This material is a collective work, conceived, designed and created by the Editorial department at Santillana, under the supervision of Teresa Grence. WRITERS Karen Ballesteros Juan Carlos Taravillo Vivian Mitchell CLIL CONSULTANT Mario Remón ILLUSTRATORS Diomedes Guilombo Carlos Alberto Salas Juan Carlos Taravillo Cristina Vidal EDITORS Elena Alfonso Dave Wile EDITORIAL MANAGERS Nuria Corredera Montserrat Herrero PROJECT DIRECTOR Lourdes Etxebarria BILINGUAL PROJECT DIRECTOR Margarita España Technical drawing Visual arts Design & Create III SECONDARY Level audio available santillana.es/clil
P r a c t i s e These activities enable you to put into practice what you have learned in the unit. By doing these worksheets, you will improve your personal, social and learning to learn competence and digital competence. Ga l l e r y This section further develops your knowledge of technical drawing. Some additional information about the use of technical drawing in other areas is also included. In addition, you will also develop your personal, social and learning to learn competence and citizenship competence. Con t e n t The contents of the unit are introduced with examples that show you step-by-step how to draw geometric constructions. These pages are intended to develop your mathematical competence and competence in science, technology and engineering. C r e a t e These activities allow you to apply and to show what you have learned about the subject. By doing these worksheets, you will improve your entrepreneurship competence and cultural awareness and expression competence. Technical drawing III Technical drawing III is organised into seven units, which aim to develop technical drawing skills. Each unit contains the following sections: Technical Drawing III 43 6 L i n e a r p e r s p e c t i v e I Linear perspective is a representation technique for representing 3-D objects and spaces on a 2-D surface. Some of the characteristics of linear perspective are: • Objects that are the same size in reality appear to be larger or smaller than each other. Their size in the drawing depends on their distance from the viewer. • Some of the lines that are parallel in reality are not drawn in parallel. They converge at a point in the distance from the viewer. These are the elements of linear perspective: – Picture plane (PP): the plane parallel to the viewer. The drawing is made on this plane. – Ground plane (GP): the viewer and the objects to be represented are located on this plane. – Horizon plane (HP): the plane at the eye level of the viewer. It is perpendicular to the picture plane. – Horizon line (HL): the line of intersection between the horizon plane and the picture plane. – Ground line (GL): the line of intersection between the ground plane and the picture plane. – Viewpoint (V): the point from which the viewer is looking. – Principal point (P): this is the point of intersection of the horizon line and the perpendicular line to it that passes through the viewpoint. The principal point indicates the position of the viewer in relation to the picture plane. In frontal perspective, the principal point is the same as the vanishing point. – Vanishing point (VP): this is a point on the horizon line where all the parallel lines converge, as long as the direction of the lines is not parallel to the projection plane. GALLERY View of the Grote Houtstraat in Haarlem, Nicolaes Hals, 1655. Many artists use linear perspective in their paintings. Elements of linear perspective PP P HL HP V GL GP Object Viewer ES0000000135176 175050 CDNO_DIB_TEC_Nivel_III_DISENA_102758.indd 43 29/1/22 14:1345 Technical Drawing III 25 CREATE Design a building and draw its views. Add the dimensions and colour it. ES0000000135176 175050 CDNO_DIB_TEC_Nivel_III_DISENA_102758.indd 25 29/1/22 14:1408 Technical Drawing III 39 PRACTISE Construct the object that has an oblique plane whose vertices are A, B, C and D. B B B A A D D C C C A D ES0000000135176 175050 CDNO_DIB_TEC_Nivel_III_DISENA_102758.indd 39 29/1/22 14:1358 GALLERY View of the Grote Houtstraat in Haarlem, Nicolaes Hals, 1655. Many artists use linear perspective in their paintings. Listen to the audio files at santillana.es/clil
