Hurvich, all others by the author.

AERIAL

DRAWING

Photograph of a d r a w i n g 3 6 " X 2 4 " m a d e for E d w a r d Sears Read, A.I Α . , showing necessity for many vanishing points for horizontals, one vanishing point for verticals

I. V I S I B L E S P A C E

RELATIONS

The window Picture Plane as a frame for making Perspectives no longer an established principle of correct draftsmanship. False impressions of visual limitations corrected by the Aldrich Viewer.

PERSPECTIVE DRAWING, since the earliest known publications, has been based on a visible horizon. The horizon line was plainly evident in nature, and when its image* was transferred to the drawing board it formed a logical foundation on which to build the geometry of perspective. On this line, as a trace of the first importance, the vanishing points of the images of all horizontals that appeared to the eye were found, and through the generations during which the science of descriptive geometry developed the images of horizontals continued to receive the favor of draftsmen, but the images of verticals remained in all constructions parallel to each other and parallel to the sides of the picture. Whether the line of sight was directed up, down, or level, the picture plane remained vertical, and consequently the images of verticals remained conveniently vertical. Even when the "bird's eye" view came into popular use, it brought no evidence that the images of verticals might share the honors of their horizontal team-mates and possess a vanishing point of their own. As long as we continue to assume that the picture plane is vertical, there will be no vanishing point for verticals. But we believe that the picture plane is not necessarily vertical; that, on the contrary, it must be assumed to stay unalterably perpendicular to the line of sight wherever our attention is directed. We also believe that our attention is more often directed up or down than level, and that in consequence our whole psychology of pictorial memories should be recorded as they habitually appear, not according to past geometric practice. Conscious of this fact, Mr. William T. Aldrich (in 1935) made it the basis of a series of lectures on Perspective, explaining the geometric process by which it could be used, and, as a demonstration for popular explanation and for his own use, invented a viewing device which shows graphically the soundness of his geometric premise. Like other viewing devices of this sort, it consists of a rectangular frame holding a net of squares lined off by strings, through which one views and measures the sight size of objects and transfers them to corresponding squares •The word "image" is used throughout this paper to denote the optical counterpart of an object as it appears to the observer.

1

on the drawing. But, unlike other viewing devices, the center of the picture is fixed by the crossing of two central strings. Most important of all, the Aldrich Viewer has an eye-piece fixed at a point consistently on a line with the center of the picture. This eye-piece may be moved to and from the frame, but is so constructed as never to be off the exact perpendicular to the center of the frame. T h e frame and the line of sight, therefore, maintain a constant rightangle relationship in whatever direction the device is focused. Together they are mounted on a tripod which has a universal joint to allow easy adjustment. W h e n this Viewer is set up so that the line of sight is not level, and when the center upright line is made to correspond with some vertical in evidence, such as the jamb of a window or the corner of a room, the images of other verticals, which appear near the sides of the frame, will not be parallel to the sides of the frame, not parallel to each other, and, of course, not parallel to the center vertical first chosen. T h e frame of the Viewer has thus furnished the eye with the limitation it needs to fix the truth of its visual impression. It discloses the fact that while the image of one vertical, the one in the direct center, is vertical in the visual picture, the images of all other verticals converge to a point. T h a t point is below the center of the picture when the line of sight is directed downward and is above the center of the picture when the line of sight is directed upward. T h e images of verticals will be parallel only when the line of sight is level. T h i s Viewer is a truly great contribution to good draftsmanship. Not only is it useful in studio and drafting room for practical application, but it trains the eye to a keener consciousness of pictorial values.

It furnishes startling

evidence that the human eye has been deceived by the assumption that the picture plane must always be a vertical window; for tests will show that it is just as absurd to expect the images of verticals to be unalterably parallel to the sides of the picture as to expect the images of horizontals to be parallel to the top and bottom. Viewers of this sort do not need to be expensive, but it pays to have them strongly built and as well arranged as possible.

M r . Aldrich has one for

studio use which stands firmly on a movable platform, and the adjustments may be made by conveniently placed ratchets. F o r outdoor use, he has one built of light material set on a folding camera tripod (see Plate I ) . T h i s type is the one I find most useful, particularly in settling on a point of sight for a perspective of a large project. A blueprint

2

P L A T E I.

