Elementary linear perspective of form and shadow ... Complete in itself ... For draftsmen, artists, and industrial drawing courses in preparatory and high schools, etc. In two parts...

Front matter

f»

V ,

*

Part Ii.— Derivative Methods; With Some Notes On Aerial Perspective

By S. EDWARD WARREN, O.E.,

FORMER PROFESSOR OP DESCRIPTIVE GE03IETRY, ETC., IN TUB RENSSELAER POLTTECHNIC INSTITUTE, ETC., ETC., AND AUTHOR OF A SERIES OF ELEMENTARY AND HIGHER TEXT-BOOKS ON MECHANICAL INDUSTRIAL DRAWING, AND DESCRIP- TIVE GEOMETRY AND ITS APPLICATIONS.

REVISED AND ENLARGED EDITION.

NEW YORK: \. V^^^ohTngT<

53 East Tenth Street. 1891.

Copyright, 1891.

By S. EDWARD WARREN,

Chapter II. — Preliminary Principles and Explanations 12 PART I

primitive methods.

Chapter I. — Defiyiitions and General Principles 1

" II. — Tlie Elements of P'ojection^ 1

" III. — The Construction of the Perspectives of Objects from, their Projections. Examples for Practice 24

" IV. — Pml Projections, and Perspectives made from them 28

Perspectives of Geometrical Solids, Art. (68.) 32

Example 1. — To Find the Perspective of a Vertical Square

Prism 32

" 2. — To Find the Perspective of a Triangular Pyramid. Examples for Practice 35

Chapter Y,— -Removal of Practical Difficulties, arising from the Confusion of

Projections and Perspectives 36

§ I. — First Method. — Translation forward of the Perspective Plane. . 36

Example 3. — To Find the Perspective of a Cube, etc 38

§ 11.— Second 3Iei7iod.—Vse of Three Planes 38

Example 4.— To Find the Perspective of an Obelisk, etc 41

Chapter VI. — Projections and Perspectives of Circles, arid of Bodies having partly

or wholly Curved Boundaries 43

11 CONTENTS.

PAGE

Example 5. — To Find the Perspective of a Circle, lying in

the horizontal plane 43

" 6. — To Find the Perspective of a Cylinder, standing on the horizontal plane 47

** 7. — To Find the Perspective of a Cone, standing

on the horizontal plane 50

" 8. — Do. of a Cone whose axis is parallel to the

ground line . 52

** 9. — Do. of a Cone whose axis is parallel to the

vertical plane only 53

** 10. — Do. of a Cone whose axis is oblique to both

planes of projection 54

" 11.— To Find the Perspective of a Sphere 58

First Method of finding the apparent contour 58 Second Method " " " 61

** 12. — To Find the Perspective of a Concave Cupola

Koof 63

Chapter VII. — Perspectives of Shadows 65

General Principles and Illustrations, Arts. (89-99.) 65

Example 13. — To Find the Perspective of the Shadow of a

Square Abacus on a Square Pillar 67

** 14. — Do. of a Triangular Pyramid upon the Horizontal Plane 69

** 15. — Do. of a Dormer Window upon a Roof 71

16. — Do. of a Convex, Four-sided Dome, Flag-staff, and their Shadows 73

PART II.

DERIVATIVE METHODS.

Chapter I. — General Principles and Illustrations 75

Example 1. — To Find the Vanishing Point of Telegraph

Wires, etc 78

" 2. — To Find the Vanishing Point of a Perpendicular and of a Diagonal 79

" 2a.— To Find the Perspective of a Perpendicular

and a Diagonal 80

Particular Derivative Methods, Arts. (118-121.) 81

CONTENTS. iii

FAGB

Example 3. — To Find the Perspective of a Straight Line, in any position oblique to both planes of projection, etc 82

