Engineering Graphics

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ENGINEERING GRAPHICS

Introduction

1. What is an Engineering Graphics and its requirements? A standardized graphic representation of physical objects and their relationship is called Engineering Graphics or Engineering Drawing or Engineering Drafting. It is prepared, based on certain basic principles, symbolic representations, standard conventions, notations, etc. A drawing can be done using freehand, instruments or computer methods. Engineering drawing is the graphic language, from which any trained person can visualize the object. It conveys the same picture to every trained person. Drawing prepared in one country may be utilized in any other country, irrespective of the language spoken there. The role of engineers is to develop products and in the process of product development, two important steps involved are – (i) deciding the specifications of the product, (ii) preparing the product’s drawings. Engineering drawing deals with the second step. The product’s sketches are prepared for manufacturing purposes. The drawings constructed by a design engineer are transferred to the manufacturing engineer. The manufacturing engineer produces the product as per the dimensions and specifications supplied by the design engineer. Thus, engineering graphics is a language of all persons involved in engineering activities.

2. Instruments & accessories used in engineering graphics 1. Drawing Board Drawing board is a desk or flat surface with proper dimensions used to support sheet and make it flat and ready to for technical drawing. More recently engineers and draftsmen use the drawing board for making and modifying drawings on paper with ink or pencil. Some drafting tables incorporate electric motors to provide the up and down and angle adjustment of the drafting table surface. 2. Mini Drafter It is an instrument used to draw geometrical shapes and figures with great precision. It does not require any other instrument like scale or set squares. The mini-drafter is a versatile tool and can be used to draw almost everything. It can be used to draw parallel lines, perpendiculars, inclined lines of any degree with unmatched speed. It consists of one long scale and one small scale and the scales are in ‘L’ shaped which can be fit on any drawing board easily. 3. Set-squares (45° – 45° & 60° – 90°)

______Darshan Institute of Engineering & Technology, Rajkot i Engineering Graphics (2110013) Introduction A set square is an object used in engineering and technical drawing, with the aim of providing a straightedge at a right angle or other particular planar angle to a baseline. These set squares come in two usual forms, both right triangles: one with 90- 45-45 degree angles, the other with 30-60-90 degree angles. 4. T-square A T-square is a technical drawing instrument used by draftsmen primarily as a guide for drawing horizontal lines on a drafting table. It may also guide a set square to draw vertical or diagonal lines. Its name comes from its resemblance to the letter T. The top of the T-square hangs off of the drafting table and is called the head; the blade is the part that stays on the drafting table. The T-square usually has a transparent edge made of plastic which should be free of nicks and cracks in order to provide smooth, straight lines. 6. Scales (Ruler) A ruler, sometimes called a rule or line gauge, is an instrument used in geometry, technical drawing, printing and engineering/building to measure distances and/or to rule straight lines. You’ll need to use this instrument frequently for making technical drawing. The scales are available in plastic, wooden and steel material. Generally steel rule is used to draw engineering drawings.

5. Protractor (180°, 360°) A protractor is a square, circular or semicircular tool, typically made of transparent plastic, for measuring angles. Most protractors measure angles in degrees (°). Radian-scale protractors measure angles in radians. Semi-circular protractor is of 180° and full circle protractor is of 360°. More advanced protractors, such as the bevel protractor, have one or two swinging arms, which can be used to help measure the angle. 7. Roll & Draw

______Darshan Institute of Engineering & Technology, Rajkot ii Introduction Engineering Graphics (2110013) Roll & Draw is most handy instrument for making charts & parallel lines at super-fast speed. It is also popular as “A draftsman’s best friend.” Furthermore, it is a multipurpose drawing instrument that lets you measure in centimeters and draw quadrants, squares, circles, angles, crosshatching, and vertical and horizontal parallel lines, quickly and easily. 8. Compass (Drafting) A compass is a technical drawing instrument that can be used for inscribing circles or arcs. A typical compass is consists of two legs. One pencil lead is added to a leg and a tip point to pivot on paper is provided to another leg. Circles can be made by fastening one leg of the compasses into the paper with the spike, putting the pencil on the paper, and moving the pencil around while keeping the hinge on the same angle. Drafting compass is available in different variants that can be selected as per drawing requirement. 9. Circle Master Circle Master is very helpful template type drawing instruments which helps to draw circles of different radius at high speed. It eliminates use of compass as circle of fixed radius can be drawn. It is a flat plate or circular plate with different sized holes punched at surface as shown in figure. If you want to draw circle of radius other than circle radius punched then it is required to use other drafting instrument as radius of circle cannot adjusted. 11. Drawing Paper & Drawing Sheet Generally a clean white paper or Sheets of sized varied from A1, A2 (most common), A3, A4 is used to make technical drawings on it. 10. French Curves A French curve is a template made out of metal, wood or plastic composed of many different curves. It is used in manual drafting to draw smooth curves of varying radii. The curve is placed on the drawing material, and a pencil or other implement is traced around its curves to produce the desired result. 12. Drawing Pencil (Lead pencil or mechanical pencil, wooden pencil) A mechanical pencil is a pencil with a replaceable and mechanically extendable solid pigment core called a lead. It is designed such that the lead can be extended as its point is worn away. The lead is usually graphite based and can easily remove and placed by pushing tip at top of mechanical pencil. Drawing pencil comes in wooden material too that are useful to draw sketches in shades. Different shades or grades of mechanical pencil are shown in figure at left. 13. Sharpener ______Darshan Institute of Engineering & Technology, Rajkot iii Engineering Graphics (2110013) Introduction A pencil sharpener is a device for sharpening a pencil’s writing point by shaving away its worn surface. It is mostly used when wooden pencil is selected to draw sketches as used writing point of wooden pencil may blur the drawing that results in black damage to paper. 14. Eraser An eraser is used for removing pencil markings on paper or sheet. It is very helpful in removing unnecessary line of drawing by rubbing it on unwanted lines and points. 15. Drawing pins & clips Drawing clips are used to hold drawings on drawing boards which remove sheet movement while working. Drawing pins are used to attach two or more sheets together. They are inserted and removed by hand, hence the terms “thumbtack” and “push pin” is also used. The term drawing pin comes from their being used to hold drawings on drawing boards. 16. Duster or handkerchief Duster or handkerchief is often used to clean paper on which drawing is going on. They are available in different variants. Maximum dust is possible when you use eraser to erase something on paper. The duster is very helpful at that time as removal of dust from hand leads to more black spots on drawing paper. Above are the basic but mostly used drawing instruments. All are easily available in stationary shops at your local place. You should use them to make precise drawings exactly as per data given.

