Arts secundary

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ARTS AND CRAFTS C
SECONDARY
Eugenio Bargueño Gómez
Mercedes Sánchez Zarco
Francisco Esquinas Romera
Begoña Sainz Fernández
Pedagogical advisor:
Paula Quílez Pons
MADRID · BUENOS AIRES · CARACAS · GUATEMALA · LISBON · MEXICO
NEW YORK · PANAMA · SAN JUAN · BOGOTA · SÃO PAULO · AUCKLAND · HAMBURG
LONDON · MILAN · MONTREAL · NEW DELHI · PARIS · SAN FRANCISCO · SYDNEY
SINGAPORE · SANT LOUIS · TOKYO · TORONTO
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HOW
TO USE
THIS BOOK
1
WARM-UP ACTIVITY
Expressing
emotions
Visual language is a medium to convey
messages using images. To create these
images, we analyse reality and convert it
into a graphic representation.
In this unit, you will learn how to express
your emotions through lines that form
shapes, textures that make sensory surfaces, light that gives a sense of spatiality
and colours that complete the composition. You will have the chance to experiment with different techniques and
compositions that will help you convey
your message faithfully.
Unit introduction
Before you begin, you will explore
the key concepts of the unit through
a warm-up activity. Go to the Online
Learning Center (OLC) to download
the worksheets for this section.
Before you begin, download and print
the worksheet ‘Expressing emotions’
from the OLC. Through this warm-up
activity, you will explore the unit’s key
concepts.
Also download this unit’s worksheet
and do the proposed activities as you
work through the topic. This will help
consolidate your knowledge and your
can-do abilities.
An image and a text introduce the contents
in all units
TOM WESSELMANN, BATHTUB COLLAGE NO. 1, 1962.
Content development
WORKSHEET
Theoretical contents are presented at an easy reading level by always offering
images, illustrations, suggestions and examples that can help you understand the
concepts. We want to teach you how ‘to look’ and how ‘to express yourself’ better
in the world of images.
6 Typography
A. Gradation
B. Cross-hatching
C. Shading with dust
You can produce value gradations with pencils.
The lines are darker if you press down and lighter
if you touch the paper lightly with the pencil.
You can create textured surfaces by drawing
crossing lines, and you can create fine drawings by combining different types of hatching
and shading. In addition to that, by using an
eraser to create highlights and shading other
areas, you can create a textured finish.
The dust produced when you use a pencil
sharpener can be used to shade a surface. Use
a tortillon or your finger to rub certain areas to
make them lighter. You can also use an eraser
to draw white lines or areas on the surface and
create freehand shapes.
To do shading or hatching, place the pencil at an
angle and move it in a zigzag motion. If you rub
hard, the shade is darker than if you rub gently.
Complementary activities distributed
throughout the different sections of
each unit. Download the worksheets
from the Online Learning Center.
Typography is the study and classification of different families of letters and symbols. A typeface
is the complete set of characters in one design,
body or style of letters.
Width and thickness
Typefaces are classified as thin, light, medium
and bold, depending on the thickness of their
lines. If a typeface is very heavy, its counters fill
up and disappear, but if it is very light it might
be difficult to distinguish from the background.
Designers tend to use medium thickness typefaces for long texts and very thick or thin ones
just occasionally.
(small letters). Texts written in upper case take up
more space and are slower to read, whereas
lower-case letters fill the text with signs that make
it easier to read.
Cap height
Chp
Ascender
X-height
Descender
6.2 Typographic families
Proportion between horizontal and vertical axes
They are classified as round (when the axes are
equal), narrow (when the horizontal is shorter
than the vertical) and wide (when the horizontal is
longer than the vertical). Narrow typefaces can be
used when there is a lot of text to fit in the space
and when you want to use several narrow columns.
D. Water-soluble pencils
If the pencils are water-soluble, you can blend
the lines you have drawn.
6.1 Characteristics of letters
Traditionally, the construction of typefaces was
made up of parts called arms, legs, tails, bars, etc.
The structure of letters is always the same, whatever the typeface. So, for example, a capital T is
always made up of a stem and a horizontal arm.
Shoulder
Ascender
Mean line
Arm
Ab Ab Ab
Oblique vertical axis
These are letters with serifs. A serif is a small line
added to a stroke to link one letter to the next
and they make it easier to read long texts.
X-height
There is a series of rules that must
be followed to achieve narrative
continuity, such as respecting the
visual axis of each frame and maintaining the same direction when an
object moves on screen. If we don't
do this, it is confusing for the viewer.
A storyboard is a graphic representation of how a video will unfold, shot
by shot.
You can use a storyboard to narrate
a story using simple illustrations in
a sequence. They are also known
as continuity sketches. Walt Disney
Studios started using them in the
production of their animated films
around 1927.
Director Chris Wedge studying the
storyboard of Ice Age, 2002.
Storyboards are made up of panels or frames which contain drawings of
the most important moments in the action. The level of representation
varies from very rudimentary to very detailed. They may be drawn in
pencil or ink, and in colour or black and white.
Drawings of stick figures
with indications of movement.
You can add cuts to move the story
forward and you can combine different types of shot to make the scene
richer. For example, you could move
from a general shot to a close-up of
the protagonist.
Stylised figures
with indications of movement.
Types of representation.
Technical scripts are made up of scenes divided into sequences and
shots. They represent images using graphic resources (arrows) to show
movements of the characters or the camera. It is also the basis for representing precise technical instructions such as the frame, position (angle
and point of view) and camera movement (travelling, crane, panning, etc.),
decoration, sound, playback, lighting and special effects.
Link
Base line
Descender counter
Tail
Stem
Loop
96 / Unit 3 / Designing
138 /ideas
Unit 4 / Telling stories
This is the size of the letter and it is measured in
typographic points. We also differentiate between
upper-case letters (capitals) and lower-case letters
TITLE
Sequence
Scene:
Frame:
Sound:
(dialogue, music, sound effects)
Notes:
Sample storyboard.
A range of different camera movements are also used as expressive resources.
For example, a panorama or a close-up can add information without using
dialogues, and rapid cuts between objects and characters can create suspense.
Pan
left
Travel
in
Travel
out
CI
NE
M
A
Pan
right Tilt
up
Tilt
down
Body
Leg
/ Arts and Crafts C / 13
This is the first typeface used in European printing and it imitates the manuscripts monks used to
produce with wide-nib quills.
These are called italic typefaces. The angle is usually 15 degrees. Italics are mainly used to highlight a
part of the text, rather than for whole blocks of text.
Ear
Counter
A typographic family is a set of typefaces that
have similar features. All the members of a
typographic family have some features in common and some that are their own. For example, we can classify them like this:
3.2 Storyboards
This family developed from Roman typefaces. The
stems are equal but thicker and the serifs are square.
Storyboard panels.
Camera movements.
/ Arts and Crafts C // Arts
97 and Crafts C / 139
C
STRUCTURES
AND FORMS
LOOK AROUND YOU
CHARACTERISTICS
OF FORMS
,
,
and
LET’S DEVELOP A PROJECT
.
Relation between the size of each part of an object and the size of the whole object.
PROPORTION
ICONIC
REPRESENTING
FORMS
between the representation and the real object.
ABSTRACT
recognizable objects.
POLYGONS
Known
LINKAGE
Known
Between
,
OVAL AND OVOID
Knowing its
CONIC SECTIONS
,
and
,
The orthogonal system is based on
are
to the picture plane.
ABSTRACT
POLYGONS
MIND MAP
LINKAGE
AXONOMETRIC
SYSTEM
,
,
PYRAMIDS
CONE
• Design an A3 poster with:
.
,
.
CHARACTERISTICS
OFRelation
FORMS
REDUCTION COEFFICIENT
between the
AND FORMS
.
.
The linear perspective is
CONIC SECTIONS
PROPORTION
THE LINEAR
SYSTEM
TYPES
.