THE ALDRICH VIEWER

T h e squares may either be made of string or lines marked on a sheet of glass of the site plan is laid on the floor, the viewing device adjusted, and the print moved into a satisfactory point of view. When this is fixed, it is a simple matter to note the height in inches from the floor and the direction in degrees for the line of sight, and to relate these to the scale of the site plan. The final construction of the results thus obtained may be drawn without guesswork. Many ways of m a k i n g this Viewer will suggest themselves to gadgeteers. I have made them from nearly everything, from a tin can to a tennis racquet. The most amusing and instructive, although crude in workmanship, is made from a coxswain's megaphone. The mouthpiece serves as an eye-piece. T h e larger end serves as the picture plane, which is fitted with stringed squares

3

P L A T E II. DEVICE T O D E M O N S T R A T E T H E R E L A T I O N O F T H E H O R I Z O N T O C O N V E R G I N G VERTICALS made fast across the opening by wedging the string in short cuts in the cardboard. Attached to a lead weight which hangs downward through a slot in the cone and pivoted on a long brad across the narrow end is an arrow at right angles to the lead weight, which acting as a level, points to the horizon line as the megaphone is slanted up or down (see Plate II). By habitual use of such Viewers, one very quickly becomes accustomed to building a visual impression of the presence of a similar sight limitation without the aid of the device itself, and the eye is trained to accept the new construction with converging verticals as a matter of common observation. As one continues to make observations through such a Viewer, other contradictions to time-worn rules of perspective will become apparent. T h e most 4

P L A T E III.

DRAWING MADE W I T H T H E ALDRICH VIEWER (Looking d o w n )

absorbing is the formation of ellipses which appear as images of circles. In the direct center of the picture, the ellipses are symmetrical with the center line of the Viewer. But when they appear at the side, it is evident that the ellipses slant perceptibly away from the vertical and away from the images of verticals. Such observations of fact furnish a draftsman with an independent means of finding out the truth in perspective for himself without reliance on geometry or on the conclusions of others. Whether or not he may care to depict the truth thus discovered is wholly a matter of his own choice. 5

PLATE IV.

DRAWING MADE W I T H T H E ALDRICH VIEWER (Looking up) 6

II. G E O M E T R Y O F S P A C E

RELATIONS

Basic Principles of geometric perspective must be revised to conform to modern point of sight. SOME KNOWLEDGE of elementary geometry is necessary fully to understand the following explanation of the vertical vanishing point and its relation to centrolinear perspective. M a k i n g drawings with three or more vanishing points in a perspective is not a new achievement. A m o n g the problems in the Victorian books on perspective, there are plates showing methods of drawing an object which has been turned away f r o m a vertical position, thereby necessitating a vanishing point for each of the three dimensions of that object.

B u t such problems are dis-

cussed as cases rarely to be encountered and without reference to the possibility that such a construction m i g h t be used as a general case. In the more recent publications, a vanishing point for the images of verticals is described for the purpose of drawing single objects.

T h e s e methods are

called " t r i c o n j u g a t e " or "three-point" as distinguished f r o m one- or two-point perspective. T h e system is expressly designed as a special case for the delineation of a single object, not as a universal one for use in a broad representation of space relations. A

universal

diagram

of

space

relations—one

which

embodies

height,

breadth, a n d depth—must be versatile enough to include not only three points but as many more as we w a n t . Most of the vanishing points in a picture are f o r the images of horizontals, while just one alone will include all the images of verticals. A realization of the fact that the images of verticals in a drawing are constant throughout, while the images of horizontals vary greatly in the same drawing, has led me to refer the space measurement in a perspective to the verticals.

H e i g h t as a measurement of depth is, therefore, a more con-

venient dimension to use than the customary length and breadth. Custom makes us reluctant to depart from geometric principles practised through

generations

and

may

be

responsible

for

our

inclination

to

stay

grounded in visual representations in spite of our rapidly g r o w i n g consciousness of the world from higher altitudes.

I refer to the quadrant

introduced

by M o n g e in the latter part of the eighteenth century and still held, as it should be held, the great axiom of descriptive geometry. It is neat and precise in plotting points in space for those w h o choose to do so from a level aspect,

7

but to one who wishes to direct his line of sight above or below the horizontal, it is contradictory to his purpose. For, by definition, one plane of the quadrant is vertical while the other is horizontal, and the vertical plane is assumed to be the plane of projection (the picture plane). If we are to be consistent in our belief that the picture plane is not necessarily vertical, then we cannot use the quadrant without redefinition. After many attempts to rearrange the quadrant into a new environment, we are obliged to set it aside and establish an entirely different group of assumptions for the execution of perspectives. ASSUMPTIONS

Assumptions on which modern perspective writers are generally agreed are that the visual rays from the eye to objects in space are regarded as a cone-like structure called the Cone of Vision. The apex of the cone is the eye or point of sight called the Station Point. The axis of the cone is the Line of Sight (see Plate V, Fig. 1). It is agreed among the most advanced that, under certain conditions, the plane of projection (the Picture Plane) may be assumed to slant or tilt away from the vertical. But it is here assumed that the picture plane is always perpendicular to the line of sight and that the point of intersection of the line of sight with the picture plane fixes the Center of the Picture. This combination of two assumptions is of the greatest importance as an emancipation from the vertical picture plane. Common practice relies on fixing the "tilt" of the picture plane which the station point is assumed to follow and accommodate itself as a special case. Such methods, designed as they are for a special purpose, are not easily adapted to a universal method. A universal method of measuring the relative breadth, depth, and height must be one that will bring the three dimensions and scattered points in space within our view to the picture plane. But before we are able to connect one to another the relative distance of points in space on the picture plane, we must first fix the vertical and horizontal position of the station point itself.