3. Drawing Standards Drawing standards are set of rules that govern how technical drawings are represented. These drawing standards are used to convey the same meaning to everyone who reads them. There are so many international standards like ANSI (USA), BS (UK), JIS (Japan), AS (Australia), DIN (Germany), ISO, etc. In India, we have BIS (Bureau of Indian Standards). It works in association with other international standard developing organizations worldwide. BIS has recommended and published various standards for technical drawings. These standards are available in the form Special Publication (SP) 46:2003, titled, “Engineering Drawing Practices for Schools and Colleges”.

______Darshan Institute of Engineering & Technology, Rajkot iv 1. Engineering Scale

1.1 Introduction There is a wide variation in sizes for engineering objects. Some are very large (eg. Aero planes, rockets, etc.) Some are very small (Wrist watch components) There is a need to reduce or enlarge while drawing the objects on paper. Some objects can be drawn to their actual size. The proportion by which the drawing of an object is enlarged or reduced is called the scale of the drawing.

1.2 Definition Scale is defined as the ratio of the linear dimensions of the object as represented in a drawing to the actual dimensions of the same. This ratio is known as Representative Factor. Representative Factor (R.F.)

1.3 Types of Scale 1. Full Scale 2. Reduced Scale 3. Enlarged Scale

1.4 Full Scale: Using Full scale means you are drawing object’s all lines and points as per in drawing data. If we study it further, Drawing is in Full Scale if dimensions in data and the dimensions in drawing are same. Let us take an example: One circle’s radius is 20 mm in data and you are drawing it of 20 mm in drawing then you are using full scale in drawing. Example: 1:1, 2:2, 3:3, 4:4, 5:5 etc.

1.5 Reduced Scale: Using reduced scale means you are drawing object’s all lines and points by reducing it to some points. Let us take an example: One circle’s radius is 20 mm in data but you are drawing it of 10 mm in drawing then you are using reduced scale in drawing. Example: 1:2, 1:3, 1:4, 1:5, 1:6 etc.

______Darshan Institute of Engineering & Technology, Rajkot 1.1 Engineering Graphics (2110013) 1. Engineering Scale 1.6 Enlarged Scale: Using Enlarged scale means you are drawing object’s all lines and points by increasing or enlarging it to some points. Let us take an example: One circle’s radius is 20 mm in data but you are drawing it of 40 mm in drawing then you are using enlarged scale in drawing. Example: 2:1, 3:1, 4:1, 5:1, 6:1 etc.

1.7 Unit conversion: 10 mm = 1 cm mm = millimeter 10 cm = 1 dm cm = centimeter 10 dm = 1 m = 100 cm dm = decimetre 10 m = 1 dam m = metre 10 dam = 1 hm = 100 m dam = decametre 10 hm = 1 km = 1000 m hm = hectometer 12 in = 1 ft km = kilometre 3 ft = 1 yd in = inch ft = feet yd = yard

1.8 Data Given to Solve the Problem 1. The R.F. of the scale. 2. The units which it must represent, for example, millimetres and centimetres, or feet and inches etc. 3. The maximum length which it must show.

The length of scale is determined by the formula:

1.9 Types of Scale:- Engineers Scale: The relation between the dimension on the drawing and the actual dimension of the object is mentioned numerically (like 10 mm = 15 m). Graphical Scale: Scale is drawn on the drawing itself. This takes care of the shrinkage of the engineer’s scale when the drawing becomes old.

1.10 Types of Graphical Scale:- 1. Plain Scale 2. Diagonal Scale 3. Vernier Scale 4. Comparative scale 5. Scale of chords

1.11 PLAIN SCALE This type of scale represents two units or a unit and its sub-division. ______Darshan Institute of Engineering & Technology, Rajkot 1.2 1. Engineering Scale Engineering Graphics (2110013) A plain scale is used to indicate the distance in a unit and its nest subdivision. A plain scale consists of a line divided into suitable number of equal units. The first unit is subdivided into smaller parts. The zero should be placed at the end of the 1st main unit. From the zero mark, the units should be numbered to the right and the sub-divisions to the left. The units and the subdivisions should be labeled clearly. The R.F. should be mentioned below the scale. EXAMPLE 1.1 On map of Ahmedabad city 1 cm represents 1 Km. Construct a plain scale to measure the distance between Gujarat Technological University and Lal Darwaja which is 6 Km. Also indicate on scale, the distance between Geeta mandir and Kankariya lake which is 3 Km and 7 hm. Given: 1cm = 1 km Maximum length to be measure = 6 km Indicate= 3 km and 7 hm =3 km + 0.7 km Solution:

=6 cm

1.12 DIAGONAL SCALE The diagonal scales give us three successive dimensions that is a unit, a subunit and a subdivision of a subunit. Through Diagonal scale, measurements can be up to second decimal places (e.g. 4.35), is used to measure distances in a unit and its immediate two subdivisions; e.g. dm, cm & mm, or yard, foot & inch. Diagonal scale can measure more accurately than the plain scale. ______Darshan Institute of Engineering & Technology, Rajkot 1.3 Engineering Graphics (2110013) 1. Engineering Scale Diagonal scale concept 1. Let the XY in figure be a subunit. 2. From Y draw a perpendicular YZ to a suitable height. 3. Join XZ. Divide YZ in to 10 equal parts. 4. Draw parallel lines to XY from all these divisions and number them as shown. 5. From geometry we know that similar triangles have their like sides proportional. 6. Consider two similar triangles XYZ and 7’ 7Z, 7. We have 7Z / YZ = 7’7 / XY (each part being one unit) Means 7’ 7 = 7 / 10. x X Y = 0.7 XY