PERSPECTIVE
Unit 3 / Activity 2
You can use any technique that your teacher
recommends. You can find interesting information in these websites:
http://www.miessociety.org/
http://farnsworthhouse.org/
72 / Unit 2 / Drawing projects
/ Arts and Crafts C / 73
ABSTRACT
Class
Date
A CD cover has the following parts:
• Front cover: The front cover usually has the
name, logo or image of the band. It gives the design its identity and represents the musical content. The cover may be made up of pages like a
book to show images of the band or song lyrics.
DEFINED BY POLYGONS
• Back cover: This is usually where you find the
track list and design or recording information.
LINKAGE
TECHNICAL
REPRESENTATION
INTERSECTION
OVAL AND OVOID
OF FORMS
MIND MAP
CONIC SECTIONS
• Spine: This shows the artist’s name and the
album’s title.
Remember this process for creating a product:
1. Initial ideas: Brainstorming and first sketches.
LOOK AROUND YOU
2. Evaluation and selection:
• Main idea.
• Form-function relationship.
STRUCTURES
AND FORMS
• Factors affecting use.
CHARACTERISTICS
OF FORMS
• Intended audience.
• Artistic style.
3. Study of materials and colour:
PROPORTION
REGULAR POLYHEDRONS DEFINED BY
TECHNICAL
REPRESENTATION
OF FORMS
PROJECTIONS
• Final construction solution.
ICONIC
REPRESENTING
FORMS
ORTHOGONAL
SYSTEM
• Technical drawing, packaging.
• Final model.
ABSTRACT
INTERSECTION
114 / Unit 3 / Designing ideas
/ Arts and Crafts C / 115
POLYGONS
LINKAGE
OVAL AND OVOID
CONIC SECTIONS
REGULAR POLYHEDRONS
REDUCTION
COEFFICIENT
AXONOMETRIC
ORTHOGONALPROJECTIONS
SYSTEM
TYPES CAVALIER
PERSPECTIVE
REGULAR POLYHEDRONS
AXONOMETRIC
LINEAR SYSTEM
THE
SYSTEM
CAVALIER
PERSPECTIVE
THE LINEAR
SYSTEM
/
METHODS
Date
Download the self-assessment sheet from
the OLC to assess your ‘can-do abilities’.
Remember that the events in the script should
follow this structure:
• Exposition or introduction of intentions.
• Climax or plot of the story which describes
the conflict.
• Resolution or conclusion of the narrative.
Download the self-assessment sheet from
the OLC to assess your ‘can-do abilities’.
Remember that you can change this order according to the type of montage you want to use and
that you should use the expressive and narrative
resources you have studied in class (types of shots,
narration, camera movements, etc.). Organise your
narrative into scenes and sequences.
Scene:
Shot:
Action:
Scene:
Shot:
Action:
MINDTYPES
MAP
The theory concludes with a Mind map
of the contents of the unit for you to
complete.
/
Class
REDUCTION COEFFICIENT
METHODS
/
Grade
Use the image below as the starting point for a
narration relating to a social topic that interests
you (such as neglect, poverty, the environment
or health). You can use all or part of it, but keep
the 16:9 screen size ratio.
TYPES
PROJECTIONS
Name
A social storyboard
INTERSECTION
REDUCTION COEFFICIENT
METHODS
Unit 4 / Activity 2
DEFINED BY
SYSTEM
/
Grade
Make a graphic design template of a CD cover
for your favourite band or singer. Remember that
graphic designers don’t just make the outer cover:
They also arrange the texts, titles and images as
well as finding appropriate typefaces. They create
coherent compositions that reflect the subject, style
or intention of the content.
Download the self-assessment sheet from
the OLC to assess your ‘can-do abilities’.
ICONIC
Name
Design a CD
PERSPECTIVE
METHODS
REPRESENTING
FORMS
LOOK AROUND YOU
– Three drawings of the project in different representation systems.
. The point these three
.
,
– A description of the house. Why is it
important?
– An introduction to Mies. Why he has
been a decisive figure in the history of
architecture?
REGULAR POLYHEDRONS
The perspective you get when you project
CAVALIER
PERSPECTIVE
The activities in this book allow you to
express yourself individually and as a
group member. We also provide you with
a space where you can freely develop
your creativity.
• Design modifications to the project and
apply them to your drawings.
.
.
PRISM
CYLINDER
Three orthogonal planes which intersect form
planes have in common is
and its axes are
• Look for information about the architect
and his project.
A research project helps you connect
and apply the unit's contents with your
everyday experience.
• Draw sketches of the house in different
representation systems.
,
between two planes forms
between line and plane determines
INTERSECTION
PROJECTIONS
STRUCTURES
OVAL AND
OVOID
.
projection. Projection rays
DEFINED BY
ORTHOGONAL
SYSTEM
Mies van der Rohe is considered one of the
20th century’s greatest architects. One of his
most important works is Farnsworth House.
This project symbolises the fullest expression
of modernist ideas and it also represents the
culmination of modernist aesthetics.
Your task is:
.
.
PLANES
L
DRAWING PROJECTS
PLANES
ON
LOOK AROUND YOU
MIND MAP
TECHNICAL
REPRESENTATION
OF FORMS
ICONIC
Activities
LOOK AROUND YOU
PLANES
G
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N
End of unit
CHARACTERISTICS
OF FORMS
PLANES
MAP
Download the self-assessment sheet from
SELF-ASSESSMENT
the OLC to assess
your ‘can-do abilities’.
Plot:
Scene:
Download the self-assessment sheet from
the OLC to assess your ‘can-do abilities’.
154 / Unit 4 / Telling stories
Action:
Shot:
Scene:
Shot:
/
Action:
/ Arts and Crafts C / 155
/
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Contents
Unit 1
Expressing emotions
5
1. Elements of visual and artistic language
2. Dry techniques
3. Colour
4. Wet techniques
5. The human body
6. Volume
7. Composition
8. Proportion
Mind map. Look around you
Activity 1. Lines box in and give shape to objects
Activity 2. Working with textures
Activity 3. The emotions of colour
Activity 4. Arte Povera
Activity 5. Colour and rhythm
Activity 6. Symmetry and the law of balance
Activity 7. The human body
Activity 8. Appreciating art. Analyse and understand
Unit 2
Drawing projects
Unit 3
Designing ideas
6
12
14
17
19
21
25
30
32
33
35
37
39
41
43
45
47
49
1. Structures of forms
2. Measurement relations. Proportion
3. Representing forms
4. Technical representation of forms
5. The orthogonal system
6. The axonometric system
7. Cavalier perspective
8. The linear system
Mind map. Look around you
Activity 1. Regular polygons
Activity 2. Linkage. Oval and spiral. Conic curves
Activity 3. The orthogonal system:
intersection of planes, folds
Activity 4. The orthogonal system:
front views and tetrahedrons
Activity 5. Isometric perspective
Activity 6. The cavalier perspective
Activity 7. One-point perspective
Activity 8. Two-point perspective
50
51
52
53
58
64
66
68
72
73
75
77
79
81
83
85
87
89
1. What is design?
2. Branches of design
3. History of design
4. Graphic design
5. Signage
6. Typography
7. Editorial design
8. Packaging design
9. Corporate Visual Identity
10. New technologies in design
11. Web design
12. Materials and techniques in design
13. Characteristics and elements of industrial design
14. The design process
15. Projects
16. Building architectural models
Mind map. Look around you
Activity 1. Raising awareness
Activity 2. Design a CD
Activity 3. Packaging
Activity 4. Your school’s corporate image
Activity 5. A dimensioned draft
Activity 6. Draft the objects around you
Activity 7. Design a paperweight
Activity 8. My room
Unit 4
Telling stories
90
91
92
94
95
96
98
99
100
102
103
104
106
107
110
111
112
113
115
117
119
121
123
125
127
129
1. Audiovisual language
2. Photography
3. Sequential images
4. Images in movement
5. Advertising
6. Multimedia communication
Mind map. Look around you
Activity 1. Think visually
Activity 2. A social storyboard
Activity 3. Photography: changing roles
Activity 4. Library: camera planning
Activity 5. Parallel montage
Activity 6. Reading an advert
Activity 7. Animated advertising project (I)
Activity 8. Animated advertising project (II)
130
133
138
141
148
150
152
153
155
157
159
161
163
165
167
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2
Drawing projects
In our everyday life, we are surrounded by
objects with many different shapes. If we try
to describe them, we might talk about their
silhouette, size, colour or texture, and we
often compare them to geometric shapes.