8

CONE,

or

VISION

Γι g. 2

FIG. 3

Abb it ε ν α τ ι ο ν s S Ρ -

S T A T I ON

PP

"ICTUR-E

-

PLANÉ

L S -

LINE

PD

ΡΚ.1 N C I P L C .

-

or

P o ι NT

Z O — PLUMB 5P

p

-

SIGHT PtSTANCe

LINE

CENTER,

AT

or

A A -

ANGLE

ALPHÌ.

AB

-

ANGLE

BETA

CP

-

CIRCLE

or

STATION

POINT

PICTURE.

THE

PICTURE

PLATE V.

FUNDAMENTALS

STATION POINT, PLANES, AND VANISHING POINTS

The station point is assumed to be at the intersection of three planes at right angles to each other. One plane is horizontal, two are vertical. The horizontal plane represents breadth and depth; one vertical plane represents breadth and height; the other vertical plane represents height and depth (see Plate V, Fig. 2). Thus, each plane may serve separately to represent two of the three dimensions of space, but no single plane the same two dimensions. At each of the intersections of these planes, however, is a line which, being common to one pair of planes, is common to the dimension found alike in that pair of planes. Thus, the line common to both vertical planes is the principal vertical and represents height. It is assumed to extend from the zenith to the earth's center and is called the ZO. The line common to the horizontal plane and the vertical plane representing height and breadth, represents breadth. The line common to the horizontal plane and the vertical plane representing height and depth, represents depth. The three lines, therefore, representing breadth, depth, and height, furnish a key to a space diagram through which any plane may be assumed to pass. The planes we are chiefly concerned with at the outset are two of the three already mentioned (see Plate V, Fig. 3 ) : (1) the horizontal plane which includes the breadth and depth lines, ( 2 ) the vertical plane which includes the depth and height lines. Each of these planes is assumed to have a protractor with the station point as a center which will later form an important part of the system. The line of sight from the station point is assumed to stay within the vertical plane in the three separate conditions of vision: that of looking up, looking level, and looking down. The picture plane follows the direction of the line of sight and is perpendicular to it at all times. The point of intersection of the line of sight with the picture plane is the center of the picture. This point is also the station point, projected to the picture plane by its shortest route, and is referred to as SP11.

10

The circle formed by the intersection of the cone of vision with the picture plane has at its center SP P and is used as the selected circular limit of the picture ( C P ) . The angle of the cone is called Beta. The limit of this angle is assumed to be no greater than 30°. The angle the line of sight makes with the horizontal is called Alpha. When the line of sight is level, Alpha is 0°. The principal distance is the distance from the station point along the line of sight to the picture plane. All the visual rays (or lines) in the cone of vision appear on the picture plane as points of undetermined distance (see Plate V, Fig. 4). The points are images of lines as they penetrate the picture plane on their way from the station point to infinity. Each point which is the image of a line is also the image of that line at infinity. This is true of any line through the station point which penetrates the picture plane, whether it is within the cone of vision or outside of it. The images of lines, however, that do not pass through the station point cannot appear as points and must, therefore, be lines. The images of parallel lines are assumed to meet at a common point of infinity.* The images of a system of parallel lines, one of which passes through the station point, must include the image of the point of infinity and converge to it. This point of convergence for the images of a system of parallel lines is called the vanishing point of that system. Thus in Plate V, Figure 4, the image of ZO is its point of intersection with the picture plane and is the vanishing point for the images of all verticals. The image of the depth line is also its point of intersection with the picture plane and is the vanishing point for all the images of horizontal lines parallel to it. The breadth line is parallel to the picture plane and does not reach the picture plane. The image of lines parallel to it will, therefore, be parallel on the picture plane. •The word "infinity" as used in perspective drawing is a localized space conception.

II

R A B A T T E M E N T OF T H E STATION P O I N T AND T H E T H R E E M A I N

DIMENSIONS

Plate V I , F i g u r e 1, shows the picture plane indicated as a d r a w i n g b o a r d s u p p o s e d to be slanted at an angle A l p h a of 3 0 ° d o w n w a r d .

T h e horizontal

a n d vertical planes are represented with the breadth a n d height lines. T h e s e planes as they penetrate the picture plane leave m a r k s or lines of intersection.

Such lines of intersection m a d e by planes passing through the

station point are called traces. at infinity.

