m

Student Wohk— State College of Washington.

SHADES AND SHADOWS

BY

DAVID C. LANGE, M.S.

INSTRUCTOR IN ARCHITECTURE IN THE WASHINGTON

STATE COLLEGE

NEW YORK

JOHN WILEY & SONS, Inc.

London: CHAPMAN & HALL, Limited

1921

:_3

Copyright, 1921

BY

DAVID C. LANGE

• PRESS OF «■

eRAUNWORTH & CO.

eOOK MANUFACTURERS

BROOKLYN, N. Y.

PREFACE

It has been the endeavor of the author to give in the

following pages sufficient information to enable one to

cast correctly the Shades and Shadows on any architectural

object.

The information contained herem was compiled with

special attention for its use as a text book on Shades and

Shadows. Architectural students in many colleges receive

their early training under Engineering teachers. An at-

tempt Vv'as therefore made to serve such students, by

assuming the point of view of their engineering training

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at the beginning of the book, and lead them by a study of

Shades and Shadows toward an appreciation of the artistic

architectural point of view, so seldom developed in a strictly

engineering course.

The author wishes to acknowledge his indebtedness to

the Faculty of the Architectural School of the University

of Pennsylvania, where he received his first impressions and

appreciation of Architectm-e, and especially to Professor

Thomas Nolan for his able assistance in a review of the

subject matter contained in the book. Also to those whose

work is shown as illustrations.

4G8462

CONTENTS

Chapter Page

I. Elementary Prinxiples of Descriptive Geometry.

Points, Lines, and Planes 1

Intersections of Solids with Planes 27

Intersection of Solids 34

II. Principles of Shades and Shadows.

General Principles of Shades and Shadows 48

Shadows of Points 58

Shadows of Lines G6

Shadows of Planes 78

Shades and Shadows of Solids 82

Genera! Methods of Finding Shades and Shadows 90

Wash Rendering 130

NOTE

Problems are referred to as plates.

No illustrations accompany the following:

Plates I to XIII Pages 36 to 44

Plates I and II Pages 63 and 64

Plates III, IV, V Pages 75 and 76

Plates XXVII, XXVIII, XXIX Page 135

SHADES AND SHADOWS

CHAPTER I

ELEMENTARY PRINCIPLES OF DESCRIPTIVE GEOMETRY

POINTS, LINES AND PLANES

1. Descriptive Geometry is the art of representing

a body in space upon two planes (called the Horizontal, H,

and the Vertical, V, Coordinates, indefinite in extent and

intersecting at right angles to each other in a line called

the Ground Line), by projecting lines, perpendicular to the

coordinates, from points of intersection of the contiguous

sides of the body and from points of its contour, and thus

solving graphically many geometrical problems involving

three dimensions. (Figs. 1, 2, and 3.)

NOTATION

The'f oUowing notation is used :

H Horizontal Coordinate Plane,

V Vertical Coordinate Plane,

P Profile Coordinate Plane,

Q, P, R, S, etc., any other planes.

HQ, HP, etc.. Horizontal Traces of any other planes,

VQ, VP, etc.. Vertical Traces of any other planes,

a, b, c, d, etc., any points in space,

a", 6", etc., Vertical Projections of any points.

PRINCIPLES OF DESCRIPTIVE GEOMETRY

2nJ Qua Irant

Vertical Coordinate

Horizontal Coordinate

16" \ch

Fig. 3

Ii«. 4

G

Fig. 5

POINTS, LINES AND PLANES 3

a!", h^, etc., Horizontal Projections of any points,

a^, b'', etc., Profile- Projections of any points,

A, B, C, D, etc., any lines in space,

A", B", etc.. Horizontal Projections of any lines,

A'', B' , etc.. Vertical Projections of any lines,

A^', 5", etc., Profile Projections of any lines,

GL, . Ground-Line.

2. The intersection of these Coordinate Planes form

four angles or quadrants, called the First, Second, Third,

and Fourth Angles, or Quadrants. (Fig. 1.)

3. In order to represent the two projections of an object

on the same sheet of paper, the upper portion of the Ver-

tical Plane is revolved backward about the Ground-Line

as an axis, until it coincides with the Horizontal Plane.

(Fig. 2.)