To represent an object, first you have to
look at it as a whole, without going into details. Then, you can break it down into its
main parts, following your particular criteria. Through this process, a different whole
emerges from the original one, because you
are representing a three-dimensional object
on a two-dimensional support.
Before you begin, download and print
the worksheet ‘Drawing projects’ from the
OLC. Through this warm-up activity, you
will explore the unit’s key concepts.
Also download this unit’s worksheet
and do the proposed activities as you
work through the topic. This will help
consolidate your knowledge and your
can-do abilities.
Aerial view of Midtown Manhattan.
www.mheducation.es
1 Structures of forms
An object’s form is its external appearance. Several elements such as contour, silhouette, size, colour
and texture influence in it. In order to understand
its form, you must observe an object's physical
appearence.
50 / Unit 2 / Drawing projects
If you look at Rietveld’s chair, you can see that
it is made up of a structure of lines and planes
that reflect the proportions of the human body.
This is an example of ergonomics (relating and
adapting objects to the human body).
Nature provides us with perfect examples of
efficiency: Birds’ wings have inspired aeroplane
wing design, and the colours of animals are the
result of nature’s survival strategies. Nothing is
superfluous in nature.
We can differentiate between two- and threedimensional forms. Two-dimensional forms are
flat shapes and they are characterised by form,
colour, texture and dimensions.
Three-dimensional forms refer to the third dimension, that is, the outside and inside of the
form, its location within its surroundings and
the point from which it is observed.
Gerrit Rietveld, Red and Blue Chair, 1918.
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2 Measurement relations. Proportion
Proportion is the relation between the size of
each part of an object and the size of the whole
object.
If the dimensions of one figure correspond to the dimensions of another in a ratio, then, the two figures
are proportional. In this case, the ratio between the
squares is equal to A = k · B, where k is the constant
of proportionality between the two figures. The
measurement relations between two figures can be:
equality, similarity and symmetry.
• Equality: Two figures are equal when all of
their sides and angles are the same.
• Similarity: Two figures are similar when their
sides are proportional and their angles are the
same.
• Symmetry: Two figures are symmetric when
they are the same but they are inverted around
an axis, a centre or a plane of symmetry.
B'
F
B
• Extension scale: The drawing is bigger than
the real object, for example: S = 5:1
3
4
E
e
D
5
D'
6
7
A
O
8
B'
A'
C'
C
A
C'
9
B'
B
D'
First, draw a right-angled triangle ABC. Side
BC measures 100 mm and AB can have any
length. Divide side BC into 5 mm parts and join
the divisions to vertex A. Divide side AB into
ten parts and draw parallel lines to the triangle’s
base. If you divide side AB into four equal parts,
you get the reduction scales S = 1:4, S = 1:2 and S = 3:4.
This lets us obtain any scale.
• Reduction scale: The drawing is smaller than
the real object, for example: S = 1:20
2
E'
D
dimension of the representation
real dimensions of an object
• Natural scale: The object has the same dimensions as the drawing: S = 1:1
1
D'
D
C
Scale =
A scale triangle is a useful graphic tool for extending or reducing figures.
A
F'
A
It is possible to represent any object, but drawing a small watch component is not the same
as drawing a building. Scale is the size ratio
between a real object and its representation.
This means the proportion between a drawn
segment and its real measurement.
2/3
C'
A'
C
B
2.2 Scale triangle
2.1 Scales
A'
B 10
11
12
S.1:10
S.2:10 - S.1:5
S.3:10
S.1:4
S.4:10 - S.2:5
P
S.5:10 - S.1:2
S.6:10 = S.3:5
S.7:10
S.3:4
S.8:10 = S.4:5
S.9:10
S.1:1 C
S.11:10
S.12:10 = S.6:5
/ Arts and Crafts C / 51
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3 Representing forms
In order to represent forms it is important to
take into account the following factors:
• Too much information makes representation
more difficult.
When you start a drawing you must have a
clear idea of the object, but as a whole. That
way you can draw the contours on the support
in proportion to reality.
• We need to make a distinction between representing real volume (natural drawing) and
drawing flat shapes.
• Representation can be objective or subjective.
A. Contour and silhouette
Contour is the closed line that surrounds a
representation of form.
The silhouette is the surface that the contour
line encloses.
Abstraction is a term with a very broad meaning.
In art, when we talk about abstraction we mean
that the represented forms, whether flat or solid,
do not represent recognisable objects. In the 20th
century, a great number of abstract art movements
emerged, but there are three main trends:
• Brancusi, in his work, offers an example of the
reduction of natural appearances to radically
simplified forms.
3.1 Iconic
representation of form
The word icon comes from the Greek eikôn
(likeness), so the term iconic indicates likeness
or similarity between the representation and the
real object. Focusing on drawing as a means of
representation, it is useful to follow some guidelines with respect to the principle mentioned
above: Too much information makes it more difficult to represent an object effectively.
3.2 Abstract
representation of form
• The creation of works of art using basic nonfigurative shapes. One of the pioneers in this
field is British artist, Ben Nicholson.
Silhouette.
Contour.
B. Negative and positive space
Whether we are looking at an image or a real
object, our vision needs to be able to blank out
the forms, and focus on the spaces they occupy
and on the empty spaces around them. In this
combination, proportion is a key element (the relationship between the measurements of the real
object and the drawing).
• Finally, this period saw the appearance of free
and spontaneous expression, such as Action
Painting. An example of this procedure is Eyes in
the Heat, by Jackson Pollock.
The two terms seem very similar, but there is
an important difference between them: With
a contour, there is too much information in or
outside it, and this makes it difficult to draw.
With a silhouette, however, the representation
is just a surface without information, making it
much easier to draw.
52 / Unit 2 / Drawing projects
Jackson Pollock, Eyes in the Heat, 1946.
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4 Technical representation of forms
4.1 Regular polygon
with a given radius
B
B
F
E
4
2 3
1
1
O
C
5
D
H
D
I
f
3
4
5
I
3
4
O
5
6
7
8
C
D
E
N 59
f
6
5
4
G
H
M
A
H
General
method to draw
any polygon
from the given
radius of its
P circumscribed
circumference.
2
A
2 3
4
O
C
P
1
5
O
E
3
30°
B
H
F
d
c
I
2
B
A
Octagon.
B
A
B
2. With the compass on A and radius AB, draw an
arc that intersects the perpendicular bisector
at point C (note that point C is the centre of
the regular hexagon with side AB). The centres
of the circumcircles to the polygons will be on
this line.
3. With your compass on C and radius AC, draw
a circumference, getting point P where it intersects the perpendicular bisector.
B
G
D
1. Draw the perpendicular bisector of AB.
Hexagon.
D
Heptagon.
C
D
A
9
8
7
J
C
1
E
I
B
E
e
B
F
A
F
F
F
H
C
4
5
E
G
3
F
Pentagon.
G
11
10
9
8
7
e
G
6
c
4
1
d
B
P ≡12
Square.
1 23 4
c
e
G
d
Heptagon.
A
1
2
O
L a
Pentagon and decagon.
B
3
K
E
A
5
1
A
Equilateral triangle.
D
2
I
O
B
B
A
F
a
1
G
5
6
8
2
A
Square and octagon.
2
1
3
D
A
A
B
g
3
4
G
Equilateral triangle
and hexagon.
H
E
2
General method to draw any polygon with one
known side. Example: regular nonagon (side: AB):
7
G
C
F
O
C
4.2 Regular polygons
with one known side
4. Using Thales’ theorem, divide radius CP into
six equal parts to get points 7, 8, 9, 10, 11 and
12. Each one is the centre of the circumcircle
to regular polygons with 7, 8, 9 or more sides.