T h e trace of a plane is the i m a g e of that plane

It is, therefore, the vanishing line for the i m a g e s of all planes

parallel to it. A trace is used as a hinge by which a plane m a y be revolved to the picture plane, revealing whatever is represented as being in that plane to the picture plane. T h i s process is called

rabattement.

T h u s , in Plate V I , F i g u r e 2, the vertical plane is rabatted sideways to the picture plane on its vertical trace ( V T ) as a hinge, revealing on the picture plane the station point ( S P V )

with its protractor; the Z O ;

the

Principal

D i s t a n c e ( P D ) ; the line of sight ( L S ) ; a n d the depth line ( H ) . In a like m a n n e r , but d o w n w a r d , the horizontal plane is rabatted on its horizontal trace ( H T ) as a hinge to the picture plane, revealing the plan view of the station point ( S P H ) with its protractor; a n d the breadth line.

The

depth line falls on the V T . W i t h these two rabattements—one an elevation view which reveals the m e a s u r e m e n t of height related to depth f r o m the station point to infinity, the other a plan view which reveals the m e a s u r e m e n t of breadth related to depth f r o m the station point to infinity—a d r a f t s m a n is able to compose a space diagram. T h e process of rabattement shown in perspective in Plate V I , F i g u r e 2, is s h o w n in Plate V I , F i g u r e 3, as it w o u l d appear on the face of a d r a w i n g board. F o r convenience of identification, the horizontal rabattements are s h o w n in dash lines, the vertical rabattements in dot a n d dash lines. T h e vertical rabattements m a y be m a d e on either side of the plane's trace, but it is well to m a k e the horizontal rabattements d o w n w a r d , for by reversing the process, the information is turned upside d o w n . T h e most easily read d i a g r a m s of space relations are m a d e u p of the i m a g e s of squares. Because of depth in space, these i m a g e s diminish in size as they near infinity.

T h e v a n i s h i n g points of the sides of squares alone will not

f u r n i s h the necessary m e a s u r e m e n t for the proportional diminution of each i m a g e . B u t since a s q u a r e m a y be constructed by u s i n g its d i a g o n a l , the i m a g e s of squares m a y be constructed by u s i n g a diagonal vanishing point with its t w o corresponding rectangular vanishing points.

12

4·5%Ρ HOR.ÌZ0NTAV.

τ (V

TR. ο VP

fi c p V \

, Ν \LS-ff \

U =

/ // /'

SP κ

BREADTH

χ

• χ

\

rim *χ

• ν /

M>R,iJONTAL R.ASATTLMt UT, :

jV * A. ' ^ / / /

VERTICAL TB.ACL (VT)

V*

Picture

PLAME

Fig. 3

PLATE VI.

RABATTEMENT 13

With the aid of the protractor which forms a part of the station point rabattement, the vanishing point for any system of parallel lines may be found. With 45° as the diagonal of the rectangles, diagonal vanishing points are plotted on their respective traces and form a part of the space construction. T h e images of squares, once established on the basis of the same unit of measure for height and breadth, may be used to build the images of cubes. With the images of cubes available, a space diagram is furnished to delineate any object or group of objects. SPACE DIAGRAM

T h e horizontal and vertical rabattements established in Plate VI are used in Plate VII to make a three-dimensional space diagram. T h e principles under which it is done will be familiar to those who use the "two point" perspective construction, with, however, one important difference. Formerly, the two vanishing points employed represented two horizontals at right angles to each other and both points were on the horizontal trace. The images of verticals, being parallel to each other, were therefore without a vanishing point. While in this diagram one vanishing point is on the vertical trace, the other vanishing point is at the junction of the horizontal and vertical traces. One is for the images of all verticals ( V V P ) ; the other is the vanishing point for the images of all horizontals parallel to the vertical plane (0° V P ) . After a satisfactory point of sight for the perspective of a given project has been selected by the aid of the Aldrich Viewer, the next step should be to draw in miniature the facts as set up by the Viewer and relate them to a definite scale. Using the unit of measure of the plan of the project, set down by scale all the essential components for the elevation rabattement ( P D ; angle Alpha; H T ; 0° VP; VVP and the distance to the ground line; see Plate VII, Fig. 1). Go through the same process for the plan rabattement (Plate VII, Fig. 2), but instead of executing these drawings separately, combine them in one drawing as shown in Plate VII, Figure 3. Keep in mind, however, that the vertical trace, VT, is serving as the picture plane for the vertical rabattement. Using the main dimensions projected in the plan view to the H T and projected in the elevation view to the VT, fix the squares to determine the horizontal and vertical sides of the cube. Build up and fix the smaller squares by using the diagonal vanishing points marked on each trace 45° VP. Interpolate the information from the plans of the project to the image of the project within the cubical space diagram.

14

Ο

A / Ν. / > "7% Α / / ^ Ν ί

Λ. / y Λ

HT

VP

-V Ν

PO Α

íf,

/Τ /•>

Ά.

