David C. (David Christoph) Lange.

. (page 1 of 5)
Online LibraryDavid C. (David Christoph) LangeShades and shadows → online text (page 1 of 5)
Font size

m

Student Wohk— State College of Washington.

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

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
object.

The information contained herem was compiled with
special attention for its use as a text book on Shades and
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
at the beginning of the book, and lead them by a study of
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

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

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

d" «"

1 3 4 5

Online LibraryDavid C. (David Christoph) LangeShades and shadows → online text (page 1 of 5)