4. All points in the First Quadrant are vertically and

horizontally projected, respectively above and below the

Ground-Line. All points in the Third Quadrant are

horizontally and vertically projected, respectively above

and below the Ground-Line. All points in the Second

Quadrant are projected above the Ground-Line. All

points in the Fourth Quadrant are projected below the

^ Ground-Line. (Figs. 2 and 3.)

5. Two projections of a point are always on one and the

same straight line, perpendicular to the Ground-Line.

(Fig. 3.)

6. Two projections are always necessary definitely to

locate a point or line in space.

7. The distance of a point in space above the Horizontal

Plane is equal to the distance of the Vertical Projection of

the point from the Ground-Line; and the distance of a

point in space, in front of the Vertical Plane, is equal to the

PRINCIPLES OF DESCRIPTIVE GEOMETRY

.vjr

I//

Fig. 6

V .Fig. 7

1 1 II

j 1 ^^ Jrf"

Fig. 9

e\ ,f"

I Jc''

Fig. 10 Fig. 11

Fig. 12 Fig. 13

Fig. 14

POINTS, LINES AND PLANES 5

distance of the Horizontal Projection of the point from the

Ground-Line. (Figs. 2 and 3.)

8. A point situated on either Coordinate Plane is its

own projection on that Plane, and its other projection is in

the Ground-Line. (Fig. 6.)

9. A straight line is determined by the points at its

extremities, and a solid is made up of lines; hence the pro-

jections of a sufficient number of points of a line or solid

determine their entire projections. (Fig. 7.)

10. Lines parallel in space have their projections par-

allel. (Fig. 8.)

11. A straight line perpendicular to either Coordinate

Plane has its projection on that plane as a point, and its

other projection is perpendicular to the Ground-Line.

(Fig. 9.)

12. A line parallel to either plane has its projection

on that plane, parallel to itself, and equal to its true length;

and its other projection is parallel to the Ground-Line.

(Figs. 10 and 11.)

13. A line parallel to both coordinates, or to the Ground-

Line, has both projections parallel to the Ground-Line.

(Fig. 12.)

14. A point in a line has its projections on the pro-

jections of the line. (Fig. 13.)

15. Projecting Planes are planes containing two or

more projecting lines.

16. The Traces of a line or plane are the intersections of

the line or the plane with the coordinates, and the two traces

of the same plane must always meet in the same point in

the Ground-Line, although sometimes at infinity, when

the plane is parallel to a coordinate. The traces of a line

lying in a plane must also be in the traces of that plane.

(Figs. 14 and 15.)

6 PRi\riPLP:s OF descriptive ceometry

POINTS, LINES AND PLANES 7

17. The Profile Plane is a plane perpendicular to the

Ground-Line and hence to the Horizontal and A'ertical

Coordinates, taken for convenience on the left side. Pro-

jections of objects on the Profile Plane are shown by revolv-

ing the Profile Plane about its Vertical Trace into the

Vertical Plane.

18. If a straight line is perpendicular to a plane, its

projectians are respectively perpendicular to the traces of

the plane. For if an assumed plane is perpendicular to

the given plane and a Coordinate Plane, and contains the

straight line, it is perpendicular to their line of intersection;

and any line in this plane is perpendicular to this line of

intersection. But this assumed plane is also a Projecting

Plane of the given line, and the lines of projection must

lie in the traces of this assumed plane. Projections of the

lines, therefore, are respectively perpendicular to the traces

of the plane. (Fig. 16.)

19. To find the traces of a given line.

Let A be the given line. (Figs. 17 and 18.)

The traces of a line are the points in which the line

pierce^ the Coordinate Planes. (Art. 16.)

The projections of these traces must, therefore, lie in

the projections of the line, and one projection oi each trace

^lies in the GL. (Arts. 14 and 8.) The other projection

lies at the intersection of a line perpendicular to the GL

through this point in the GL, and the projection of the line.

(Art. 6.)

To obtain the H Trace of the line, the line is continued

to the H Plane, which is shown by the Vertical Projection

intersecting the GL. This is the V Projection of the H

Trace; and the H Projection is the intersection of a per-

pendicular to the GL at this point with the H Projection

of the line. The V Trace is found in the same way.

8 PHIXCIPLE!^ OF DESCRIPTIVE GEOMETRY

Fig. 22

POINTS, LINES AND PLANES 9

20. To determine the true length of a hne joining two

points in space.