*' 4. — To Find the Perspective of a Tower and Spire. 84

Practical Remarks, Art. (123.) 86

" 5. — To Find the Perspective of a Cross and Pedestal 89

Chapter II. — Ih'spectivts of Shadows 92

Example 6. — To Find the Vanishing Point of Rays, and of

their Horizontal Projections 92

" 7. — To Find the Perspective of the Shadow of any

Vertical Line upon the Horizontal Plane . . 93

" 7a.— To Find Shadows on Side-walls 95

Chapter III. — ^fiscellaiuoics Problems 96

Example 8, — To Find the Perspective of a Pavement of Squares, whose sides are parallel to the

ground line 96

" 9. — Do. of a Pavement of Hexagons, whose sides make angles of 30° and 90° with the ground

line 98

" 10.— Do. of an Interior 98

** 11. — Do. of the Shadows in an Interior 102

" 12.— Do. of a Cabin 104

" 13. — Do. of the Shadow of a Chimney on a Roof.. 108

Chapter IV. — Pictures, and Aerial PRrspeetive 110 Landscape Outlines 110

Landscape Details.— Trees. Hills. Valleys Ill

Ascent and Descent. Level of the Eye 112

Reflections in Water. Location of the Centre of the Picture 113 Location of the Perspective Plane. Shadows of Trees, and

other Vertical Objects. Time of a Given Aspect 114

Light and Shade 115

Edges. Color 116

Appendix

§ 1. — Description of Plates 117

§ 2. — Generalization of the Horizontal Plane and Horizon 119

§ 3, — Perspectives from Perspective Plans 121

To instruct the young in the principles of geometry is to render them the greatest service. Not only does this science develop the intelligence, but it makes precision habitual, thanks to the exact knowledge which it gives of the dimensions of all sorts of bodies, considered under different aspects. Without knowledge of geometry, the artist, uncertain of his result, works blindly. Led by ignorance, he drags the young confided to his care from error to error. — From Baclielier's ' ' Memoir on the Boyal Free Drawing School of Paris" written a hundred yea/rs ago.

PKEFACE.

For several years, while teaching a comparatively advanced course on Perspective, I purposed to compose an elementary perspective, which should be interesting and popular, without being empirical ; and, on the other hand, perfectly demonstrative, without being too elevated. In other words, I have sought to make my work elementary, not in the sense of merely stating perspective facts without adequate explanation, but in the sense of selecting simple, yet widely and always useful examples, and then fully explaining, in easy order, the few plain principles necessary to the solution of such examples.

An exact knowledge of perspective is indispensable to those who would make exact representations, for industrial purposes, of architectural or mechanical structures as they appear. It is highly useful to those, even, who practise perspective as an ornamental art, in the making of pictures ; inasmuch as it enables them to know scientifically, as well as feel sensibly, whether their drawings are correct or incorrect. It is also interesting to the amateur judges and admirers of pictures, as well as to their makers ; and, finally, it is useless to none who are in any manner engaged with the arts of graphical representation or design.

It is a part of the price of truth, w^hereby we discover its worth, that we must discover the truth concerning propriety of arrangement in any subject, as an indispensable condition for its successful treatment.

'' Projections/' which, usually for industrial purposes, repre-

VI PREFACE.

sent objects as they are, in form and size, naturally precede, in a course of exact study, ^' Perspective," which, usually for pictorial effect, represents objects as they appear. Perspectives, or drawings of objects as they appear, are made, then, from Projections, or drawings of objects as they are; and which, therefore, are competent representatives of those objects.

The study of projections thus properly preceding that of perspective, as its natural foundation, disadvantages will unavoidably arise from attempts to treat of exact perspective, without a formal preliminary treatment of projections. Hence, this work, while complete in itself, is the natural successor, for those who use both, of my ^^ Elementary Projection Drawing/' in which objects are shown in projection only.

It is unfortunate for learners that a subject so simple, useful, and attractive, as Perspective is, when properly treated, should come to be regarded with aversion, merely owing to defects in its treatment ; the chief of which defects is, perhaps, the failure fully to exhibit its foundation in ^^ projections.''^

The present volume is an attempt to expressly present Perspective, as founded on ^^ Projections," and in this respect it differs, more or less noticeably, from numerous elementary works on the subject. - I accordingly hope for such results, in respect to ready and interested understanding of the subject, as the improved treatment of it, which I have endeavored to give, leads me to anticipate.

Only a few, and quite simple^ problems in perspectives of shadows have been inserted. Yet they are believed to be sufficient.

The tried and proved feature, so obviously important in text-books of every grade, and in every branch, of applied mathematics, of examples for practice, additional to those explained in the text, has been incorporated in this as in my other volumes, generally.