5. In our example, our centre is point 9 and its
radius is magnitude A9. Draw out the circumference and, starting at A, transfer onto it the
value of AB with your compass as many times
as the number of sides the polygon must have.
6. Finally, join the points to complete the polygon.
/ Arts and Crafts C / 53
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4.3 Linkage
Types of linkage depending on the elements to link.
Linking two lines, r and s, with the arc of a
circumference while knowing tangency point T
T1
T
r
Linking two given circumferences O and O’
with an external arc of the given radius r
1. From O and O’ draw two arcs with a radius that
is the sum of the given radius r and the sum of
the given circumference’s radius (s + r and t + r).
The point where these arcs cross is the centre
O” of the circumference you are trying to draw.
Join O” with O and O’ and find tangency points
M and N.
1. The process is similar to the previous one. In this
case, the centre is at O and you draw an arc of
radius r – s.
2. Draw the circumference you have been asked
for, with the centre at O” and with radius r.
O
s
2. Draw another arc with centre at O’ and radius r – t.
The point where both arcs cross is O”, which is
the centre of the circumference you have been
asked to draw. Find points M and N by joining
O” to O and O’.
3. Finally, draw the circumference with the centre
at O”.
r1
r
Linking an arc of radius r1
and a line using an arc of
radius r
r 1+
r
O1
r
s+r
O
O
r–s
T
N
O''
O
O"
M
r
Linking an arc of radius
r with a line r’ knowing
the tangency point T
54 / Unit 2 / Drawing projects
Linking two given circumferences O and O’
with an internal arc that has the known radius r
O'
O
r
T
O'
r–t
M
O
N
t+ r
O"
r
M
O''
O1
T
O
O'
T
r
N
O'
N
r
r–s
r–t
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4.4 Oval and ovoid
Drawing an oval knowing its
major axis AB
Drawing an oval knowing its
minor axis CD
Drawing an ovoid knowing its
major axis AB
Drawing an ovoid with a known
minor axis
1. Divide the major axis AB into
three equal parts in order to find
points O1 , O2 and 3 (that coincides with B).
1. Draw the bisector of line CD to find
point O.
1. Divide AB into six equal parts. Drop
a perpendicular to AB through point
2. With the centre at 2, draw the circumference to find points C and D.
1. Get point O by drawing the perpendicular bisector of the known
axis AB.
2. With the centre at O1 and radius
O1O2, draw a circumference. Do
the same thing with the centre at
O2 and a radius O2 O3 . This last circumference cuts the previous one
at points O3 and O4.
3. Draw lines from O3 to O1 and O2,
and from O4 to O1 and O2. The
points where these lines cross the
circumference are tangency points
T1, T2, T3 and T4.
4. Finally, with the centre at O3 join T3
and T4, and with the centre at O4,
join points T1 and T2.
A
2. With the centre at O and radius
OC, draw a circumference that
cuts the bisector and gives points
O1 and O2. Join these points with C
and D as in the illustration.
2. Transfer the length AB from C and
D to find points O3 and O4. Join
these points to point 5.
3. With centre at C and radius CD,
draw an arc until it crosses the lines
previously drawn. Do the same
with the centre at D and a radius
DC to find points T1, T2, T3 and T4.
3. With the centre at O3 and radius
O3C, and with the centre at O4
and radius O4D, draw arcs until
they cross the lines at points T1
and T2.
3. Join points A and B to P to give us
the lines r and s.
4. Finally, with the centre at O1 join
points T1 and T4, and with the centre
at O2, join points T2 and T3 .
4. Finally, with the centre at 5, draw a
circumference that joins points T1
and T2.
5. With P as its centre and radius PM
or PM’, draw the last arc to finish
the ovoid.
O4
C
D
O1
O2
1
2
3
B
A
O1
O
O3
T2
A
O2
B
O4
2
C
O1
O
A
1
T2
O3 C
B
B
T3
T1
T1
A
O4 D
T4
T4
A
4. Draw two arcs with radius AB and
centres at points A and B to get the
points M and M’.
A
B
T3
2. With its centre at O and radius OA,
draw a circumference that intersects the perpendicular bisector at
point P.
D
B
O3
3
4
T1
O2
5
6
B
M
T2
s
P
M’
r
/ Arts and Crafts C / 55
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4.5 Conic sections
We can get conic sections by cutting a cone at different planes.
B. Hyperbola
C. Parabola
A. Ellipse
You get a hyperbola by cutting a
cone with a plane parallel to its axis.
It is a flat, opened curve in which the
difference between the distances
from points on the hyperbola to other
fixed points, called foci, are always
constant.
We draw a parabola by making a
parallel cut to the slant of a cone. It
is a flat, open curved line. It is defined as the geometric place of the
plane’s points that are at the same
distance to a fixed point F, called
focus, and a line d called directrix.
The vertex of the curve is point V.
The symmetry axis passes through F
and is perpendicular to the directrix.
Ellipses are flat, closed curves obtained by cutting a cone with an oblique
plane. Ellipses have two perpendicular axes: the major axis AB and the minor
axis CD. The major axis contains two points called foci and the distance between
them is called the foci distance. The radius is the distance from any point of an
ellipse to one of the foci. The sum of the distances from any point of the ellipse
(point P) to any of the two fixed points F and F’, called foci, is always constant.
Drawing an ellipse with two
known axes
Drawing a hyperbola knowing
its axes and vertex
Drawing an ellipse using a grid
Once you know both axes, draw a
rectangle and divide lines OA and
AE in equal parts (5 parts in this case).
The points where rays C1, C2, C3 and
C4 cross rays D1, D2, D3 and D4, give
you different points of the ellipse.
Knowing that the AB axis is 2a and
that CD is 2b, with the centre at C or
D and a radius OA, we draw an arc
that cuts the major axis at F and F’.
These points are the foci of the curve.
Take any point N of the major axis
and with centre at F and a radius AN,
draw arc 1. With centre at F’ and radius
NB, draw arc 2. These two arcs cross
at point M. Finally, obtain the ellipse
by repeating this with other points of
the major axis and the foci, and join the
obtained points with a continuous line.
C
E
4
3
2
1
A
1
2
3
4
O
B
You start from one of the axes. With
the centre at O and a radius CA,
place foci F and F’. Draw points 1, 2,
3, etc. and the symmetric points 1’,
2’, 3’, etc. Draw arcs with the centre
at F’ and radius 1B, 2B, 3B, etc., that
will cross the arcs drawn from F and
radius 1A, 2A, 3A, and so on. Repeat
the process to find more points of the
hyperbola. Join them with a continuous line. Do the same to obtain the
opposite hyperbola.
M3
C
C
a
2
A
M
F
d
r1
D
1
A
b
N
a
56 / Unit 2 / Drawing projects
O
r'
F'
B
O
F
F'
D
M2
M1
P
r
tv
V
L
r
F
N
M
b
D
A
1 2 3
B
We already know directrix d, the axis
and focus F. The vertex is the midpoint of AF. Drop a perpendicular line
through point 1 on the axis. With the
centre at F and radius A1 = r, cross
the perpendicular line to find point P.
Repeat this several times to obtain
more points on the curve.
N
M
r
b
Drawing a parabola knowing its
directrix, symmetry axis and focus
1
p
p
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4.6 Scientific drawing
B. Drawing symbols: Celtic knots
Celtic symbols date back to the 5th century and
they originally became known because monks
used them to decorate their manuscripts. One of
the most famous of these is the Book of Kells. Celts
used these designs as symbols of protection, to
give courage to warriors before battle and, sometimes, to attract love.
Scientific drawing is a type of graphic expression
that requires a very rigorous method of representation. As such, it is a way to create images that
are used a lot by scientific communities such as
doctors, biologists and archaeologists to represent the bodies, organisms and objects that are
important to their work.
The Celts passed on their knowledge through
engravings. Some of them even survive today
carved in stone, iron or bronze. The geometric
designs encode their beliefs, rituals and magical
powers, but very few people can decipher them.