Similarly 1’ – 1 = 0.1 XY 2’ – 2 = 0.2 XY Thus, it is very clear that, the sides of small triangles, which are parallel to divided lines, become progressively shorter in length by 0.1 XY. Example 1.2 The distance between Ahmedabad and Bombay is 500 km. It is represented on a railway map by 10 cm. Construct a diagonal scale to measure kilometers. Show on scale the distance between Ahmedabad and Surat which is 237 km. Given: 10 cm = 500 km Maximum length to be measure = 500 km Indicate= 237 km.=200 km +30 km + 7 km Solution:

______Darshan Institute of Engineering & Technology, Rajkot 1.4 1. Engineering Scale Engineering Graphics (2110013)

=10 cm

1.13 ISOMETRIC SCALE When one holds the object in such a way that all three dimensions are visible then in the process all dimensions become proportionally inclined to observer’s eye sight and hence appear apparent in lengths. This reduction is 0.815 or 9 / 11 (approx.) it forms a reducing scale which is used to draw isometric drawings and is called Isometric scale. In practice, while drawing isometric projection, it is necessary to convert true lengths into isometric lengths for measuring and marking the sizes. Derivation of isometric scale:

______Darshan Institute of Engineering & Technology, Rajkot 1.5 Engineering Graphics (2110013) 1. Engineering Scale

From Fig:

Hence,

EXAMPLE: 1.3 Construct the isometric scale for 120 mm long line. Given: Actual length = 120 mm Solution:

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2. Engineering Curve

2.1 Conic Curves (Conics) When a cone is cut by a cutting plane with different positions of the plane relative to the axis of cone, it gives various types of curves like Circle, ellipse, parabola, and hyperbola. These curves are known as conic sections. Right circular cone is a cone that has a circular base and the axis is inclined at 900 to the base and passes through the center of the base. Figure 2.1 show a right cone and the various conic curves that can be obtained from a cone by sectioning the cone at various conditions.

Figure 2.1 A right cone and the various conic curves that can be obtained from a cone by sectioning the cone at various conditions

2.2 Conics Conic is defined as the locus of a point moving in a plane such that the ratio of its distance from a fixed point and a fixed straight line is always constant. Fixed point is called Focus. Fixed line is called Directrix

When eccentricity e < 1 Ellipse e =1 Parabola e > 1 Hyperbola

2.3 Ellipse An ellipse is obtained when a section plane, inclined to the axis of the cone, cuts all the generators of the cone as shown in figure 2.2.

______Darshan Institute of Engineering & Technology, Rajkot 2.1 Engineering Graphics (2110013) 2. Engineering Curve

Figure 2.2 Section of cone

2.4 Methods to Construct Ellipse The following methods are generally used to construct the ellipse: a. Focus directrix method. b. Arc of circle method c. Concetric circle method d. Oblong method

2.5 Focus-Directrix or Eccentricity Method Ellipse is the locus of a point which moves in a plane so that the ratio of its distances from a fixed point (Focus) and a fixed straight line (Directrix) is constant and less than one. Example 2.1: Draw an ellipse if the distance of focus from the directrix is 35 mm and the eccentricity is 3/4. Solution: 1. Draw the directrix DD’ and axis CC’ 2. Mark F on CC’ such that CF = 35 mm. 3. Divide CF into (3+4) 7 equal parts and mark V at the third division from O. Now, e = FV/ CV = 3/4. 4. At V, draw a perpendicular VA = VF. Join CA. 5. Through F, draw a line at 45° to meet CA produced at B. Through B, draw a perpendicular BV’ on CC’. Mark O at the midpoint of V– V’. 6. With F as a center and radius = 1–1’, cut two arcs on the perpendicular through 1 to locate P1 and P1’. 7. Similarly, with F as center and radii = 2–2’, 3–3’, etc., cut arcs on the corresponding perpendiculars to locate P2 and P2’, P3 and P3’, etc. 8. Also, cut similar arcs on the perpendicular through O to locate V1 and V1’. 9. Draw a smooth closed curve passing through V, P1, P/2, P/3, …, V1, …, V’, …, V1’, … P/3’, P/2’, P1’. 10. Mark F’ on CC’ such that V’ F’ = VF.

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Figure 2.3 Ellipse by focus directrix method Normal and Tangent 1. Take any point S on the curve 2. Join the point S with focus F 3. Draw the line perpendicular to SF which intersect the directrix at point M 4. Draw tangent T-T’ from the point M and passing through P 5. Draw the normal N-N’ perpendicular to tangent T-T’ as the normal and tangent are perpendicular to each other

Note:- CF = Distance between focus and directrix = CV + VF VF = Distance between vertex V and focus F, (Distance of the point on ellipse from vertex) CV = Distance between directrix and vertex V (Distance of the point on ellipse from directrix)

2.6 Arcs of Circle Method An ellipse is also the set of all points in a plane for which the sum of the distances from the two fixed points (the foci) in the plane is constant. This is clear from figure 2.4.

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Figure 2.4 Another definition of ellipse An ellipse is also defined as a curve traced by a point, moving in a plane such that the sum of its distances from two fixed points is always the constant and equal to major axis.

As per definition F1P1 + F2P1 = F1P2 + F2P2 = F1C + F2C = F1D + F2D = constant = major axis

So, F1C = F2C = F1D = F2D = half of the major axis = OA The arc of circle method of drawing an ellipse is generally used when I. The major axis and minor axis are known, II. The major axis and the distance between the foci are know, and III. The minor axis and the distance between the foci are know.

Example 2.2: Construct an ellipse by arcs of circle method. The major and minor axes are 140 mm & 100 mm respectively. Also draw the tangent and normal to the ellipse at any suitable point Solution: 1. Draw a horizontal major axis of the length 140 mm and give the notations A & B as shown in the figure 2.5. And mark a midpoint O on it. 2. Draw a vertical axis, perpendicular to the horizontal axis & passing through the point O; of the length equal to the length of minor axis, which is 100 mm and give the notations C & D as shown in the figure 2.5. 3. With the center C or D and length equal to the half of the length of major axis (OA); which is

70 mm; cut the major axis on two sides of the minor axis and give the notations F1& F2 respectively as shown in the figure 2.5. These are the focal points of the ellipse. 4. Divide the distance between F & O into five equal divisions and give the notations 1,2,3 etc.