V PLMtL TR-KCE.

V PLANI. TILUCL r λ VVP PLAN

ElXVATION R-ABATTtMENT

RABATTEMENT FIG.

Fig. 1

/V / >. / ^ N - A

' V

I

HT

'A/

* V > / V

V PLUKt ΤΚ-kCE AND PP roR_ Vt-¿Tic.M-s ,

ELEVATION

AND

PLAN

RABATTEMENT

COMBiNLD FIG. 3

PLATE VII.

SPACE DIAGRAM 15

RIGHT TRIHEDRON

To draw objects which are oblique to the picture plane in their three dimensions, one should be familiar with the properties of a right trihedron. A trihedron is a solid figure having three sides and a triangular base. If the three sides of a trihedron are at right angles to each other (like the" corner of a box), it is a right trihedron. The edges of a right trihedron are perpendicular to each other. Plate VIII, Figure 1, shows a right trihedron with a pin thrust through the apex O, perpendicular to the base ABC. The mark that the pin makes on the base is called the ortho center of the base. Plate VIII, Figure 2, shows each of the three faces of the trihedron rabatted to the paper, revealing the true size and shape of each right triangle. A line from each corner of the base through the ortho center ( O C ) to the opposite rabattement of the apex will be perpendicular to the trace of the rabattement, thus (Plate VIII, Fig. 3 ) : A through OC to O 3 is perpendicular to CB. Β through OC to O 1 is perpendicular to AC. C through OC to O 2 is perpendicular to AB. The apex of each rabattement will fall on a circle made by using that side of the base as a diameter. 0 1 is on circle from diameter AC. 0 2 is on circle from diameter AB. 0 3 is on circle from diameter BC. In perspective, the apex O is assumed to be the SP. T h e ortho center ( O C ) the SP P .

16

PLATE VIII. RIGHT TRIHEDRON 17

REPEATED VERTICAL MEASUREMENT

Plate IX shows a space diagram similar to that in Plate VII, Figure 3, and it is assumed that an object is turned at another angle from the space diagram. T h e degrees selected are 25° from the 0° VP on the left and 65° from 0° on the right, as indicated on the plan rabattement SP h . These lines are brought to the H T together with the diagonal, which, in this case, is the 20° VP. A line from the 25° VP to the V V P will be the trace of the plane from the SP to the PP. Using this trace between 25° VP and the V V P as a diameter, make a semicircle. A line from SP P perpendicular to the 25° plane trace will intersect the semicircle. This point of intersection is the rabattement of the SP. T h e protractor, the ZO, the horizontal line, and the diagonal VP are then drawn, giving complete information in which to draw the image of receding squares in that plane. This process repeated for the 65° plane trace will complete the trihedron construction for objects in that inclination. T h e scale of heights already established in the original space diagram may be used throughout the drawing to transfer the proper scale to the image of any other vertical.

18

PLATE IX. REPEATED VERTICAL MEASUREMENT 19

SUN'S

SHADOWS

Plate X , Figure 1, sho'vs the use of protractor rabattement in determining the sun's shadows. It is assumed that the invisible shadow or portion of space from which light is excluded is the umbra. T h u s the umbra of a point is a line, and the umbra of a line is a plane. Since the sun's rays are parallel, and since these rays may be treated as any other system of parallel lines, they will be referred to the umbra of the station point and the umbra of the ZO. T h e direction of the umbra plane is plotted with the aid of the protractor at S P H ( 7 0 ° from the V T ) . T h e intersection at the H T is the V P for the direction of the sun's shadows. T h e vanishing point for shadows on all horizontal surfaces must be on the horizontal trace and must also be within the umbra plane. T h e vanishing point of the sun's shadows ( V P S S ) must, therefore, be at the intersection of the umbra plane and the H T . Since the umbra plane includes the ZO, it must also include the V V P . A line, therefore, between the V P S S and the V V P will be the trace of the umbra plane. On this trace between V P S S and the V V P , rabatte the S P and locate the protractor. In this case, the altitude of the sun is assumed to be 2 0 ° . Plot this angle, and the intersection with the umbra trace will be the vanishing point of the sun's rays ( V P S R ) . T h e intersection of lines to the V P S S and lines to V P S R will furnish the direction and length of shadows. Plate X , Figure 2, is another example for finding the vanishing points for the sun's shadow and the sun's rays by protractor rabattement, but the sun is assumed to be in front of the station point, not behind it as in Plate X , Figure 1.