Let A be the given Hne. (Figs. 19 and 20.)

A line is seen in its true length upon that coordinate

to which it is parallel or in which it lies. By revolving

the line, therefore, into or parallel to either coordinate,

the true length of the line in space is determined. This

is done in two ways: (1) By revolving the line parallel

to a Coordinate Plane, by means of revolving the Pro-

jecting Plane parallel to the opposite coordinate. (2) By

i revolving the Projecting Plane about its intersection with

i the same Coordinate Plane, as an axis, into that coordinate.

! 21. To pass a plane through two intersecting or par-

allel lines.

Let A and B be two intersecting lines. (Figs. 21 and

22.)

The traces of a plane must contain the traces of the

lines. The traces of the given lines, therefore, are deter-

1 mined, and like traces connected. These lines are the traces

i of the required plane, and meet in the GL. (Arts. 16 and

19.)

A plane may be passed through any three points by

!^ passing two intersecting lines through the three points.

22. Given one projection of a line or point lying in a

plane, to find the other projection.

' Let a^b'' be the H Projection of the line, and c^ be the

; H Projection of the point, lying in the plane P. (Figs.

I 23 and 24.)

The traces of the line in the given plane containing the

given point must lie in the traces of the plane. The pro-

jections of the traces are, therefore, determined, and from

them the unknown projection. (Arts. 16 and 19.)

23. Given the projections of a point or line lying in a

10

l'inX(II'Li:S OF DESCRIPTIVE (lEOMETRY

POINTS, LINES AND PLANES 11

plane, tc find its j)()8iti()n when the plane is revolved al)()ut

its trace to coincide with either coordinate.

Let ab be the line and c the point, lying in the plane

P, and shown by their projections. (Fig. 25.)

The axis about which the point or points of the line

revolve must lie in the plane into which the point or points

of the line is revolved. The revolving points describe a

circle whose plane is perpendicular to the line of inter-

section of the given plane with the Coordinate Plane. The

intersections of this circle or circles with the coordinate

are the required positions of the point or points in the line.

Through the given point on the plane a plane is passed,

perpendicular to the trace of the given plane, about which

trace the given plane is to be revolved. The trace of the

Auxiliary Plane is perpendicular to the axis of the revolving

plane. On the trace of the Auxiliary Plane a point is laid

ofT at a distance from the axis and trace of the given plane

equal to the hypotenuse of a right triangle, one side of

which is the distance from the projection of the point to

the axis on the revolving plane, and the other side of which is

equal to the distance of the point from the same projection.

24. To find the true size of an angle made by two inter-

secting lines. (Fig. 26.)

Let A and B be the tw^o intersecting lines.

The angle betv/een intersecting lines may be measured

when a plane containing the lines has been revolved to

coincide with one of the Coordinate Planes. A plane is,

therefore, passed through the two intersecting lines and

its traces determined. This plane, with its lines, is then

revolved about either of its traces as an axis, until it coin-

cides with that Coordinate Plane in which the trace lies.

The angle made by the intersecting lines, in the revolved

position, is the required angle. (Art. 23.)

12

PRINCIPLES OF DESCRIPTIVE GEOMETRY

POINTS, LINES AND PLANES 13

25. To find the true size and shape of any plane surface.

Let abed be the plane surface. (Fig. 27.)

This plane surface appears in its true size and shape

^vhen the plane containing it is revolved to coincide with

a Coordinate Plane. The plane, therefore, containing the

given plane surface, is revolved into the coordinate plane

about one of its traces as an axis, and the revolved position

of the plane surface constructed. This is the true size

and shape of the plane surface. (Arts. 23 and 24.)

26. To find the shortest distance from a point to a line,

and to draw the projections of it in its position in space.

Let A be the line and a the point. (Fig. 28.)

The shortest distance from a point to the line is a per-

pendicular from the point to the line. A plane, therefore,

is passed through the line and point. (Art. 2L) This

plane, containing the line and point, is revolved about its

trace into the Coordinate Plane. This determines the

revolved position of the line and point. (Art. 23.) A

perpendicular line is now drawn from the revolved position

of the given point to the revolved position of the given

line. This is the true length. This perpendicular, revolved

back, determines its projections.