5. Now with the distance equal to A1 and center F1 draw an arc of sufficient length, and with

the distance equal to B1 and center F2 cut the previously drawn arc on two sides of the

major axis and give the notations P1 & P1 as shown in the figure 2.5.

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6. Similarly draw arcs with the distances A2 – B2, A3 – B3, A4 – B4 etc. and center F1 - F2 respectively, which should intersect with each other respectively as shown in the figure 2.5.

And give the notations as P2-P2, P3-P3, P4-P4 etc.

7. Draw a smooth free hand medium dark curve through the points P1, P2, P3 etc. as shown in the figure 2.5; in sequence, so the resulting curve is the ellipse.

Figure 2.5 Ellipse by arcs of circle method

Figurer 2.6 Steps to draw an ellipse by arcs of circle method

______Darshan Institute of Engineering & Technology, Rajkot 2.5 Engineering Graphics (2110013) 2. Engineering Curve Normal and Tangent 1. Now mark a point anywhere on the ellipse; i.e., the point S, and connect this point with the

focal points F1 & F2 with straight lines. Then bisect the angle F1SF2 and draw a line of suitable length and give the notations N – N’ as shown in the figure. This is normal passing through the point S on the ellipse. 2. Draw a line which is perpendicular to the previously draw normal and give the notations T- T’, this is tangent passing through the point S on ellipse. 3. Give the dimensions by any one method of dimensions and give the name of the components by leader lines wherever necessary.

2.7 Concentric Circle Method Concentric circle method is used when the major axis and minor axis of the ellipse are given. This method is illustrated in figure 2.7. Example 2.3: Construct an ellipse by concentric circle method. The major and minor axes are 130 mm & 70 mm respectively. Also draw the tangent and normal to the ellipse at any suitable point

Figure 2.7 Ellipse by concentric circle method Solution: 1. Draw the major axis AB = 130 mm and minor axis CD = 70 mm, bisecting each other at right angles at O. 2. Draw two circles with AB and CD as diameters. 3. Draw two circles with AB and CD as diameters. Divide both the circles into 12 equal parts and number the divisions as 0, 1, 2 … 12 and 0’, 1’, 2’….12’. ______Darshan Institute of Engineering & Technology, Rajkot 2.6 2. Engineering Curve Engineering Graphics (2110013) 4. From all points of outer circle draw vertical lines downwards and upwards respectively. 5. From all points of inner circle draw horizontal lines to intersect those vertical lines. 6. Mark all intersecting points properly as those are the points on ellipse. 7. Join all these points along with the ends of both axes in smooth possible curve. It is required ellipse.

Normal and Tangent Method to draw normal and tangent to ellipse is similar those discussed in arc of circle method.

2.8 Oblong method or Rectangle method This method is used when the ellipse is required to be drawn in the given rectangle. The longer side of the rectangle is considered as the major axis and the shorter side of the rectangle is considered as minor axis. Example 2.4: Draw an ellipse with a 130 mm long major axis and a 80 mm long minor axis. or Draw an ellipse circumscribing a rectangle having sides 130 mm and 80 mm. Solution: 1. Draw the major axis AB = 130 mm and minor axis CD = 80 mm, bisecting each other at right angles at O. 2. Draw a rectangle EFGH such that EF = AB and FG = CD. 3. Divide AO and AE into same number of equal parts, say 5. Number the divisions as 1, 2, 3 and 1’, 2’, 3’ as shown in figure. 4. Join C with 1, 2 and 3.

5. Join D with 1’ and extend it to meet C–1 at P1. Similarly, join D with 2’ and 3’ and extend

them to meet C–2 and C–3 respectively to locate P2 and P3. 6. Do similar construction in right side part.

Figure 2.8 Ellipse by Rectangle method ______Darshan Institute of Engineering & Technology, Rajkot 2.7 Engineering Graphics (2110013) 2. Engineering Curve Normal and Tangent Method to draw normal and tangent to ellipse is similar those discussed in arc of circle method.

2.9 Parallelogram Method Example 2.5: Two points A and B are 100 mm apart. Third point C is 75mm from A and 50mm from B. Draw an ellipse passing through A, B &C. Solution: To draw ellipse passing through A, B and C, construct parallelogram in such a manner that points A, B, and C become mid-point of side of the parallelogram as shown in figure 2.9. STEPS ARE SIMILAR TO THE PREVIOUS CASE (RECTANGLE METHOD) ONLY IN PLACE OF RECTANGLE, HERE IS A PARALLELOGRAM.

Figure 2.9 Ellipse by parallelogram method

2.10 Application of Ellipse

Figure 2.10a Application of Ellipse

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Figure 2.10b Application of Ellipse

2.11 Parabola A parabola is obtained when a section plane, parallel to one of the generators cuts the cone. This is illustrated in figure 2.11.

Figure 2.11 Cutting of a cone to obtain parabola Methods to Construct Parabola The following methods are used to construct parabola 1. Focus and Directrix Method 2. Rectangle Method 3. Parallelogram Method 4. Tangent Method ______Darshan Institute of Engineering & Technology, Rajkot 2.9 Engineering Graphics (2110013) 2. Engineering Curve 2.12 Focus and Directrix Method Parabola is the locus of a point which moves in a plane so that the ratio of its distances from a fixed point (Focus) and a fixed straight line (Directrix) is constant and equal to one. This method is used when distance between focus and directrix is given. Example 2.6: Draw parabola if distance of focus from directrix is 60 mm. Solution: Method to draw parabola by focus and directrix method is similar to those for ellipse. Only difference is, eccentricity is equal to one instead of less than one.

Figure 2.12 Construction of parabola by eccentricity method

2.13 Rectangle Method Example 2.7: Construct the parabola of the base 100 mm and the axis length 50 mm using rectangle method. Solution: 1. Draw the rectangle with base AB = base of the parabola and height AD = height of the parabola (AB = 100 mm and CD = 50 mm). 2. Draw the vertical line OV from mid – point of the AB as shown in figure 2.13. 3. Divide the line AD and AO into equal number of parts say 5. 4. Join the points 1, 2, 3 …. 5, with V.

