www.mheducation.es www.mheducation.es 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 www.mheducation.es 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 www.mheducation.es 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 / www.mheducation.es 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 www.mheducation.es 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. www.mheducation.es 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 www.mheducation.es 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. www.mheducation.es 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 www.mheducation.es 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 www.mheducation.es 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 www.mheducation.es 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 www.mheducation.es 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 www.mheducation.es 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 www.mheducation.es 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 www.mheducation.es 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 www.mheducation.es 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 www.mheducation.es 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 www.mheducation.es 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 www.mheducation.es 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 www.mheducation.es 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 www.mheducation.es 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’ www.mheducation.es 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 www.mheducation.es 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 www.mheducation.es 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 www.mheducation.es 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 www.mheducation.es 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 www.mheducation.es 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’. www.mheducation.es 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 www.mheducation.es Try out your ideas 74 / Unit 2 / Drawing projects www.mheducation.es 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 / Arts and Crafts C / 75 www.mheducation.es Try out your ideas 76 / Unit 2 / Drawing projects www.mheducation.es 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 / Arts and Crafts C / 77 www.mheducation.es Try out your ideas 78 / Unit 2 / Drawing projects www.mheducation.es 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 / Arts and Crafts C / 79 www.mheducation.es Try out your ideas 80 / Unit 2 / Drawing projects www.mheducation.es 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 / Arts and Crafts C / 81 www.mheducation.es Try out your ideas 82 / Unit 2 / Drawing projects www.mheducation.es 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 / Arts and Crafts C / 83 www.mheducation.es Try out your ideas 84 / Unit 2 / Drawing projects www.mheducation.es 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 / Arts and Crafts C / 85 www.mheducation.es Try out your ideas 86 / Unit 2 / Drawing projects www.mheducation.es 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 / Arts and Crafts C / 87 www.mheducation.es Try out your ideas 88 / Unit 2 / Drawing projects www.mheducation.es www.mheducation.es www.mheducation.es www.mheducation.es