A. Natural science
One way to demonstrate the importance of this
type of illustration is to look at the field notebooks used by biologists and archaeologists in
their research. They use this essential tool as a
place to record notes and drawings of anything
relevant to the study they are carrying out, and
later, to make their conclusions.
Today, Celtic knots are popular symbols in the designs of tattooists, artisans, jewellers and illustrators.
Copy one of these knots, using what you have
learned about polygons and linkage. Use a pencil,
set squares and compass.
Look at how these examples are drawn. Do your
own drawings of plants (leaves, flowers, etc.). You
can use a magnifying glass to study more closely
what you are going to draw.
Apex
Petal
Pollen
Corolla
Veins
Lamina
Pistil (female)
Midrib
Stamen (male)
Sepals
Margin
Ovary
Ovules
Base
Petiole
Calyx
/ Arts and Crafts C / 57
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5 The orthogonal system
The orthogonal system is based on a cylindrical
orthogonal projection. Projection rays are perpendicular to the picture plane.
The projection on the horizontal plane is the top
view, the one on the frontal plane is the front view
and the profile view is on the profile plane.
B
5.1 Representing planes. Important lines of a plane
We can define a plane by three unaligned
points, by a line and an external point, by two
converging lines or by two parallel lines. The
orthogonal system represents the plane according to the intersection of the plane with its
projection planes.
vα
α
1
r2
r
Principal lines of the plane
A
C
B'
A'
C'
Profile plane
Frontal plane
Horizontal plane
58 / Unit 2 / Drawing projects
Although planes contain an infinite number of
lines, we will define four of them to make it easier
to work with this geometric element (intersections, folds, etc.). These lines are:
r1
vα
r2
2
r
1. Horizontal line of a plane: This is one of the
plane’s lines that is parallel to a horizontal plane.
The top view of line r1 is parallel to the top view
of plane hα . The front view of r2 is parallel to the
ground line.
2. Front line of a plane: This is one of the plane’s
lines that is parallel to the frontal projection.
The top view of line r1 is parallel to the ground
line. The front view of r2 is parallel to the front
view of plane vα .
3. Line with the widest angle to the horizontal
plane: This is one of the plane’s lines that forms
the widest angle possible with the horizontal
plane. The top view of line r1 is perpendicular to
the top view hα of the plane.
4. Line with the widest angle to the front
plane: This is one of the plane’s lines that
forms the widest angle possible with the front
plane. The frontal projection of line r2 is perpendicular to the front view of plane vα .
hα
α
r1
hα
vα
3
r
α
r2
r1
hα
vα
4
α
r2
r
r1
hα
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5.2 Intersection
A. Intersection between planes
B. Intersection between line and plane
If we take a look at these two planes, we can see they are like two sheets of
paper that cross in space. The intersection between them forms a line.
If we look at this figure we can see that the arrow intersects the plane and,
as a result, we get a point.
The intersection between two planes always forms a line. Because this
line belongs to both planes, it must follow the conditions belonging to
both planes. The views of the line should be on the views of the planes.
The intersection between a line and a plane is always a point, except when
they are parallel. To determine the intersection between a line and a plane in
the orthogonal system it is necessary to:
Given planes α and β:
• Draw the views of plane β, which contains line r.
• The intersection of the top views of planes α1 and hβ determines the
top view, Hr, of the line of intersection r.
• Determine line s, which is the intersection line between the two planes.
• The intersection of the front view vα and vβ determines the front view
Vr of line r.
β
We place the front view A2 of an intersection point where the front views
cross r2 and s2 .
Since the views s1 and r1 coincide, we determine view A1 by dropping the
perpendicular line to the ground line from its front view A2 .
vα
vβ
vα
vβ
r2
Vr
A2
s2
r2
r1
V
A1
r
α
A
r
Hr
H
hβ
s1 = r1 = hβ
hα
hα
/ Arts and Crafts C / 59
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5.3 Folds
In the following figure you can see a plane with
a point contained in it that acts as a hinge. The
surface falls onto the horizontal plane. This is
called folding.
A. Folding a point
Point A is a point contained in a plane α. To fold point A onto the
horizontal plane, we follow these steps:
r2
A2
vr
1. Draw the parallel and perpendicular views to the hinge hα
through point A1 .
2. From A1 , we transfer the length A1 A” given by the height
of the front view of point A.
c
c
hα = hinge
3. With the centre on A’ and a radius of A’A”, draw an arc that
crosses through the perpendicular line at the hinge. This
gives us folding point A0.
A1
hα
A0
vr
c
r1
c
A1
1. Choose a point of line A and fold it in the way explained above.
A''
A'
r0
hr
A0
2. Drop the perpendicular α through view A1 which coincides with the hinge.
vα
A2
O
A1
3. With the centre on O and a radius OA2 , draw an arc that
crosses the perpendicular line at point A0 .
4. We join A0 with the vertex of the plane O to get the folded
front view α0.
s1
hα
C. Folding a plane
1. We choose any point of the front view A. These points
always have their top view A1 on the ground line.
s2
A2
To fold a plane α and the line r contained in it:
When we represent a plane, the front view is one of the lines
of the plane that has been folded using the explanation above.
r2
vs
To fold a line contained into one of the planes, it is only necessary to fold two points of the line.
In the orthogonal system, when folding a plane α
onto the horizontal plane, the hinge is the top view
of hα . If using the front plane, it would be vα .
r1
vα
B. Folding a line
2. Join the folded point A0 with the top view of line hr, which
is a double point. We draw folded line r0 .
A''
A'
To fold a point onto the front plane, we follow the same
steps as above but now the hinge is the view of the plane vα .
Folding a plane onto a fixed plane consists of
making them coincide by turning it around the
intersection line. This intersection line is a folding line called a hinge.
60 / Unit 2 / Drawing projects
vα
hα
α0
A0
These geometric solids are formed by a straight
line, called the generatrix, that moves in parallel to itself or around a curve or polygon (such as
in cylinders and prisms), or around a fixed point
(such as in pyramids and cones). They all have
their base on a flat, horizontal projection plane.
• Right pyramid: The line joining
the apex with the centre of the
base is perpendicular to the base
plane.
• Regular pyramid: A right pyramid
where the polygon forming the
base, or directrix, is regular.
The dihedral system representation of a pyramid on
a horizontal plane is given by the following process:
A cone is formed by a straight line or generatrix
that passes through a fixed point called the apex
and its base is a curved line called the directrix.
2. Find the horizontal projection of the apex V,
which joins each corner of the base polygon
(that is, V1 with A1 , B1 , C1 , D1 and E1). This gives
the horizontal representation of the pyramid.
3. In the same way, V2 is joined with A2 , B2 , C2 ,
D2 and E2 to give the vertical representation
of the pyramid.
V
Lateral edge
Height
and axis
As the directrix, or base, is on the horizontal
plane, the circumference is projected onto this
plane in its true form.
Once you have positioned the vertical projection
of the apex, V2, you can join it with the projections
A2 and B2, which are the generatrices that form the
contour of the vertical representation. Either one
of them gives the true size of any of the others.
V
Generatrix
Base or
directrix
V3
V2
α
Dihedral projections
Height
Base or
directrix
Generatrix
If the directrix of this type of solid is a circumference, it is called a circular cone.
V3
V2
O
O
α
.
Types of pyramids
B. Cone
1. Position the base given on the horizontal plane
in its true form. Here we are using a regular
pentagon as the base, so the vertical projection of the pyramid is on the ground line.
A. Pyramids
A pyramid is a geometric solid formed by a straight
line that passes through a fixed point, the apex,
and its base is a polygon called the directrix. The
straight lines that converge at the apex and the
sides of the directrix polygon are called lateral
edges, and any line that joins the apex to any
point on the polygon, except the lateral edges, is
a generatrix.
Dihedral projections
V. M
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5.4 Radial surfaces
D2
D’1
E2 O2
C2 A2
E1
D’1
B2
A1
V1 = O1
D1
C1
A2
A3 E3 O3
B1
B3 D 3
C3
O2 = C2 = D2
B2
D3
O3 = A3 = B3
C1
V1 = O1
A1
C3
B1
D1
/ Arts and Crafts C / 61
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C. Prism
D. Cylinder
A prism is formed by a straight line called the generatrix, which travels in
parallel to itself and whose base is a polygon, or directrix. The positions of
the generatrix at the corners of the polygon are known as the edges.