5. Draw the vertical line from 1’ which meet the line V1 at P1. Similarly locate all the points. 6. Join all P points by drawing smooth curve. ______Darshan Institute of Engineering & Technology, Rajkot 2.10 2. Engineering Curve Engineering Graphics (2110013)

Figure 2.13 Construction of parabola by rectangle method Normal and Tangent 1. Mark the given point S on the curve. Draw line perpendicular to axis and meet the axis at M. 2. Extend the axis OV up to M’ such that VM = VM’. 3. Draw a line T-T’ from M and passing through the S, which is required tangent. 4. Draw the normal N-N’ perpendicular to tangent at S.

2.14 Tangent Method Example 2.8: Construct a parabola having base length 100 mm and axis height 60 mm by tangent method. Solution: 1. Draw a triangle with base AB = base of the parabola and axis OC = 2 X height of the parabola (AB = 100 mm, OC = 2 X 60 = 120 mm)

Figure 2.14 Construction of parabola by tangent method

______Darshan Institute of Engineering & Technology, Rajkot 2.11 Engineering Graphics (2110013) 2. Engineering Curve 2. Divide the line AC and BC into equal number of parts say 12 and number the points as shown in figure. 3. Draw the lines joining 1 - 1, 2 – 2, 3 – 3…….. 12 – 12. 4. Draw the smooth curve in such a way that all the line drawn through previous step become tangent to the curve. 5. The curve will start from point A and end at point B. Point V is the vertex of the curve.

2.15 Application of Parabola There are a large number of applications for parabolic shapes. Some of these are in search light mirrors, telescopic mirrors, a beam of uniform strength in design applications, the trajectory of the weight less flight, etc. These are shown in figure 2.15.

Figure 2.15 Few applications of parabolic shapes

______Darshan Institute of Engineering & Technology, Rajkot 2.12 2. Engineering Curve Engineering Graphics (2110013) 2.16 Hyperbola Hyperbola is the locus of a point which moves in a plane so that the ratio of its distances from a fixed point (Focus) and a fixed straight line (Directrix) is constant and greater than one. A Hyperbola is obtained when a section plane, parallel/inclined to the axis cuts the cone on one side of the axis. This is illustrated in figure 2.16. A Rectangular Hyperbola is obtained when a section, parallel to the axis cuts the cone on one side of the axis.

Figure 2.16 Cutting of a cone to obtain parabola Methods to Construct Hyperbola The following methods are used to construct hyperbola: i. Focus and Directrix Method ii. Rectangle Method

2.17 Focus and Directrix Method This method is used when distance between focus and directrix is given. Example 2.9: Draw a hyperbola having eccentricity 3:2, if distance of focus from the directrix is 50 mm. Solution: Method to draw hyperbola by focus and directrix method is similar to those for ellipse. Only difference is, eccentricity is greater than one instead of less than one.

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Figure 2.17 Construction of hyperbola by eccentricity method

2.18 Rectangle Method This method is used to draw the rectangle hyperbola. The curve is generated by a point which moves in such a way that the product of its distance from two fixed lines is constant. The fixed lines are called the asymptotes, which are at right angle in rectangular hyperbola. Example 2.10: A point P is 45 mm and 30 mm respectively from two straight lines which are at right angles to each other. Draw a rectangular hyperbola from P within 6mm distance from each asymptote line. Solution: 1. Draw the asymptotes OA and OB at right angle to each other. 2. Draw the line CD parallel to line OA, 45 mm away from it. 3. Draw the line EF parallel to line OB, 30 mm away from it.

4. Locate the point P0 (30, 45) at the intersection of above lines.

5. Take number of points on lines P0E and P0C as shown in figure. Keep the distance of last points 6 mm from two asymptotes as given in data. 6. Draw a line from O which passing through 1 and meet the line CD at 1’. 7. Draw lines parallel to OB and parallel to OA, from 1’ and 1 respectively.

8. Mark the point P1 at intersection of above lines. 9. Similarly marks the all points as shown in figure 2.18.

Normal and Tangent 1. Take a point S on the curve.

______Darshan Institute of Engineering & Technology, Rajkot 2.14 2. Engineering Curve Engineering Graphics (2110013) 2. Draw lines parallel to OA and OB from S as shown in figure. Mark the points G and G’ by keeping the distance X and Y constant. 3. Draw a line GG’ which is tangent. Draw normal N-N’ perpendicular to tangent at S.

Figure 2.18 Construction of hyperbola by rectangle method

2.19 Hyperbola (Applications) Hyperbolic shapes finds large number of industrial applications like the shape of cooling towers, mirrors used for long distance telescopes, etc.

Figure 2.19 Application of hyperbolic shapes in engineering

______Darshan Institute of Engineering & Technology, Rajkot 2.15 Engineering Graphics (2110013) 2. Engineering Curve 2.20 Cycloid These curves are generated by the fixed point on the circumference of a circle which rolls without slipping on a fixed straight line or another fixed circle. The rolling circle is called the generating circle. The fixed straight line is called directing line. The fixed circle is called directing circle. Cycloid can be divided into three groups. 1. Cycloid 2. Epicycloid 3. Hypocycloid

Cycloid: cycloid is a curve generated by the point on the circumference of the rolling circle when it rolls among the fixed straight line called the directing line. Epicycloid: Epicycloid is a curve generated by the point on the circumference of the rolling circle when the circle rolls outside another circle called directing circle. Hypocycloid: Hypocycloid is a curve generated by the point on the circumference of the rolling circle when the circle rolls inside another circle called directing circle.

2.21 Construction of Cycloid Example 2.11: A wheel of the diameter 50 mm rolls over the straight line without slipping for one rotation. Draw the path traced by the point P which is initially at the point of the contact between the wheel and the straight line. Name the path traced. Name the curve and also draw tangent and normal to the curve at suitable point on the curve. Solution:

1. Draw a circle of diameter 50 mm with center C0. 2. Divide the circle into 12 equal parts and give the number as shown in figure 2.20. 3. Draw a horizontal line of length 2r from 0 (distance travel by circle in one rotation).