The simplest conception, and resulting definition, of the per-

PREFACE. Vll

spective of a point is, that it is where the " visual ray " through the point pierces the plane of the picture ; i.e., the "perspective plane."

The method of construction just indicated, and adopted for Part I., does away with the whole machinery of "vanishing points/' "perpendiculars," "diagonals," etc.; and, accordingly, these, with their advantages, are briefly explained and illustrated in Part II., giving rise to derivative methods of construction, and aiding the reader in understanding the methods usually employed by writers on perspective.

In my larger work on " Higher Perspective," and to some extent, also, in my " Elements of Descriptive Geometry, Shadows, and Perspective," I have endeavored to exhibit more fully a systematic arrangement of all its methods, and interesting peculiarities and details.

In a brief Appendix will be found two plates, indicating such work as a mastery of the principles of this volume will enable one to execute ; also, a few hints which may aid the student in his further practice.

Finally, while the subject, as here treated, has been, and may still be, sometimes found sufficient for the purposes of Engineering, or other Technical Schools, yet, with the greatly increased attention given to preparation for skilled industrial life, this volume is intended to be suited to the wants of Preparatory Scientific Courses in Academies, High Schools, etc.

Newton, Mass., 1891.

NOTE TO TEACHERS.

While no one method of using this volume can be declared to be absolutely the best for all cases, yet it is suggested to teachers, as the result of long experience in the use of this and the other volumes on kindred subjects, that a very efficient method of doing the most and the best work, in a given time, is to have exercises of mingled interrogation, or blackboard demonstration and drawing, from one to two hours in length, according to the grade of the class ; which may most conveniently consist of twelve to twenty working together at once — each at his drawing, except while called to explain a principle, or point of construction, from his drawing, or at the blackboard.

LINEAR PERSPECTIVE.

INTRODUCTION,

Chapter I. Instruments, Materials, And Methods. '

1. Paper. — For elementary practice, thick unruled writing, or tough printing paper, will answer. For nicer work,- German cartoon, or English smooth drawing paper, will be convenient ; and for exact constructions, in lines or tints of Indian Ink, Whatman^s drawing paper will be best.

2. Support of the Paper. — For slight pencil or ink sketches, the paper may lie flat on an atlas, or any firm, but not rigid surface. For larger and exa^t ink drawings, the paper must be well wet, and then fastened round the edges with gum-arabic, or mucilage, to a smooth and accurately square-cornered drawing board. When dry, it will be found to be tightly stretched. Care should be taken to keep paper flat and smooth, when not stretched as just described.

3. Pencils. — For sketching, use a hard pencil, as No. 4, or 5, of Faber's, and make only the faintest lines. For finishing up pencil drawings, use softer and blacker pencils, as InTo. 3 for well defined objects, and Nos. 1 and 2 for shadows, foliage, &c.

In pencilling a drawing which is to be inked, use a pencil sharpened on a fine file, to a thin edge, rather than a round point, since it will thus keep sharp much longer.

4. Rulers. — For drawing on stretched paper, use a T rule, for drawing all lines from right to left ; and a right angled triangle, for drawing lines perpendicular to these. With loose paper^ use a common ruler and right angled triangle.

5. To draw parallels in any oblique position^ by a ruler and tri angle. To draw through p^ for example, a parallel to ah. Place a side of the triangle on ah^ and bring up the ruler ac, as shown in Fig. 1. Hold the ruler fast, and slide down the triangle to the position (2) when pd will be parallel to ah.

6. To draw perpendicidars in ohlique positions. — Slide the triangle, as before, till de passes through jo, then d being a right angle, a line can be drawn through p^ and perpendicular to ah.

7. Compasses. — For drawing ink or pencil circles, the compasses should have movable legs, which may be replaced by a drawing pen, or pencil-holder.

8. In using the compasses^ hold them by the joint, with the thumb and forefinger. Then, in setting off distances on a line, turn them, alternately, on one side and the other of the line, never taking both points at once from the paper, till the operation is finished. This method is most expeditions and accurate. Like wise, in describing a circle, the whole can be accomplished with quick and uninterrupted motion.

9. Irregular Curves. — For drawing other curves than circles, points of which have been previously constructed, use the thin plate of wood with variously curved edges and openings, and called an irregular curve.