20

PLATE X. SUN'S SHADOWS

21

REFLECTIONS

When a ray of light falls on a bounding surface, like a mirror or smooth water, part of it is turned back or reflected and part of it is scattered. In perspective, we are concerned with the part that is reflected. T h e law is: T h e angle of incidence equals the angle of reflection. T h u s in Plate X I , Figure 1, a pair of cubes is supposed to stand on a mirror. T h e angle of incidence of the verticals, being perpendicular to the mirror, is 9 0 ° . T h e reflection of a vertical will, therefore, be vertical and each vertical is the same length as its reflected image, and vice versa. T h e same vanishing points are used to find the reflected image that were used to find the image of the object that cast the reflection. This, in general, is the system to use in simple reflections in water with, of course, due allowance made for displacement and for the roughened surface of the water itself. T h e reflection of objects which are affected by the elements, such as the direction of smoke and waving flags in the wind, is subject to the same method as the images of any other set of parallel lines and may be found by protractor rabattement. T h e example chosen in Plate X I , Figure 2, is that of tide direction in water. It is assumed that the reflections of buoys leaning by the force of tide are to be drawn. Since the angle of incidence and the angle of reflection are in the same plane, the vanishing point of the buoys and their reflections may be treated in the same manner as the umbra plane for finding the sun's shadows. T h e direction of the tide is plotted by the aid of the protractor at the S P H . T h e intersection at the H T is the V P for the direction of the tide. A line from this V P to the V V P will be the trace of the plane of the tide direction. Rabatte the station point and protractor to SP V . Lay out the angle of incidence and the angle of reflection, which, in this case, is supposed to be 2 0 ° . With their vanishing points, the images of the buoys and their reflections are obtained. T h e length of the reflection related to the length of the buoy is found by the intersection of the images of verticals.

22

PLATE XI.

REFLECTIONS 23

PERSPECTIVE DRAWING TABLE

The question naturally arises as to whether or not the vertical vanishing point increases the labor of making perspectives. The answer is yes because it takes more time to make accurate drawings than it does to make inaccurate drawings. Perspectives that call for the horizon on or near the horizontal center of the picture should be made with parallel verticals under the same system that has been practiced for generations. But perspectives that call for the horizon near the top or bottom of the picture, or completely off the picture, should be made with converging verticals. The vanishing point for converging verticals is almost invariably off the drawing board and all off-the-board vanishing points are troublesome. A great deal of ingenuity and mechanical talent has been exercised in simplifying the labor of using long straight edges. The most useful tools are cardboard arcs or light wooden battens temporarily nailed to the board to furnish a guide for a straight edge. Centrolineads in varying sizes are all indispensable in making good perspectives. T h e fact, however, that the vertical vanishing point is always on the vertical center of the drawing makes it possible to fix a permanent guide at the edge of a drawing table. If the guide is made adjustable, it may be used for any perspective (Plate X I I ) . Note that the center of the guide is fixed, and that there are thumbscrews right and left to adjust the guide to the proper arc. In the design of the thumbscrew, be sure that some provision is made to take up the torque caused by the swing of the guide away from the axis of the nut. This may be accomplished either by putting the nut on a swivel or by allowing contact at the guide to slide on a specially prepared groove. Whatever method is used, be sure, in the preparation of such a guide, that it is strong enough to withstand the swinging hips of impetuous visitors. Also arrange some way of locking or concealing the thumbscrews. It is surprising how the sight of a thumbscrew will tempt idle hands into maladjustment. The drawing shows a set-up for a looking-down perspective, but the guide may be used as easily for perspectives looking up by working on the opposite side of the board. In general, a perspective table should be a large one, at least three feet by six feet, placed so as to allow plenty of room around the table. Cramped space encourages poor draftsmanship. The mistake commonly made by draftsmen is to put all the vanishing points on a small board. The result is a series of painful impossibilities. The practice of making distorted perspectives is so prevalent that the method has become an accepted convention and is a great detriment to the progress of architectural presentation. The analysis of drawings of two competitions recently published proves that the drawings were made with a principal distance of less than half the width of the drawing and an angle of vision between 75° and 110°. It is also 24

P L A T E XII.

PERSPECTIVE TABLE

regrettable to note in passing that the few competitors w h o had gone to the trouble of m a k i n g legible perspectives were placed a m o n g the honorable mentions, not in the prize-winning class. T o m a k e first-class legible perspectives, a draftsman should: 1. E q u i p himself with a large table and all the tools possible for off-theboard vanishing points. 2. Select a point of sight using the Aldrich Viewer. a. Principal Distance three or more times the width of the picture. b. Angle of Vision no more than 30°. 3. Make a careful miniature of the results of his decisions and locate all the vanishing points, including those of the sun's shadows and rays. 4. Transpose f r o m the miniature to the d r a w i n g all the principal points and lines. 5. D o not hurry. Concentrate

on good draftsmanship. 25

P L A T E XIII.

P E R S P E C T I V E W I T H L O W HORIZON

Note the reflection of verticals in the water 26

P L A T E XIV.