A cylinder is formed by a straight line called the generatrix which travels
in parallel to itself on a curved base or directrix.
In a right cylinder, the edges are perpendicular to its supporting plane,
that is, the base.
Types of prisms
• Right prism: The edges are perpendicular to the supporting plane, that is,
the plane holding the base.
Dihedral projections
1. The horizontal representations of the two bases, the first supported on
the horizontal plane and the second parallel to the first, appear in their
true form, that is, a circumference.
• Regular prism: A right prism where the base polygon or directrix is regular.
Dihedral projections
2. In the vertical projection of the bases, one base appears on the ground
line and the other parallel to the first at the height of the cylinder.
1. Position the base given on the horizontal plane of projection. The projected prism has a square as its base, so the base should be placed at an
angle to avoid confusion, and in its true form. Its corners are the horizontal
representations of the edges.
2. The vertical projections of the base are positioned on the ground line, one
parallel to it and one at the height of the prism.
3. The edges are projected onto the vertical plane in their true form, as they
are parallel to it.
Upper base
E2
G2
H2
G3
E3
H3
Height
and axis
Height
and axis
A2
B2
D2
C2
B1= G1
O
62 / Unit 2 / Drawing projects
O’
Generatrix
Base or
directrix
A1 = E1
C1 = H1
D1 = F1
B3
A3
C3
J2
C2
F3
O’
Edge or
generatrix
α
F2
J3
G3
H3
Upper
base
Base or
directrix
D3
A2
α
D2
O
O2
B2
E3
E1 = G1
C1 = A 1
O1 = J1
F1 = H 1
B1= D1
O3
F3
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5.5 Regular polyhedrons
C. Octahedron
Polyhedrons are geometric solids delimited by flat polygonal planes. They include some, called regular
polyhedrons, whose faces, edges and angles are all equal. There are five types: tetrahedron, hexahedron
or cube, octahedron, dodecahedron and icosahedron. Here, we will study the dihedral projections of
the first three.
A. Tetrahedron
B. Hexahedron
A tetrahedron supported on the horizontal plane is
represented in the dihedral system as follows:
A regular hexahedron, or cube, is a polyhedron
formed by six square faces. It has eight vertices
and twelve edges. It also has four diagonals that
cross at their midpoints.
You know the length of the edge, so you can draw
the projection of the supporting face in its true
form: an equilateral triangle.
1. Draw perpendicular lines to each side through
the opposite vertex. Where the perpendicular
lines cross each other is point V1.
2. From here, you can easily find the height of the
tetrahedron, h: Draw the perpendicular A1V1
from V1 . Then, draw an arc with its centre at A1
and a radius equal to the side of the triangle
that crosses the perpendicular at point P.
3. Finally, draw length V1P from O2 to complete the
vertical projection of the tetrahedron.
C
In the dihedral system, a hexahedron supported
on the horizontal projection plane is represented
as follows:
1. Construct a square, A1B1C1D1, in the horizontal
projection with sides of equal length to the edges
of the hexahedron.
2. In the vertical projection, A2, B2, C2 and D2 are
positioned on the ground line. The hexahedron’s
height is equal to edge a, so the face opposite
the base is positioned on a horizontal plane at a
height equal to the length of the cube’s sides.
G2
V2
h
H
C2
B2
O2 A2
E
A1
O
Using the dihedral system, we are going to represent an octahedron so that its diagonal is perpendicular to the horizontal projection plane, and we
start by knowing the length of its edges.
The horizontal projection simply involves drawing
a square, A1, B1, C1 and D1. Its diagonals cross at
point M1, which coincides with O1, and determine
the four faces of the upper half of the octahedron.
The other four faces are hidden underneath them.
Vertex M2 is positioned on the ground line at a
height equal to the length of the square’s diagonal.
The remaining projections, points A2, B2, C2 and D2,
are positioned at half of the height A1C1, and the
projection O2 is on the ground line.
M2
E2
F2
C2
M
V
B
H2
A regular octahedron is a polyhedron with eight
faces which are all equilateral triangles. It has six
vertices, twelve edges and three equal diagonals
that cross perpendicular to one another at their
midpoints.
G
F
C2
D2
B2
D1= H1
A2
C
D
A
D2
D1
B
B2
A2
O2
D
α
V1
A
P
α A
A1= E1
α
O
M1
B
O1
C1
A1
C
h
B1
G1= C1
O1
C1
F1= B1
B1
/ Arts and Crafts C / 63
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6 The axonometric system
6.1 System fundamentals
6.3 Drawing flat planes
Z
If we take a look at the corner of any room, we
can see that it is formed by two walls that are
perpendicular to each other and the floor. This
means that three orthogonal planes which intersect and form the three principal axes make
up the room. The point these three planes have
in common is called the origin O, and its axes
are X, Y and Z.
YOZ
plane
XOZ
plane
O
Picture
X
Y
plane (PP)
The best way to draw complex flat planes is to inscribe them in other simpler
ones, such as squares and rectangles. Draw the perspectives of these base
figures and position the important points on them, such as vertices, centres or
the important points of the curves in the figure you want to represent.
Study the graphic process followed here to represent a pentagon in isometric
drawing on the different projection planes (ZOX, XOY and ZOY).
XOY
plane
If we place a plane that cuts these three planes,
we obtain the representation of the three axes
on the new plane. The projection plane is called
the picture plane, and the projection of the
three axes on the picture plane are the axes of
the representation system (OX, OY, OZ). The
triangle that intersects the picture plane with
the trihedron is called its triangle view.
E
D
Z
Z
Z
O
Y
X
E
C
Y
X
3. Position the vertices of the square on the grid and construct it.
4. From vertices A1 , B1 , C1 and D1 , draw parallels to the axis Z, and extend
them to the distance d to get vertices A, B, C and D.
Z
Z
C2 D2 B2 A2
C
X
Y
120°
Isometric.
64 / Unit 2 / Drawing projects
130°
Dimetric.
Y
D
A1
Y
X
110°
Trimetric.
C1
Y
A
X
D1
O
X
X
2. Draw the grid in perspective, drawing parallels to the axes.
O
O
B
1. Place the square on an orthogonal grid.
120°
130°
120°
X Y
B
Let’s look at how to represent a flat figure (a square) in isometric drawing,
given in the dihedral system:
Z
130°
C
A
O
Y
A
B
In this system, we will study three types of projection: isometric, dimetric and
trimetric.
Z
Z
C
O
O
D
B
In cylindrical projections, the projections are parallel to each other. If they
are perpendicular to the picture plane, we obtain a cylindrical, orthogonal
projection. If they are oblique, it is a cylindrical, oblique projection.
O
E
A
D
C
If we place an object within the four planes and we represent all of the points
orthogonal to the picture plane, PP, we obtain the representation or image
on the picture plane. This is called an orthogonal axonometric representation.
120°
D
A
6.2 Types of projections
100°
Z
E
B1
B
Y
O
D1
C1
B1
X
A1
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6.4 Drawing a
circumference
6.5 Drawing solids
To draw solid shapes in isometric perspective, it
is a good idea to start with the most important
information about the 3D object. The information
may be given using the dihedral system in the
form of a plan, front view and elevation.
The representation of a circumference in isometric
perspective is an ellipse. However, in an isometric
drawing, we can replace an isometric ellipse with
an oval inscribed in a rhombus, since the square
that circumscribes a circle becomes a rhombus in
isometric drawing.
The processes for obtaining isometric perspectives
of a circumference located in the planes ZOX, XOY
and ZOY are as follows:
To convert the representation of a solid in the dihedral system to an isometric drawing, you have
to represent the position of the solid in relation to
the projection planes. The isometric axes should
coincide with the coordinates in the dihedral
representation.
1. You always start with a square that circumscribes
the circumference, A, B, C and D. This square becomes the rhombus A’, B’, C’ and D’, by following
the method given before for drawing flat figures.