(r 27. To find the line of intersection of any two planes.

Case L To find the intersection of two planes inter-

secting within the limits of the drawing.

Let X and P be the intersecting planes. (Figs. 29 to

32.)

The intersection of the Horizontal Traces must be a

point common to both planes, and therefore a point common

to their line of intersection. The Vertical Projection of

this point lies in the GL. Projections of another point

in this line of intersection may be determined in the same

way from the intersection of the Vertical Traces. (Art. 8.)

14 PRINCIPLES OF DESCRIPTIVE GEOMETRY

POINTS, LINES AND PLANES

15

16 PRINCIPLES OF DESCRIPTIVE GEOMETRY

Having given, therefore, the projections of two points

in the hne of intersection, the projections of the Une itself

are easily determined. (Art. 9.)

Case II . To find the intersection of two planes when

the traces do not intersect within the limits of the drawing.

(Figs. 33 to 37.)

A series of Auxiliary Planes are passed parallel to a

Coordinate Plane. These Auxiliary Planes cut from the

given intersecting planes straight lines which are parallel

to the Coordinate Plane, and which intersect in points

common to both the given planes, and lie, therefore, in

their line of intersection.

The following is another proof for the same problem:

The line of intersection between the two planes is com-

mon to each plane, and the traces of the line of intersection

must, therefore, lie in the traces of each plane. (Art. 16.)

Hence the point of intersection of the V Traces of the planes

is the Vertical Trace of the required line of intersection,

with its H Projection in the GL. Another point in the

line of intersection is the intersection of the H Traces of

the planes, and its Vertical Projection is in the GL. Having

then the two projections of the two points in the line of

intersection, the projections of the line of intersection

itself is easily determined. (Art. 6.)

Case III. To find the intersection of two planes when

both intersecting planes are parallel to the GL, or when

one of the intersecting planes contains the GL.

Let P and be the two planes parallel to the GL. (Figs.

36 and 37.)

On the Profile Plane the traces of the intersecting planes

parallel to the GL are determined. A point common to

both traces is a point in the intersection of the two planes.

Since the intersecting planes are perpendicular to the Pro-

POINTS, LINES AND PLANES

17

c)

VP

I

vo

A.

A''

\

^\

HO

Fig. 36

18 PEINCIPLEvS OF DESCRIPTIVE GEOiMETRY

POINTS, LINES AND PLANES 19

tile Plane, so also must the line of intersection be perpen-

dicular. The V and H Projections, therefore, of this line

are the projections of the line of intersection of the two

planes parallel to the GL.

28. To find where a line pierces a plane.

Let P be the given Plane and A the given line. (Figs.

3S to 41.)

The given line must intersect the given plane in the

line in which any Auxiliary Plane containing the given

line intersects the given plane, at a point v>here the given

line crosses the line of intersection of the Auxiliary Plane

and the given plane.

An Auxiliary Plane is, therefore, passed through the

line to intersect the given plane. (Art. 16.)

The Hne of intersection is then determined between

the given plane and the Auxiliary Plane. (Art. 27.)

The required point lies on this line of intersection and

on the given line, or at the intersection of these two lines.

There are several cases, as follows:

Case I. WTien any Auxiliary Plane containing the line

is used. (Fig. 39.)

Case II. When the H or V Projecting Plane is used

as the Auxiliary Plane. (Fig. 38.)

Case III. ^Yhen the line is parallel to the Profile Plane,

necessitating the use of the Profile Plane. (Fig. 40.)

Case IV. WTien the plane is defined by two inter-

secting lines. (Fig. 41.)

29. To find the shortest distance from a point to a plane.

Let P be the given plane and a the given point. (Fig.

42.)

The shortest distance must be measured along a line

from the point perpendicular to the given plane, and the

projections of this line are perpendicular to the traces of

20 I'RINCIPLKS OF DESCRIPTIVE GEOMETRY

POINTS, LINES AND PLANES

21

22 1>K]N(1PLE.S OF DESCRIPTIVE GEOMETRY

the given plane. (Art. 18.) The point in which this per-

pendicuhir j^erces the gi\'cn plane is then determined.

(Art. 28.)

To find, therefore, the length of the perpendicular, a

projecting plane containing the perpendicular line is revolved

about its trace into the coordinate, where its true length

is shown.