Figure 2.20 Construction of cycloid 4. Divide the line into 12 equal parts because circle is divided into 12 equal parts and give the numbers. [If circle is divided into 8 equal parts then divide the line into 8 equal parts]. ______Darshan Institute of Engineering & Technology, Rajkot 2.16 2. Engineering Curve Engineering Graphics (2110013) 5. Draw vertical line from all 12 points as shown in figure 2.20 and give the numbers. [These all numbers indicate the position of the center of roller when it rolls on a directing line.] 6. From all the points on circle draw horizontal line.

7. With radius 25 mm (radius of roller) and center C1 mark point P1 on horizontal line drawn from 1.

8. Repeat the procedure from C2, C3, C4 up to C12 as centers. Mark points P2, P3, P4 up to P12 on horizontal line drawn from 2, 3, 4……12 respectively. 9. Join all these points by curve. Name the curve is cycloid.

Normal and Tangent 1. Mark the point S on the curve 2. Take S as center and radius equal to radius of rolling circle, mark pint C on center line. 3. Draw vertical line from C which meets directing line. 4. Draw line N - N’ from the point obtained in previous step, which passes through point S. 5. Draw tangent which is perpendicular to normal N – N’ at S.

2.22 Construction of Epicycloid Example 2.12: A circle of 50 mm diameter rolls along the circumference of another circle of 150 mm diameter from outside. Draw the path of a point P on the circumference of the rolling circle for one complete revolution and name the curve. Solution: When rolling circle rolls over the directing circle for one rotation, it covers the arc of the length equal to the circumference of the rolling circle. It is very difficult to mark the same arc length on the directing circle so the corresponding subtended angle is calculated as follows:

Where, r = radius of rolling circle R = Radius of directing circle 1. Take O as center and radius = radius of directing circle = 75 mm to draw part of directing circle having subtended angle = (25/75) X 360 = 120. Extend the lines showing subtended

angle up to C0 and C12. [OC0 = OC12 = radius of rolling circle + radius of directing circle = 25 + 75 = 100 mm]

2. Draw a circle with C0 as center and radius equal to radius of rolling circle = 25 mm such that it touches directing circle from outside.

3. Mark the point P0 at the contact point of two circles. 4. Divide the circle into any number of equal parts, say 12 parts and give the number 1, 2, 3 ….12 in clockwise direction.

5. Take O as center and radius equal to OC0 draw the arc.

______Darshan Institute of Engineering & Technology, Rajkot 2.17 Engineering Graphics (2110013) 2. Engineering Curve

6. Divide the arc C0 C12 into same number of equal parts, 12 parts and give the number C1, C2,

C3 ….. C12 as shown in figure. (These points show the different position of center of rolling circle when it rolls on directing circle) 7. Draw the arcs with O as center and radius equal to O-1, O-2, O-3 ….. O-6 as shown.

8. Take center C1 and radius equal to radius of roller = 25 mm mark the point P1 on the arc drawn from point 1.

9. Repeat the procedure from C2, C3, C4 up to C12 as centers. Mark points P2, P3, P4 up to P12 on arc drawn from 2, 3, 4……12 respectively. 10. Join all the point by drawing smooth curve. The curve is required epicycloids.

Figure 2.21 Construction of epicycloid Normal and Tangent 1. Mark the point S on the curve

2. Take S as center and radius equal to radius of rolling circle, mark pint C on center line C0 C12. 3. Draw line from C which meets center of directing circle O. Mark the point M at intersection of line OC with directing circle. 4. Join the M with S which is normal N - N’. 5. Draw tangent which is perpendicular to normal N – N’ at S.

2.23 Construction of Hypocycloid 1. Here, smaller circle is rolling inside the larger circle. It has to rotate anticlockwise to move ahead. 2. Same steps should be taken as in case of EPI – CYCLOID. Only change is in numbering direction of 12 numbers of equal parts on the smaller circle. 3. Further all steps are that of epi – cycloid. This is called HYPO – CYCLOID.

______Darshan Institute of Engineering & Technology, Rajkot 2.18 2. Engineering Curve Engineering Graphics (2110013)

Figure 2.22 Construction of hypocycloid

2.24 Application of Cycloid

Figure 2.23 Application of cycloid shapes in engineering

2.25 Involute An Involute is a curve traced by the free end of a thread unwound from a circle or a polygon in such a way that the thread is always tight and tangential to the circle or side of the polygon. Construction of Involute of circle Example 2.13: Construct the Involute of circle of 40 mm diameter for one turn. Draw tangent and normal to the involute at any point on it. Solution: 1. Draw the circle with O as center and radius equal to 20 mm. ______Darshan Institute of Engineering & Technology, Rajkot 2.19 Engineering Graphics (2110013) 2. Engineering Curve 2. Divide the circle into any number of equal parts says 12 parts. Number them as 1, 2…..12 3. Draw line 012’ = 2r, tangent to the circle at P [length of the string unwound from circle for one turn will be equal to the circumference of the circle.] 4. Divide the line 012’ into same number of equal parts i.e. 12 parts and number as 1΄, 2΄…..12’ 5. Draw tangents to the circle at 1, 2,3….12

6. Locate points P1, P2 such that 1-P1 = 1΄12’, 2-P2 = 2΄12’….

7. Join P0, P1, P2….

Figure 2.24 Construction of Involute of circle Normal and Tangent 1. Mark the point S on the curve 2. Draw line SO by joining the point S with center O of the circle. 3. Take the mid point of the line SO as C. 4. Take C as center and radius equal to OC. Draw a semicircle. 5. [The semicircle should be drawn towards the side where the larger area is covered by the involute.] 6. Marks the point M at intersection of semicircle with circle. 7. Draw the from M passing through the S which is required normal NN’ 8. Draw tangent TT’ at point S which is perpendicular to normal.

Note:- When length of the string is less or more than D.

______Darshan Institute of Engineering & Technology, Rajkot 2.20 2. Engineering Curve Engineering Graphics (2110013) Whatever may be the length of string, mark D distance horizontal i.e. along the string and divide it in 12 numbers of equal parts, and not any other distance. Rest all steps are same as previous INVOLUTE. Draw the curve completely. Involute of Regular Polygon Example 2.14: Draw the involute to a regular hexagonal plate of 25 mm size. Solution: 1. Draw hexagon of side 25 mm and give the number 1, 2, 3 …6.