Contents UNITS 5 I n t e r p r e t i ng v i ews o f 3 - D ob j e c t s 35 • Isometric perspective. Interpreting three views • Interpreting views of other 3-D objects: with inclined planes, with oblique planes, with curved surfaces and with different surfaces 6 L i n e a r p e r s p e c t i v e I 43 • Definition and elements of linear perspective • Frontal perspective. Construction of frontal perspective 7 L i n e a r p e r s p e c t i v e I I 51 • Oblique perspective • Construction of figures in oblique perspective, given the measuring points • Construction of a pyramid in oblique perspective F i na l a c t i v i t i e s 57 • Activities bringing together all the points covered in this book • Digital resources: draw a camera with LibreCAD G l o s s a r y 63 Technical drawing terms that are used in this material UNITS 1 Po l ygon s and a r c l i nk s 5 • Definition of a polygon • Elements of a regular polygon • Classification of polygons, triangles and quadrilaterals • Construction of a regular heptagon • Construction of a regular nonagon • Construction of arc links • Construction of arches 2 P r o j e c t i v e g e ome t r y 15 • Projective geometry and representation techniques • Representation of objects using projective geometry: description of the dihedral system • Nomenclature in the dihedral system. Representation of a point in the dihedral system • Representation of a line in the dihedral system 3 Vi ews 21 • Definition of views • The European system • The American system • Selection of views. Construct the main views of an object 4 Ne t s o f g e ome t r i c bod i e s 27 • Definition of geometric bodies and their nets • The nets of geometric bodies formed by flat surfaces: a cube or hexahedron and a tetrahedron • The net of a right cone • The net of an oblique prism
4 Instruments and recommendations for Technical drawing How to prepare the drawing instruments • Keep the pencils sharpened. • Clean the set squares and rulers before and after their use. • Sharpen the lead point of the compass to get clear and precise measurements. • Make sure the rubber has clean and sharp edges. The drawing process • Read the instructions carefully and refer to the images. • Remember that each step of the process is given in a logical order. • Draw the guidelines with a hard pencil (H), pressing lightly. • Complete the final drawing lines with a soft pencil (B). • Do not rub out the guidelines until the drawing is completely finished. Take note • Read the directions and instructions before starting the worksheet. • Sign your worksheets with your name. Use technical lettering. • Keep your desk and worksheets clean. Protractor French curve Set squares Rubber Pencil Coloured pencils T-square Compass Ruler Pencil sharpener
Technical Drawing III 5 1 Po l ygon s and a r c l i nk s GALLERY A polygon is a flat shape formed by three or more straight lines that join at points called vertices. The resulting line segments are known as sides. The word polygon is a compound word of Greek origin. It is formed by poly, which means many, and gon, meaning angle. So it is a flat shape with many angles. Classification of polygons Classification of triangles Classification of quadrilaterals Elements of a regular polygon: • Centre (O): the interior point that is the same distance from all of the vertices. It is the centre of the circumscribed and inscribed circumferences of the polygon. • Vertex (V ): the point where two adjacent sides meet. • Side (s): one of the segments that form the polygon. • Radius (r): a segment that connects a vertex to the centre of the polygon. • Apothem (a): a segment that connects the centre to the midpoint of each side. • Diagonal (D): a segment that connects two non-consecutive vertices of a polygon. NUMBER OF SIDES TRIANGLE QUADRILATERAL PENTAGON HEXAGON HEPTAGON OCTAGON NONAGON DECAGON HENDECAGON DODECAGON SIDES EQUILATERAL ISOSCELES SCALENE ANGLES RIGHT-ANGLED ACUTE OBTUSE PARALLELOGRAMS SQUARE RECTANGLE DIAMOND RHOMBOID ANGLES CONCAVE. A polygon that has at least one interior angle of more than 180º. CONVEX. A polygon whose interior angles are each less than 180°. SIDES AND ANGLES REGULAR. A polygon whose sides and angles are equal. IRREGULAR. A polygon whose sides and angles are not equal. AREA OR PERIMETER OPEN. A polygon whose endpoints do not connect at the same point. CLOSED. A polygon whose endpoints connect at the same point. D s V r O a TRAPEZIUMS RIGHT-ANGLED TRAPEZIUM ISOSCELES TRAPEZIUM SCALENE TRAPEZIUM TRAPEZOID