30. To pass a plane through a given point and parallel

to a g'ven plane.

Let P be the given plane and a the given point. (Fig.

43.)

The traces of the required plane are parallel to the cor-

responding traces of the given plane, and are fully known

when one point in each trace of the required plane is

determined.

A straight line, therefore, through the given point, and

parallel to either trace of the given plane, is a line in the

required plane, and intersects a Coordinate Plane in a point

in the trace of the required plane. Thus the required

plane has its traces through this point and parallel to the

traces of the given plane. (Art. 10.)

31. To pass a plane through a given point, perpen-

dicular to a given line.

Let a be the given point and A the given line. (Fig.

44.)

The V and H Traces of the required plane are per-

pendicular to the V and H Projections of the given line.

(Art. 18.)

The direction of each of the required traces is, there-

fore, known; and if a straight line is drawn through the

point, and ])aral!el to either of these traces, it is a line of

the reciuired plane. Unless parallel to the GL, it inter-

sects one of th(> i)lanes of projection at a point in the trace

24 PRINCIPLES OF DESCRIPTIVE GEOMETRY

of the required plane. (Art. 16.) Therefore, a trace

through the point thus found, and perpendicular to the

corresponding projection of the given line, is one of the

required traces of the required plane. The other trace

meets it in the GL, and is perpendicular to the other pro-

jection of the line. (Art. 16.)

32. To pass a plane through a given line, parallel to

another given line.

Let A and B be the given lines. (Fig. 45.)

The required plane contains one of the given lines,

and a line intersecting this given line which is parallel to

the second given line.

Through any point, therefore, in the first line, a line

is passed parallel to the second given line. A plane con-

taining these intersecting lines is the required plane. (Art.

21.)

33. To pass a plane through a given point parallel to

two given lines.

Let A and B be the given lines, and a the given point.

(Fig. 46.)

If through the given point lines are passed parallel

respectively to the given lines, and a plane passed through

these intersecting lines, this plane is the required plane.

(Art. 31.)

34. To pass a plane through a given line, perpendicular

to a given plane.

Let A be the given line and P the given plane. (Fig. 47.)

The required plane Q, contains two intersecting lines,

namely, the given line, and an intersecting line B, per-

pendicular to the given plane. (Arts. 18, 21, and 19.)

35. To construct the projections of the shortest line that

can be drawn, terminating in two straight lines not in the

same plane.

POINTS. LINES AND PLANES

25

26

PRINCIPLES OF DESC^'UPTIVE GEOMETRY

is the required angle which can be

seen in its true size by revolving it

and the plane O containing it about

its traces into either coordinate.

INTERSECTIONS OF SOLIDS WITH PLANES 27

Let A and B be two straight lines not in the same plane.

(Fig. 48.)

The shortest distance between two points not in the

same plane is the perpendicular distance between them,

and only one perpendicular can be drawn terminating in

these two lines.

Through one of the given lines a plane is passed parallel

to the second line. (Art. 32.) The second given line is

projected on this plane. (Arts. 12 and 29.) This pro-

jection of the second given line on the Auxiliary Plane,

intersects the first given line.

At their intersection a line E, perpendicular to the

Auxiliary Plane, is dra\\Ti. (Art. 18.) It intersects the

second given line because it is a Projecting Line, and is

the shortest distance between the lines. (Art. 26.)

36. To find the angle made by any two intersecting

planes.

Let P and R be two intersecting planes. (Fig. 49.)

A plane is passed which is perpendicular to the line of

intersection of the two planes. It is perpendicular to both

'of the intersecting planes, and cuts from each a line per-

pendicular to the line of intersection. (Art. 18.) The

I angle made by these lines cut out by the perpendicular

i;^lane is the required angle, and can be seen in its true size

Iby revolving the plane containing it, about its trace, into

jpne of the Coordinates. (Art. 24.)

INTERSECTIONS OF SOLIDS WITH PLANES

36. A solid is a magnitude that has length, breadth, and

|thickness, as a Cylinder, Cone, or Sphere.

A Cylinder is a solid generated by a straight line called

the Generatrix, moving with all its positions parallel along

curved lines called the Directrix, the two curved lines lie

28

PRINCIPLES OF DESCRIPTIVE GEOMETRY

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