2. Extend the line 5-6 such that 6P0 is equal to periphery of the hexagon (25 X 6 = 150 mm) 3. Extend all the lines 01, 12, 23….45.

4. Draw arc P0P1 with 0 as center and radius equal to 0P0.

5. Draw arc P1P2 with 1 as center and radius equal to 1P1.

6. Similarly draw all arcs P2P3, P3P4, P4P5 & P5P6 with center 2, 3, 4 & 5 and radius equal to 2P2,

3P3, 4P4 & 5P5 respectively.

Figure 2.25 Construction of Involute of hexagon

2.26 Application of Involute

Figure 2.26 Application of involute shapes in engineering ______Darshan Institute of Engineering & Technology, Rajkot 2.21 Engineering Graphics (2110013) 2. Engineering Curve 2.27 Spirals A spiral is a curve traced by a point moving along a line in one direction, while the line is rotating in a plane about one of its ends or any point on it. In other words it is the locus of a point which moves around a center, called the pole, while moving towards or away from the center. The point which generates the curve is called the generating point or tracing point. The point will move along a line called the radius vector while the line itself rotates about one of its end points. The angle between the radius vector and the initial position of the line is known as vectorial angle. The ratio of the difference in length of any two radii vectors and the corresponding angle between the radius vectors is called the constant of the curve. The angle is to be taken in radians. One complete rotation of the line which generates the curve is called convolution. A spiral may complete any number of convolutions before reaching the pole of the curve. Construction of Archemedian Spiral Example 2.15: Construct an Archimedean spiral of one convolution given the greatest and shortest radii as 84 mm and the 28 mm respectively. Draw the tangent and normal at any point on the curve. Solution: 1. Draw two concentric circle with center O and radius 84 mm & 28 mm. 2. Divide the outer circle into 8 equal parts and give number 1, 2, 3. . . . 8.

Figure 2.27 Construction of spiral

______Darshan Institute of Engineering & Technology, Rajkot 2.22 2. Engineering Curve Engineering Graphics (2110013) 3. Divide the length between the smallest and largest radii into 12 equal parts and give the number 1’, 2’, 3’ . . . . 8’ [For one convolution angular displacement is divided into 8 parts so; line is divided into same 8 equal parts. If number of convolution is 2, angular displacement becomes 16 divisions so line is divided into 16 parts.] [No of parts on line = No of parts of circle X No of convolution]

4. Take O as center and radius equal to O1’, draw the arc to get point P1 on radial line O-1.

5. Similarly take O as center and radius equal to O2’, O3’ . . . O8’, draw arcs to get points P2, P3

. . . . P8 on corresponding radial line O-2, O-3 . . . . O-8.

Normal and Tangent 1. Mark the point S on the curve. 2. Draw radius vector OS by joining point S with center O. 3. Calculate the constant on curve using following equation:

4. Draw a circle with O as center and radius equal to constant of curve. 5. Draw a line perpendicular to OS which intersect the constant of curve circle at M. [The perpendicular line should be drawn on that side of the radius vector where the curve is going away from the pole O.] 6. Join the point M with S which is normal to the curve. 7. Draw tangent T-T’ at S which is perpendicular to normal N-N’.

2.28 Application of Spiral

Figure 2.28 Application of spiral shapes in engineering

______Darshan Institute of Engineering & Technology, Rajkot 2.23

3. Projections of Points and Lines

3.1 Introduction Projecting the image of an object to the plane of projection is known as projection. The object may be a point, line, plane, solid, machine component or a building. Consider the following illustration to project the image of an object on to a plane.

Figure 3.1 Projection of an object In engineering drawing practice two principal planes are used to get the projection of an object as shown in Fig.3.2. They are: Vertical plane (VP) which is assumed to be placed vertically, the front view of the object is projected onto this plane. Horizontal plane (HP) which is assumed to be placed horizontally, the top view of the object is projected onto this plane. These principal planes are also known as reference planes or co-ordinate planes. The planes considered are imaginary, transparent and dimensionless. The reference planes VP and HP are placed in such a way that they intersect each other at right angles. As a result of intersection, an intersecting line is obtained which is known as the reference line or XY line.

Figure 3.2 System of projection

______Darshan Institute of Engineering & Technology, Rajkot 3.1 Engineering Graphics (2110013) 3. Projections of Points and Lines Observe in Fig. 3.2 the reference planes (VP & HP) forming four quadrants namely, (i) First quadrant (iii) Third quadrant (ii) Second quadrant (iv) Fourth quadrant It is assumed that the observer always stands at the right side of the reference planes. The image of the object is projected and obtained on HP by observing it from top of the object and is called top view or plan of the object. The image is projected and obtained on VP by seeing the object from front and is called front view or elevation. As per the general rule of drawing, the HP is rotated clockwise through 90°. After rotation, the images are drawn on HP and VP.

3.2 Conventional Representation Table 3.1 Conventional representation Object Point A Line AB It’s Top view A Ab It’s Front view a’ a’b’ It’s Side view a’’ a’’b’’

3.3 Projections of a Point in Different Quadrant

Figure 3.3 Projection of points in different quadrant

______Darshan Institute of Engineering & Technology, Rajkot 3.2 3. Projections of Points and Lines Engineering Graphics (2110013) 3.4 Important Conclusions about F.V. of a Point If a point is above the H.P., its F.V. will be above XY line. If a point is below the H.P., its F.V. will be below XY line. If a point is on the H.P., its F.V. will be on the XY line.

3.5 Important Conclusions about T.V. of a Point If a point is infront of the V.P., its T.V. will be below the XY line. If a point is behind the V.P., its T.V. will be above the XY line. If a point is on the V.P., its T.V. will be on the XY line.

Figure 3.4 Projection of points

3.6 Basics of Straight Line A straight line is the shortest route to join any two given points. It is a one-dimensional object having only length (l). The projections of a straight line are obtained by joining the top and front views of the respective end points of the line. The actual length of the straight line is known as true length (TL).