10. Indian Ini^. — This color, when of good quality, is of a brownish black, and is prepared in polished or gilded cakes, finegrained, and usually scented with musk or camphor. It is prepared for use, like other water colors, by touching the end to water, an^^

INSTRUMENTS AND MATERIALS. 11

rubbing on an earthen plate or tile. When enough has been ground off, wipe the cake dry to prevent its crumbling.

This ink, when thick, may be applied in a drawing-pen, or brush, so as to make black lines, or surfaces. When diluted with a quantity of water, tints of any degree of lightness may be quickly laid on the paper (stretched) by a rapid use of a goosequill sized, or larger camel's hair brush.

11. Drawings made by students, both for gaining and illustrating a knowledge of principleSf should retain in ink all the auxiliary projections from which the perspectives are made. But in actual practice, and in the plates made by students for pictorial effect, where the perspective alone is intended to occupy the given sheet or plate, these auxiliary figures should be pencilled only, on separate sheets ; which may conveniently be of buff Manila office paper, temporarily attached to the picture sheet, while the perspective figure on the latter is drawn.

By methods quite artificial, and useful in advanced practice, but not obvious to a learner, and therefore increasing the liability to error and hinderance,* these auxiliary constructions can be wholly or partly dispensed with. But the elementary methods of this volume, in using which error is impossible with ordinary care, are recommended, by preference, for common use as sufficient and satisfactory.

\\a. The finished perspectives mentioned in the last article are finished in various ways. Objects of an architectural character being those most frequently represented, they may, especially when of masonry, be finished in free-hand pen work (see Plate I., Fig. 1), or in India-ink, sepia, or in colors, with more or less of landscape accessories. It is thus a great advantage to the perspective draughtsman to have some practical acquaintance with free-hand sketching, and water-color tinting and drawing, f

* Examples of which may be found in my " Higher Perspective "; and in my "Elements of Descriptive Geometry, Shadows, and Perspective."

t See my " Elementary Geometrical Free-Hand Drawing," and '' Drafting In- Btruments and Operations"; also, any practical works on Topographical and Water-color Drawing.

12 LIJfEAE PERSPECTIVE.

CHAPTER n.

PEWJMINARY PRINCIPLES AND EXPLA N^ATIONS.

12. Sitting by a window, you may fix your attention on all that you see through one of its panes — buildings and parts thereof, trees, roads, fields, woods, streams and clouds.

As soon, however, as you give attention, both to the pane and to what you see through it, you will find that, by looking with each eye separately, you will see partly different sights through the same pane. Hence, to see definitely both the pane and what you see through it, you must close one eye.

13. This being done, you might paint upon the glass everything that you see through it, just where you see it, and of the same shade and color. A perfect pictui'e, in every respect, of all seen through the glass, from one point of sight, might thus be made on the pane. Such a picture would be called the perspective of the view seen through the pane.

14. I. Hence ^ perspective \?> a picture which shows one or more objects just as they appear, in respect both to form and color, and as seen from one fixed point of sight. See Fig. A (24a).

15. If seated quite near the window, you will observe that you cannot see all that is to be seen through it, without turning the head ; while each new direction of sight gives you at least a partly new view. Also each new position of the eye gives, evidently, a different view through the same pane.

16. H. Hence any single perspective drawing should embrace no more than one view, that is, no more than can really be seen when looking in one direction from one fixed point of sight.

17. The chief exceptions to this rule are in panoramic and architectural interior scene painting, w^hich, being intended to please large assemblies, are painted from several points of sight, or from one quite remote one.

18. All this being understood, suppose you are in a field, and dewing a distant tree through a framed pane of glass, held at a

axed distance from the eye. As you approach the tree it appears to occupy a larger and larger portion of the glass ; while, as you recede from it, a contrary effect is produced.

PRELIMINARY PRINCIPLES AND EXPLANATIONS. 13

in. Hence the size of the object, in a picture^ depends on its distance from the eye. See Figs. A and B (24«).

19. Again; if your distance from the tree is fixed, the nearei the pane is carried to the tree the more completely will the view of the tree fill it. That is—

IV. The comparative size of an object in the picture, and the whole picture, depends also on the distance of the picture plane from the object. See Fig. A (24a).

20. Further, if two trees, at equal distances, and of difierent sizes, be viewed at once through the same pane, and from the same fixed position, the larger one will cover a larger space on the pane, as seen through it.