6 Con s t r u c t i on o f a r e gu l a r h e p t agon Construct a heptagon, given one side To construct a regular heptagon, given the length of one of its sides, follow these steps: Construct a heptagon, given the circumscribed circle To construct a regular heptagon, given the radius of the circumscribed circle, follow these steps: Step 1. From vertex A, draw an angle of 30º. Step 2. From B, draw a perpendicular line to AB. Where it intersects the 30º angle line, we obtain point N. Step 3. Use centre A and radius AN. Draw an arc. Step 4. Draw the perpendicular bisector of AB. At the point of intersection with the arc in step 3, we get point O. This is the centre of the circumscribed circumference of the heptagon. Step 5. Use centre O and radius OA. Draw the circumscribed circumference. Step 6. To complete the drawing, copy the length of AB consecutively around the circumference. Step 1. Draw the perpendicular diameters of the circumference, AB and CD. Step 2. Draw the perpendicular bisector of one of the radii. For example, use OD. This obtains point M. Extend the bisector to obtain point N. Step 3. The segment MN is the length of the side of the regular heptagon. From vertex A, translate this segment consecutively around the circumference. A A A A B A A B B B B B N N N N N O O O 30º A A A B B O M M N N B C C C D D D
Technical Drawing III 9 L i nk i ng c i r c umf e r e n c e s and s t r a i gh t l i n e s Link a circumference and a straight line with an arc whose radius is less than the radius of the circumference Given a circumference with radius m, the line AB and the arc radius r, follow these steps to link the circumference and the line: Step 1. Draw a parallel line to AB at a distance r. Step 2. Use centre point O and a radius equal to the sum of the radii m and r. Draw an arc that intersects the parallel line at point C. Step 3. Draw the segment OC that intersects the circumference at D. Step 4. Draw a perpendicular line to the parallel line that passes through point C. Label point E. Step 5. Use centre point C and radius r. Draw an arc that links the points D and E. Link a circumference and a straight line with an arc whose radius is greater than the radius of the circumference Given a circumference with centre F and radius p, the line AB and the arc radius q, follow these steps to link the circumference and the line: Step 1. Draw a line parallel to AB at a distance q. Step 2. Use centre point F and a radius equal to the difference between q and p. Draw an arc that intersects the parallel line at point G. Step 3. Draw the line FG that intersects the circumference at H. Step 4. Draw a perpendicular line to the parallel line that passes through point G. Label point J. Step 5. Use centre point G and radius q. Draw an arc that links the points H and J. Link two circumferences with an arc of a smaller radius Given two circumferences with centres N and M, and radii n and m, respectively, and the arc radius r, follow these steps to link the two circumferences: Step 1. Use centre point N and a radius equal to the sum of the radii n and r. Draw an arc. Step 2. Use centre point M and a radius equal to the sum of m and r. Draw an arc that intersects the previous arc at point A. Step 3. Draw the rays NA and MA. Where the lines intersect the circumferences, label the points B and C, respectively. Step 4. Use centre point A and radius r. Draw an arc that links the points B and C. Link two circumferences with an arc of a greater radius Given two circumferences with centres X and Y, and radii x and y, respectively; and the arc radius r, follow these steps: Step 1. Use centre point X and a radius equal to the difference between radii r and x. Draw an arc. Step 2. Use centre point Y and a radius equal to the difference between radii r and y. Draw an arc that intersects the previous arc at point Z. Step 3. Draw the lines XZ and YZ . Where the lines intersect the circumferences, label the points P and Q, respectively. Step 4. Use centre point Z and radius r. Draw an arc that links the points P and Q. r Z x X P y Y Q q G B J A H F p r m O D C B E A r m n C B M A N
10 PRACTISE Link the rays that form the angle DEF, given the arc radius p. Link the perpendicular rays AB and BC, given the arc radius r. Link the parallel lines GH and MN. Link the rays that form the angle KLM, given the radius q. r A B C G H M N p q E D F K L M