PERSPECTIVE W I T H H O R I Z O N N E A R T H E T O P OF T H E PICTURE

Note the miniature construction at the bottom of the drawing. Note also the vertical scale 27

PLATE XV.

PERSPECTIVE W I T H HORIZON NEAR T H E BOTTOM OF T H E PICTURE

Note the use of a plan for perspective construction 28

PLATE XVI.

HORIZON OFF T H E

PICTURE

Note the m i n i a t u r e in the upper right-hand corner s h o w i n g perspective table w i t h p e r m a n e n t batten for verticals a n d a centrolinead for the left-hand vanishing point 29

mi *fν*.*,

R-oboVOeo CflroM-

4

P L A T E XVII.

HORIZON OFF T H E PICTURE

Note the vertical scale shown as cubes outside the picture circle 30

III. P H O T O G R A P H I C Photography—a

PERSPECTIVE

substantiation oj the geometric principles

described.

OBSERVATIONS on L i n e a r Perspective would not be complete without a comparison of the results obtained by geometric construction and the perspective made evident by photography. connection

with

this work,

T h e comparison is all the more important in

since photography

substantiates

the

foregoing

geometric principles and the fundamental perspective results are the

same,

except that photography shows existing objects and geometric perspective may serve to show those that are to exist in the future. T h e methods of arriving at the results are, of course, different in that the picture plane in geometric perspective is between the station point and objects in space. T h e picture of the objects is, therefore, as the objects appear to the eye in a positive view. In the perspective obtained by photography, the sensitized plate which represents the picture plane is behind the station point ( o r lens of the c a m e r a ) , so that the perspective, which is transmitted to the picture plane, is negative.

It is then made positive by printing.

The

fundamental

physical principles which produce these results are similar in that the focal length of the camera corresponds to the principal distance of geometry

(the

distance from the station point to the perpendicular center of the picture p l a n e ) . T h e width of the angle of the lens in a camera has its counterpart in the angular diameter of the cone of vision, which has been previously referred to as the angle Beta.

B o t h processes are capable of producing accurate perspec-

tives, and both may have their facilities extended so far beyond c o m m o n sense as to create perspectives incapable of furnishing comfortable h u m a n legibility although correct in construction. As a part of legibility, it is evident in geometric perspectives that when the principal distance is established excessively near the picture plane, the eye will attempt to place itself excessively near the picture. A converse tendency to pull away from the picture is experienced when the principal distance is excessively long. Although the eye adjusts itself more readily to a long distance than to a short one, a reasonable average for a comfortable location of the eye must be established, depending on the size of the picture and the use to which it is to be put. A vital part of establishing a comfortable view point in addition to an average principal distance is the limitation of the angle Beta. A limit of angle

31

P L A T E XVIII.

CAMDEN, MAINE. DRAWING MADE FROM PHOTOGRAPH

Only center area of photograph was used has been assumed in the past as wide as 60°, but it will be found by trial that such an angle is greater than eye comfort demands, and it is here assumed that while the angle may be as much less than 30° as one chooses, it must never be more. In geometric perspective, the principal distance and the angle Beta are determined according to the will of the delineator. His taste and judgment control the geometric processes at his disposal. In photography, however, the principal distance and the angle Beta are, within certain limitations, established by the construction of the camera itself. The demand for the shortest possible focal length and the greatest possible field has given the average camera an angle Beta of more than 60°, or a focal length about equal to the width of the negative. From the size of a given snapshot, therefore, one may roughly deduce the relative focal length of the camera used. T h u s to be in a correct viewing position for a 3 " χ 2 " picture, 32

one should put the eye at the absurdly close distance of 3 inches. T h e habit of making optical allowances for such pictures, and the fact that the prints are invariably so small that the eye, as it views them, is not forced to excessive adjustments,

put these photographs

in a class by themselves as

valuable

pictorial records. W h e n pictures are taken out of this class and enlarged for illustration or hung on the wall, great care should be exercised to check both the angle Beta and the principal distance. If the subject of a small photograph is in itself interesting and is made up mainly of curved objects, the perspective faults, although there, are so to speak latent and not noticeable until enlarged. Also, if the objects are at a distance, the extremely wide angle of the camera lens is barely detectable, but if the subjects are rectilinear in form and show objects of three readable dimensions, such as the interior or exterior views of buildings, the geometric perspective construction

becomes immediately evident.

particularly noticeable when the camera has been tilted up or down.