In the representation on the right, you can follow
the drawing process that has been used to create
an isometric drawing on the basis of a dihedral
representation.
2. Join vertex A’ with points 2 and 3, and join vertex C’ with points 1 and 4. These segments cross
each other at points O and O’ which, along with
A’ and C’, are the centres of the four arcs that
form the oval.
1. Start with the projections in the dihedral
system.
3. These coordinates become the isometric axes.
Transfer the measurements taken from the dihedral projections to the isometric drawing.
4. Finally, draw two arcs with radius lengths O3
and O’1.
4. Draw the lateral edges of the solid to their corresponding heights and complete the drawing.
Following a similar process, draw the perspectives
of the circumference on the ZOX and ZOY planes.
1
2
4
D
Z
Z
1
4
D’
Y
3
C
O A’
O
1
O’
2
3
C’
4
B’
D’
X
Y
2
A
A’ 1
B
2
B’ 4
5
C’
6
D’ 8
Y
9
O A’
O
A’
1
O’
2
3
C’
O’
B’ 4
O
X
Y
3
D’
B’
Z A’
2
4
3
Y
D’
C’
B’
O
C’
X
Y
O
D’
B’
2
3
O
C’
X
C X
7
A
A’
8
9
B
1
4
1
O’
3
Z
Y
B’
D’
C’
4
7
A
B
1
9
C
2
5
8
X
3
6
A’
4
C
2
5
Z
Z
B
Z
2. Draw a system of coordinates to position points
1 to 9 on the base of the solid.
3. Draw two arcs with radius lengths A2 and C’4
respectively.
A
1
3
6
X
7
/ Arts and Crafts C / 65
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7 Cavalier perspective
7.2 Drawing flat shapes
Cavalier perspective is the perspective you get when you project a point,
a flat shape or a solid body in space onto the plane of a picture or drawing
using oblique cylindrical projection.
The process of representing flat shapes in cavalier perspective is the same as
with isometric drawing in that you inscribe the shapes within simple, geometric
shapes such as squares or rectangles. These shapes are drawn in perspective
and the important points of the original figure, usually the vertices, are marked
on them.
This perspective is based on the use of a trihedron with three right angles,
whose lines (X, Y and Z) are taken as axes of reference and measurement in
the system. The axes that express the measurements of height Z, and width
X of a figure keep their true dimensions because the plane ZOX is parallel to
or forms part of the picture plane.
As an example, you will see below the cavalier perspective of an irregular polygon
positioned on the ZOX and XOY planes of the trirectangular trihedron.
Axis Y, however, which is perpendicular to that plane, expresses depth, and
its dimensions change in representation as we apply a reduction coefficient
to make the graphic representation of the object give a real sense of its true
proportions.
Z
Z
7.1 Reduction coefficient
O
When we project the axes onto the picture plane, the Y axis does not stay in
its true form. There is a proportional relationship between the true dimensions
and the dimensions of the drawing when the former are projected. This metric
relationship is known as the reduction coefficient and the person drawing
usually determines it so as to achieve greater clarity or precision, or simply by
aesthetic criteria.
O’
Y’ A’
A
B
O
C
X
B’
X’
C’
66 / Unit 2 / Drawing projects
PP
y
Y
225°
X
Reduction coefficient,
relationship between
Ol and Ol’
F
Z
B
B
A
C
C
2
F
D
D
X
E
D’
E = E’
F’
I
Y
X
315°
O
True size
I’
O
X
1
O
Z
Y
A
Size in
perspective
Z’
135°
Z
Y
Z
Z
Y
45°
The coefficient may be shown graphically or numerically. The most common
values are 1/2, 2/3 and 3/4, but any other fraction smaller than the unit can be
used without creating disproportion in the drawing.
Y
X
X
O
C’
Y
A’
B’
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7.3 Drawing a circumference
7.4 Representing solids in cavalier perspective
The cavalier perspective of a circumference appears in its true
form on the ZOX plane and any plane parallel to it. However,
it becomes an ellipse on the XOY and YOZ planes, and can be
drawn using the eight-point method, which was explained in
the process to draw ellipses.
A cavalier perspective is defined when we set the position of the Y axis, that is, the angle
between the X and Y axes, and the reduction coefficient of the axis.
Below, you can see some solved exercises in perspective, starting with projections of the solid
in the dihedral system.
As you can see, the circumference has been inscribed in a
square and its projection is constructed from there.
R =1/2
Z
Y
135°
Z
32
Z
R = 2/3
X
O
O’
O
O
Y
135°
X
26
42
r
X
O
X
11
X
77
Z
Z
O2
Y
11
Y
55
X
O
Z
Y
O3
X
X
O
X
O1
Y
Y
Y
/ Arts and Crafts C / 67
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8 The linear system
Panofsky gave the best definition for linear perspective: ‘When you look at an object through
a window and trace the lines that define it, you
obtain its linear perspective. Do not forget to
close one eye to get only one point of view and to
stay still while you trace.’ Although these are not
Panofsky’s exact words, this definition shows the
essence of linear perspective.
8.1 System elements
Look carefully at this drawing. See the structure
of this type of perspective. The most difficult concepts become simple if we understand the three
dimensions surrounding the object.
these rays with the picture plane gives us the linear
perspective.
• Principal distance (PD): The distance between the view point and the picture plane.
• Principal point (P): The orthogonal projection
of the view point on the picture plane.
• Ground line (GL): The line of intersection between the ground plane and the picture plane.
• Horizontal plane (HP): The parallel plane to
the ground plane which contains the PV.
HL
F'
O
HP
P
PD
• Ground plane (GP): This is the horizontal
plane where the object stands.
• Picture plane (PP): This is the vertical projection
plane. It is normally placed between the object
and the observer but it can also be behind the
object.
• Observer point (OP): This is the place where
the observer’s feet are standing.
• Visual rays (VR): These are visual lines that start
at the PV, cross the picture plane and join with the
points that define the object. The intersection of
68 / Unit 2 / Drawing projects
• Vanishing points (VP): These points are on
the horizon line. They are the place where
all the parallel lines contained on the same
plane converge (F, F’). The vanishing point of a
straight line, r, can be found by drawing another
line from the PV, parallel to r, until it crosses the
HL. The resulting intersection is the vanishing
point (F) of line r and all of its parallels.
PP
• Object (O): It can be any representable object:
The object in the figure is a cube.
• Point of view (PV): The observer is looking
with only one eye and has only one point of
view. The height from the view point to the
ground plane (the floor) is around 1.70 m. We
should place the view point in front of the object so that the representation can fulfil certain
conditions.
• Horizon line (HL): The intersection between
the horizontal plane and the picture plane. It is
at the same level as the VP. Sometimes it is not
visible.
PV
F
GP
OP
GL
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8.2 Graphical perspective
8.3 Perspective methods
It is important to follow these guidelines to create an accurate perspective:
A. One-point perspective
This type of perspective positions the point of
view so that the picture plane, PP, is parallel
to the object. In other words, most of the segments that make up the object are parallel or
perpendicular to the drawing. There is just one
vanishing point on the horizon line.
F1
• The observer’s point of view should be in a position that lets us see the most important side of
the object.
• The line from the observer to the object should be at a visual angle of 60° or 90°.
• Drawing lines parallel to the object’s edges, between the picture plane and the point of view in
the plan, gives the vanishing points, which are found on the horizon line, LH.
A. Drawing heights
To determine or measure a specific height, h, in
graphical perspective, you have to transfer it to
the picture plane, PP, i.e. position it in its true
form, and then, use parallel lines to transfer it
to the perspective.
(B)
HL
Here, the point of view, PV, is positioned so that
the picture plane, PP, is oblique in relation to the
object, so the segments that make up the object
are oblique to the drawing.
F1
F2
Contained in the geometric plane
Study the figure to see the drawing method.
Contained in the vertical plane
The method for drawing this perspective is
based on positioning it in the PP, that is, in its
true form and, from there, draw its eight points
in perspective.