12 An arch is a structure that covers part of the space between two walls or between two columns. It serves as a decorative element in buildings. Arches are built with wedge-shaped stones called voussoirs. These are placed on top of structural supports. The parts of an arch are the span, the rise and the springing line. The voussoir at the top of the arch is called the keystone. It receives the weight of the wall and distributes it sideways. Round arch The round arch or Roman arch is one that is shaped like half a circumference. Its centre point is on the springing line. Segmental arch The segmental arch is one that is shaped like half a circumference. Its centre point is below the springing line. Equilateral pointed arch In an equilateral pointed arch, the centres of the arcs are the points of support called the springing points. To construct a pointed arch, given the springing points A and B, follow these steps: Step 1. Use centre point A and radius AB. Draw an arc. Step 2. Use centre point B and radius AB. Draw an arc that intersects the previous arc, forming a pointed arch. Round Segmental Pointed Horseshoe Three-centred Ogee Trefoil GALLERY The round arch is the basic element of the barrel vault. The pointed arch is one of the characteristics of Gothic architecture. Construction of a round arch Construction of a segmental arch span rise springing line A B keystone voussoirs Con s t r u c t i on o f a r c h e s I Construction of an equilateral pointed arch Each arch has a different name, depending on its shape:
Technical Drawing III 13 M H R S I J K L A B M C N G F P Horseshoe arch The horseshoe arch consists of an arc of a circumference. It is longer than the corresponding half a circumference from the same springing points. So, its shape looks like a horseshoe. To construct a horseshoe arch from its springing points A and B, and with radius r, follow these steps: Step 1. Use centre point A and radius r. Draw an arc. Step 2. Use centre point B and radius r. Draw an arc that intersects the previous arc at point C. Step 3. Use centre point C and radius r. Draw the horseshoe arch from point A to point B. Three-centred arch The three-centred arch is formed by a segmental arch, with two arcs of a circumference at its ends. To construct a three-centred arch from its springing points A and B, follow these steps: Step 1. Draw the perpendicular bisector of the springing line AB. Label the midpoint C. Step 2. Find the midpoint of AC and label it M. Then find the midpoint of CB and label it N. Step 3. Use centre point M and radius MN. Draw an arc. Then use centre point N and the same radius. Draw an arc that intersects the previous arc at point P. Step 4. Draw the equilateral triangle MNP. Draw the rays PM and PN. Step 5. Use centre point M and radius MA. Draw an arc from A that intersects the ray PM at point F. Step 6. Use centre point N and radius CN. Draw an arc from B that intersects the ray PN at point G. Step 7. Use centre point P and radius PF. Draw the arc FG that completes the three-centred arch. Ogee arch The ogee arch consists of four arcs of a circumference. Two of the arcs meet at a point at the top in the centre. To construct an ogee arch from its springing points A and E, follow these steps: Step 1. Divide the segment AE into four equal parts. This determines the points B, C and D. Step 2. Draw perpendicular lines to AE at points B and D. Step 3. Use centre point B and radius AC. Draw an arc to determine point F. Step 4. Use centre point D and radius AC. Draw an arc to determine point G. Step 5. Use centre point B and radius AB. Draw the arc AM. Then use centre point D and the same radius. Draw arc NE. Step 6. Use centre point F and radius AB. Draw arc MP. Then use centre point G and the same radius. Draw the arc PN that completes the ogee arch. Trefoil arch The trefoil arch consists of three arcs of a circumference that form the shape of a leaf of clover. To construct a trefoil arch from its springing points H and L, follow these steps: Step 1. Divide the line HL into four equal parts. This determines the points I, J and K. Step 2. Use centre point I and radius IK. Draw an arc. Then use centre point K and radius KI. Draw an arc that intersects the previous arc at point M. Step 3. Draw the equilateral triangle MIK. Step 4. Use centre point I and radius IH. Draw an arc from H that intersects MI at R. Then use centre point K and the same radius. Draw an arc from L that intersects MK at point S. Step 5. Use centre point M and radius MR. Draw the arc RS that completes the trefoil arch. N G F M A B C D E P Con s t r u c t i on o f a r c h e s I I r A B C
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