It is The

horizon, in consequence, is either above or below the center of the photograph and, as a natural result, the images of verticals (all except one that occurs in the exact center) are not parallel to the vertical edges of the picture. N o fault can be found with the mechanical accuracy of the perspective, for it is as near perfect as possible within the limitations of the angle Beta of that particular camera and its principal distance. Cutting.the edges of the picture to reduce the angle Beta will not increase the length of the principal distance, and the perspective construction made evident by images of rectilinear lines is a miniature in every respect. All the images of parallel lines converge in miniature to their respective vanishing points close to the picture itself—so close, in fact, that the eye, without knowing the cause, will attempt from the geometric base evident in the perspective to build a minute principal distance. Consequently, the eye is incapable of putting itself as near the photograph as the focal length of the camera demands, and fails in its attempt to bring its normal

reading

distance into accord with the principal distance of the picture. Many such photographs, because of a right-line subject, have been regarded as photographically distorted in perspective when in fact the error was common to all other pictures from the same camera. T w o major operations are necessary to correct the picture: first, substantially to reduce the angle Beta by cutting the field of the negative; and second, to enlarge that small remaining part of the negative to reconcile as far as possible the short focal length of the camera to the comfortable reading distance of the eye.

33

T h e same care and tact which the draftsman must use in m a k i n g geometric perspectives should be exercised in the enlargement of photographs for painting or drawing. A n enlargement to 30 inches square from a 5 " χ 5 " photograph is sure to he a failure because the principal distance of a 5 " χ 5 " photograph is probably no more than 5 inches. W h e n , therefore, the perspective is enlarged six times from the photograph, it is not reasonable to expect the principal distance thus increased to no more than 30 inches to furnish a satisfactory aspect from the customary picture-viewing distance of five or more feet. W h e n enlarging from a photograph, make sure, therefore, that only a very limited part of the center area of the negative is used, and if it is to be hung as a picture, make sure also that the enlargement is truly large. The above deductions are based on the assumption that the camera used is one which has the lens along the line of the perpendicular at the center of the plate. W h e n photography came into use, it introduced a sound logical linear perspective which should have proved the erroneousness of the accepted geometric theory. Instead, the theoretical perspective premise remained unchallenged, and the photograph itself was distorted to the accommodation of conventional practice. Since the time of Dürer, linear perspective principles have been based on the assumption that the images of all verticals in a picture must be vertical, parallel to each other and parallel to the vertical edges of the picture. Other possibilities had been set aside as distortions and when, therefore, the first cameras produced pictures that plainly indicated that the images of verticals were not always parallel to the vertical edges, so strong was the traditional perspective practice that this feature was regarded as a fault of photography which must be corrected. In order to do so, the plate holder was cunningly equipped with a hinge so that if the line of sight were directed upward or downward, the plate holder and the plate could be adjusted to maintain a position vertical to the level ground. Thus, the images of verticals are recorded on the plate as parallel to each other (Plate X I X ) . W i t h this device, called a "swing back," the images of verticals which the camera was vainly trying to record correctly were, as the photographer's expression goes, "corrected" to a distorted perspective. In spite of the difficulty of focusing a plate which is not perpendicular to the line of sight, the practice persists today. Many examples of "corrected" verticals are shown in architects' offices and in magazines. This fault, coupled with the popular use of a wide-angle

34

PLATE XIX. CORRECT AND INCORRECT PERSPECTIVE COMPARED TO CORRECT AND INCORRECT PHOTOGRAPHY

35

PLATE XX.

MEMORIAL HALL, CAMBRIDGE. FROM PHOTOGRAPH

DRAWING MADE

Only center area of photograph was used 36

lens, renders the reproductions, which in other respects are marvels of technique and photographic skill, absurd caricatures of architectural subjects. T h e perspective furnished by a photograph when taken by a camera without tricks, such as swing back or wide-angle lens, is consistent with geometric perspective and is absolutely accurate. Its only fault is that the camera has been forced into showing more than it ought and, therefore, more than the arrested eye will comfortably encompass. In consequence, it is suggested that, unless the picture is taken by a camera with a lens of unusually long focal length, the print be limited to the center of the picture, about one-half its original area. Also, the focal length of the reproduction of photographs should never be less than 6 inches, preferably more than 12 inches. T h e emphasis that has been made throughout these chapters on the vertical vanishing point must not be taken to imply that the images of verticals in all perspectives should converge. O n the contrary, man's point of view of space and his pictures of space relations are so frequently level that there is no occasion to change the perspective balance. T h e old masters, with few exceptions, held to a level concept of space relations, and the perspective convention developed in the past is good today when the line of sight is level. But, in the present age, with the habit of looking up at fabulously tall buildings and the habit of looking down at all nature from the sky, man's abstract idea of verticality has changed. W h i l e he has not discarded the old perspective point of view, he has unconsciously developed an important revision. H e now accepts without notice the converging verticals both in photographic periodicals and in moving pictures.

Indeed, the very presence of a

constantly changing perspective construction adds dramatic reality to his visual reactions. T h e art of drawing, however, has been complacently static, and, in consequence, has not kept up with photography as a means of recording visual truths. T h e hope is, therefore, that this booklet will stimulate draftsmen and painters to cultivate an accurate and versatile linear-perspective convention.

37