V. Hence, other things being the same, the size of an object, in the picture, depends on its actual size.

21. Once more, by moving, together with the pane, from side to side, or up and down, the tree will be seen through diiferent portions of the pane, when seen from the different positions so taken.

YI. That is, the place of an object in a given picture, its size and distance being also given, depends on its direction fiom the observer.

22. VII. From the last four particulars, w^e now conclude that, in order to represent a given object truly, its dimensions^ distance from the picture^ distance from, the eye, and direction must all be known.

In other words, the relative position of the eye, the picture^ and the object^ and the size of the latter^ must be known.

23. Returning now to the picture painted on the window pane, each point of that picture is in a straight line, from the point represented, to the eye. Such a line is called a visual ray.

VIII. Hence the perspective of any point is where the visual ray from that point meets the surface of the picture.

Finally, the following general principles may serve to connect his introductory sketch, which embraces the primary facts of per spective, given by the testimony of the senses, with the more exact treatment of the subject, which succeeds, and in which the principles of perspective, based upon these facts, are demonstrated.

24. Science is a complete body of truth, whose parts are naturally related to each other ; and hence may be expressed by a systematic and connected statement of successive particulars, proceeding in natural order from primary elements to complete results.

Perspective science is such a body of truth, relating to the

LINEAR PERSPECTIVE.

manner of representing objects as they appear. This science is founded on the simple facts of vision already described, and which are learned by observing what and Jioto we see.

24^?. Let MT, Fig. A, represent a maple-tree, E the eye, or point of sight, and PQ the perspective or picture plane. Then, ME and TE being the visual rays (23) from the foot and top of the tree, mt is the perspective (14) of the tree on the plane PQ.

Again, PQ, Fig. B, is at the same distance from the eye as in Fig. A, but is nearer the tree MT. Hence (18) the perspective, mt, both is and appears larger than mt in Fig. A. It is larger, because PQ is nearer the tree. It appears larger, because the eye is nearer the tree, and therefore the visual angle, MET, which determines the apparent size of the object, is larger than in Fig. A.

Finally, in Fig. A, while the picture m't' is larger than mt (19), it does not appear larger to the eye at E, because, the distance of the eye from the tree being unchanged, the size of the visual angle MET is unchanged.

We now proceed to unfold the elements of Perspective from the preceding simple facts of vision, and to apply those elements to practical exercises in perspective drawing.

— M

Fig. B

DEFINITIONS AND GENERAL PKINCIPLES. If

PAET I.

PRIMITIVE METHODS,

CHAPTER I.

DEFINITIONS AND GENERAL PRINCIPLES.

26. The complete perspective of an object, is a picture of it, which, when viewed from a certain point, produces the same image upon tlie eye that the object itself does, when viewed from the same point.

Each point and line of such a picture, must, when suitably placed between the eye and the object, exactly cover and conceal from view the corresponding points and lines of that object. It must also, as truly as art will allow, present,, at each point, the same shade and color that is exhibited by the same object.

26. Hence perspective embraces two branches: the persjoective of /brm, called linear perspective ; and the perspective oi color and gradations of shade^ called atrial perspective.

27. Aerial perspective is an imitative art^ founded- on extensive observation of nature, and on the science of optics.

Linear perspective is either an imitative art, or an art of exact geometrical construction, according as the outlines of pictures of given objects are traced by the eye, or constructed with instruments, according to geometrical principles.

28. In point of fact, linear perspective is practised as a constructive art, chiefly in its application to regular objects. It is practised as an imitative art, mainly in the drawing of irregular, or pic turesque objects, such as trees, animals, hills, streams, and old buildings.

We will next inquire into the natural principles, which lead to the exact construction of the perspectives of objects.

29. The eyes are so related, that in attempting to see, with both of them together, objects at different instances, distinctly and at once, w^e see these objects partly double (12).

Hence in making an exact picture of any object, we suppose it

IC LINEAR PERSPECTIVE.

to be Viewed with one eye, or that the two eyes are reduced to a single seeing point, called the point of sight (14).

30. Objects become visible by means of rays of light, reflected from them to the eye, and called visual rays (23).

31. Rays of light proceed in straight lines ; as is proved by the fact that we can see nothing through an opaque bent tube.