B
B. Two-point perspective
B. Circumferences in graphical
perspective
F
5=D
h
5
C
6
3=4
P
A = A1 = B1
GL
HL
4
3
7
7=3
(A)
2
45°
1
8
8=2
5
D
B
4
6
3
7
8
1
1=A
7=A
8=6
C
2
45°
GL
B=3
2=4
/ Arts and Crafts C / 69
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D. Prolongation method for two-point perspective
C. Projection method for one-point
perspective
The method we use to determine a figure in
graphical perspective is known as the projection
method. To follow this method, you start with the
plan of the solid projected onto the picture plane.
You will see how simple it is to use this method to
draw a perspective of any figure. All you have to do
is to position the points of distance, D, and the plan
of the solid on the picture plane, PP, as mentioned
above.
V2
LH
This is one of the simplest and quickest methods. It involves determining the most important points on
a solid using two straight lines that cross at each key point and that we represent in perspective.
Let’s look at an example.
1. Start by positioning the figure or solid and
the key perspective elements in the dihedral
system: the principal point, P, and the vanishing
points, F and F’, etc.
4. You draw the perspective of the solid’s plan
by extending lines from each point towards its
respective vanishing point: Points 1, 2 and 3 to
F, and points 3, 4 and 5 to F’.
2. Prolong or extend the lines that contain the segments forming the base of the solid to find where
they intersect with the picture plane PP at points
1, 2, 3, etc. Transfer these lengths to the ground
line, GL, where you will draw the perspective.
5. Complete the representation of the solid by
extending each of its vertices to the heights
that correspond. To find them, follow the same
steps as in previous methods: Transfer it to its
true form, with the measurements given in the
dihedral front view on the PP. By transferring
them to their corresponding vanishing points
you get the dimensions in perspective.
3. Next, position the principal point, P, and the vanishing points, F and F’, on the horizon line, HL.
HL
GL
P
30 °
V2
P2
HL
(V)
P
F'
F
LH
V1
HL
P
D'
h
45 °
70 / Unit 2 / Drawing projects
GL
LT
F
D
1
5
P1
GL
F'
1
2
rip
St
r
pe
3
a
fp
3
Strip of paper
4
h'
2
o
V1
P1 4
5
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8.4 Linear perspective in photography and paintings
It is common for painters to base their work on photographic compositions,
especially if they have a realistic style of expression. This creative technique
is not as modern as it may seem, since similar knowledge was already in use
in the 17th century. Artists already knew of the advantages of the camera
obscura in creating images in perspective. This was a dark camera in which
artists could see a real image projected.
Photographs capture reality as graphical perspective does.
Painters like Jean Vermeer and Giovanni Canaletto often placed drawings
of the same topic together in order to create a spacious scene. The result is
similar to the vision we get if we join several photographs together to simulate
a large space.
Just like photographers, Canaletto used a camera obscura to create many compositions like the ones on this page. Then, he would move the camera to get two
adjacent views, achieving a realism that caught the perspective of the work of art.
Giovanni Antonio Canaletto, St Mark’s Square, 1723.
/ Arts and Crafts C / 71
STRUCTURES
AND FORMS
PROPORTION
REPRESENTING
FORMS
TECHNICAL
REPRESENTATION
OF FORMS
ORTHOGONAL
SYSTEM
AXONOMETRIC
SYSTEM
CAVALIER
PERSPECTIVE
THE LINEAR
SYSTEM
72 / Unit 2 / Drawing projects
DRAWING PROJECTS
LOOK AROUND YOU
CHARACTERISTICS
OF FORMS
, , and .
Relation between the size of each part of an object and the size of the whole object.
ICONIC
between the representation and the real object.
ABSTRACT
recognizable objects.
POLYGONS
Known LINKAGE
Known Between , and .
OVAL AND OVOID
Knowing its .
CONIC SECTIONS
, , .
DEFINED BY
, , , .
etween two planes forms .
b
between line and plane determines .
INTERSECTION
PYRAMIDS
CONE
REGULAR POLYHEDRONS
PRISM
CYLINDER
Three orthogonal planes which intersect form . The point these three
planes have in common is and its axes are .
PROJECTIONS
, , .
The perspective you get when you project .
REDUCTION COEFFICIENT
Relation between the .
The linear perspective is .
TYPES
METHODS
PERSPECTIVE
LET’S DEVELOP A PROJECT
Mies van der Rohe is considered one of the
20th century’s greatest architects. One of his
most important works is Farnsworth House.
This project symbolises the fullest expression
of modernist ideas and it also represents the
culmination of modernist aesthetics.
Your task is:
• Look for information about the architect
and his project.
• Draw sketches of the house in different
representation systems.
The orthogonal system is based on projection. Projection rays
are to the picture plane.
PLANES
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MIND MAP
• Design modifications to the project and
apply them to your drawings.
• Design an A3 poster with:
–– A description of the house. Why is it
important?
–– An introduction to Mies. Why he has
been a decisive figure in the history of
architecture?
–– Three drawings of the project in different representation systems.
You can use any technique that your teacher
recommends. You can find interesting information in these websites:
http://www.miessociety.org/
http://farnsworthhouse.org/
PERSPECTIVE
Download the self-assessment sheet from
the OLC to assess your ‘can-do abilities’.
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Unit 2 / Activity 1
Name
Grade
Class
Date
Regular polygons inside a circumference
Draw the following regular polygons within the given circumferences:
O
O
Right-angled triangle.
O
Square.
Regular pentagon.
Regular polygons with a given side
Draw the following regular polygons with the given side:
A
B
Right-angled triangle.
A
B
Square.
A
B
Regular pentagon.
/ Arts and Crafts C / 73
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Unit 2 / Activity 2
Name
Grade
Class
Date
Linkage. Oval and spiral. Conic curves
Use an external arc with the known radius r to join these two circumferences:
O
r
O
O'
Draw an ellipse with its axes AB and CD:
A
Draw an oval with major axis AB and minor axis CD:
Draw a parabola with its known directrix d, its symmetry axis and its focus:
C
d
O
O
B
F
e
D
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Unit 2 / Activity 3
Name
Grade
Class
Date
The orthogonal system: intersection of planes, folds
Fold point A contained in plane α onto the horizontal plane:
Find the intersection line of the following planes:
vα
β2
α2
β1
α1
hα
Find the intersection point of the given lines with the following plane:
Fold line r contained in plane β onto the front plane:
vβ
r2
r1
α2
α1
hβ
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Unit 2 / Activity 4
Name
Grade
Class
Date
The orthogonal system: front views and tetrahedrons
Draw the front view of an oblique prism with a regular polygonal base that is
85 mm tall. The base FGHIJ is standing on the horizontal plane:
Draw a regular tetrahedron in the orthogonal system that stands on one of
its sides on the horizontal plane. The length of its edges is line AB:
F1
A1
G1
B1
R1
E1
S1
B1
J1
H1
C1
D1
A1
I1
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Unit 2 / Activity 5
Name
Grade
Class
Date
Isometric perspective
30
10 10
40
20
Using the two views of the solid in the figures, draw
the object in isometric perspective. The lengths
are in millimetres. Use a scale of 3/2.
Z
10
20
50
40
50
10
O
Y
Z
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Unit 2 / Activity 6
Name
Grade
Class
Date
The cavalier perspective
Using the three views of the figure, draw the
solid in cavalier perspective. Remember that
the reduction ratio is 2:3. You should double the
measurements on the views.
Z
O
X
Y
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Unit 2 / Activity 7
Name
Grade
One-point perspective
Class
Date
V2 = PP2
Starting with the given data, draw the solid in one-point perspective, using
the projection method. Take the measurements directly from the views and
triple them.
HL
Front view
PP1
HL
PC
Plan
V1
GL
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Unit 2 / Activity 8
Name
Grade
Two-point perspective
Starting with the given data, draw the solid in
two-point perspective, using the prolongation
method. Take the measurements directly from
the views and triple them.
Class
V2
HL
Date
PP2
Front view
Plan
PP1
HL
30°
V1
PP
GL
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