32. The visible boundary of an object is called its apparent contour. The perspective of this contour, is the linear perspective of the object.

33. Any body, having a vertex, and plane sides, is, in a generii' sense, called a Pyramid.

Any curved surface, having a vertex, and therefore containing straight lines drawn through that vertex, is a Cone^ in the general sense of the term.

Hence visual rays, from all points of the apparent contour of an object to the eye, form a pyramid.^ or a cone — whose vertex is the point of sight — according as the object is bounded by straight or by curved lines. This being understood, this pyi-amid and cone are, for the sake of brevity, called indifferently the visual cone.

34. Next, conceive a plane to intersect the visual cone, anywhere between its vertex, that is the eye, and its base, that is the object. This plane will cut from the cone a figure which will exactly conceal from the eye at its vertex, the apparent contour of the original object. That is (25) this figure will be the linear perspective (27) of that contour, and hence of the object (32).

35. In like manner, the intersection of the visual ray (23) from any one point of the given object, with the given plane, is the perspective of the point from which that ray proceeded.

36. The given plane is therefore called the perspective plane ^ and is understood to be vertical, unless the contrary is mentioned.

Illustration. — In Fig. 2, let E be the position of the eye, ABC the object, as a wood or paper triangle, to be represented ; and PQ, the perspective plane. Then AE, BE, and CE, represent visual rays from the corners or vertices of the given triangle. Now let a, 5, and c represent the points in which these visual rays jnerce the perspective plane PQ, then ahc will be the perspective of ABC.

37. It now clearly appears, that, in order to find the perspective of an object, three things must be given ; the object itself, the position of the eye^ and the perspective plane (22). Observe here, also, that, as lines from the visible points of the object to the eye are visual rays, these rays become known as soon as the positions of the

I eye and of the object are given.

DEFINITIONS AND GENERAL PRINCIPLES.

If either of the lines, as a5, of the perspective, M'cre prolonged either way, or both ways, it would be called the indefinite 'perspective of the original line as AB.

38. Any angle, as AEC, formed at the eye by two visual rayt* s

Fig. 2.

is called a visual angle. It now appears that any line, as AC, and its perspective, ac, subtend the same visual angle. The reason, therefore, why an object and its perspective present the same appearance (25) to the eye, is, that they subtend the same visual angle ; for the apparent size of any object depends on the size of the visual angle which includes it. Hence if two equal lines be in parallel positions, but at unequal distances from the eye, the further one will subtend the smaller visual angle and will therefore appear the shorter (18). Also, if a line or a surface be viewed obliquely, instead of directly, it will appear of diminished size, and is said to he foreshortened.

39. The position, E, of the eye, and the form and position of the object ABC remaining fixed, there will be as many different sizes and forms of the perspective, abc, as there may be different distances and positions of the perspective plane, between E and ABC And these various forms and sizes of abc will all be true perspectives of ABC. To understand this completely, it is only necessary to remember, 1°: That all these forms of abc are sections of the same visual pyramid ABC— E (34), and 2*=* : That the definition of a perspective is not, a figure that is as the original object appears; but, only, one that appears as that objett does, when viewed from the sane point (25).

IB

LINEAR PERSPECTIVE.

CHAPTER n.

THE ELEMENTS OF PROJECTIONS.

40. It is evident from a consideration of Fig. 2, that we cannot; practically, find a perspective picture directly according to (35) ^. e. directly from objects themselves. Visual rays are invisible and intangible, and we cannot conveniently substitute for them, threads from every point of an object, as a house, to a fixed point, as the top of a stake, taken to represent the place of the eye, and then find where all the threads pierce a paper plane, set up between the object and the place of the eye.

41. What then can be used in place of the actual object, from which to make its perspective, as truly as if found mechanically, as above described? We employ auxiliary drawings, which show the positions, forms, and dimensions of the original objects, just a^? they really are, and from such drawings, with similar representations of the visual rays, we construct the perspectives, which shovv those objects as they appear,

42. These auxifiary drawings, which show the given object, and

its visual rays, as they really are, in respect to form and relative position, are called projections. To the explanation and construction of projections, we therefore turn, as the next thing in order.

43. lUicstration. — Let HH', Fig. 3, represent a level plane, called the horizontal plane, and VV an upright plane, at right angles to IIH', and hence; called the vertical plane.