______Darshan Institute of Engineering & Technology, Rajkot 3.3 Engineering Graphics (2110013) 3. Projections of Points and Lines 3.7 Simple Cases in Projection of Line A straight line is placed with reference to the planes of projections in the following positions. 1. Line perpendicular to HP and parallel to VP 2. Line perpendicular to VP and parallel to HP 3. Line parallel to both HP and VP 4. Line inclined to HP and parallel to VP 5. Line inclined to VP and parallel to HP 6. Line inclined to both HP and VP

3.8 Line Perpendicular to HP and Parallel to VP

Figure 3.5 Projection of line perpendicular to HP and parallel to VP

3.9 Line Perpendicular to VP and Parallel to HP

Figure 3.6 Projection of line perpendicular to VP and parallel to HP

______Darshan Institute of Engineering & Technology, Rajkot 3.4 3. Projections of Points and Lines Engineering Graphics (2110013) 3.10 Line Parallel to both HP and VP

Figure 3.7 Projection of line perpendicular parallel to both the reference plane

3.11 Line Inclined to HP and Parallel to VP

Figure 3.8 Projection of line inclined to HP and parallel to VP

3.12 Line Inclined to VP and Parallel to HP

Figure 3.9 Projection of line inclined to VP and parallel to HP ______Darshan Institute of Engineering & Technology, Rajkot 3.5 Engineering Graphics (2110013) 3. Projections of Points and Lines 3.13 Line Inclined to both HP and VP

Figure 3.10 Projection of line inclined to both reference planes 3.14 Master Problem for Projection of Line

Figure 3.11 Master problem

______Darshan Institute of Engineering & Technology, Rajkot 3.6 3. Projections of Points and Lines Engineering Graphics (2110013) 3.15 Solved Problem of projection of line Problem 3.1: Line AB is 75 mm long and it is 300 & 400 Inclined to Hp & VP respectively. End A is 12mm above HP and 10 mm in front of VP. Draw projections. Line is in 1st quadrant. Solution:

Figure 3.12 Problem no. 3.1 Problem 3.2: FV of line AB is 500 inclined to XY and measures 55 mm long while it’s TV is 600 inclined to XY line. If end A is 10 mm above HP and 15 mm in front of VP, draw it’s projections, find TL, inclinations of line with Hp & VP. Solution:

Figure 3.13 Problem no. 3.2 ______Darshan Institute of Engineering & Technology, Rajkot 3.7

4. Projections of Planes

4.1 Introduction A plane is a two dimensional object having length and breadth only. Its thickness is always neglected. Various shapes of plane figures are considered such as square, rectangle, circle, pentagon, hexagon, etc.

Figure 4.1 Various Planes Types of Planes Perpendicular planes which have their surface perpendicular to any one of the reference planes and parallel or inclined to the other reference plane. Oblique planes which have their surface inclined to both the reference planes Traces of a Plane The trace of a plane is the line of intersection or meeting of the plane surface with the Reference plane; if necessary the plane surface is extended to intersect the reference plane. The intersection line of the plane surface with HP is called the Horizontal Trace (HT) and That of VP is called the Vertical Trace (VT).

4.2 Positions of a Plane A plane figure is positioned with reference to the reference planes by referring its surface in the following possible positions. 1. Surface of the plane kept perpendicular to HP and parallel to VP 2. Surface of the plane kept perpendicular to VP and parallel to HP 3. Surface of the plane kept perpendicular to both HP and VP 4. Surface of the plane kept inclined to HP and perpendicular to VP 5. Surface of the plane kept inclined to VP and perpendicular to HP 6. Surface of the plane kept inclined to both HP and VP ______Darshan Institute of Engineering & Technology, Rajkot 4.1 Engineering Graphics (2110013) 4. Projections of Planes 4.3 Projections of a Plane with its Surface Perpendicular to HP and Parallel to VP Consider a square plane ABCD having its surface perpendicular to HP and parallel to VP as in Fig. 4.2(i).

Figure 4.2 (i) Surface Perpendicular to HP and Parallel to VP The front view is projected onto VP which is a square a’,b’,c’,d’ having true shape and size. The top view is projected onto HP and is a line ab(c) (d) parallel to XY. The invisible corners are enclosed in. The plane surface is extended to meet HP to get the HT which coincides with the top view of the plane. It does not have a VT because the plane is parallel to VP. The projections and traces obtained are drawn with reference to the XY line as shown in Figure.

4.4 Projections of a Plane with its Surface Perpendicular to VP and Parallel to HP Consider a square plane ABCD with its surface perpendicular to VP and parallel to HP as in Fig. 4.2(ii).

Figure 4.2 (ii) Surface Perpendicular to VP and Parallel to HP ______Darshan Institute of Engineering & Technology, Rajkot 4.2 4. Projections of Planes Engineering Graphics (2110013) The top view is projected onto HP which is a square a b c d having true shape and size. The front view is projected onto VP and is a line a’b’ (c’) (d’) parallel to XY. The invisible corners are enclosed in ( ). The plane surface is extended to meet VP to get the VT which coincides with the front view of the plane. It does not have a HT because the plane is parallel to HP. The projections and traces obtained are drawn with reference to the XY line as shown in Fig. 4.2 (ii).

4.5 Projections of a Plane with its Surface Perpendicular to both HP and VP Consider a square plane ABCD having its surface perpendicular to both HP and VP as in Fig. 4.2(iii).

Figure 4.2 (iii) Surface Perpendicular to both HP and VP The front view and top view (a’) b c (d’) are projected onto VP and HP respectively. Both the views are lines perpendicular to the XY line. The true shape of the plane is obtained in the side view which is projected onto a profile plane (PP) which is perpendicular to both HP and VP. In this case, the left side view a’’ b’’ c’’ d’’ is obtained on the PP which is at the right side of the given object (plane). The plane surface is extended to meet HP and VP to get HT and VT which coincide with the top and front views respectively. The projections and traces obtained are drawn with reference to the XY line as shown in Fig. 4.2 (iii).

4.6 Projections of a Plane with its Surface Inclined to HP and Perpendicular to VP Consider a square plane ABCD with its surface inclined at an angle q to HP and Perpendicular to VP as in Fig. 4.2(iv). The top view abcd is projected onto HP. It is smaller than the true shape and size. The front view is projected onto VP and is a line a’b’(¢) (d) inclined at an angle q to XY. The invisible corners are enclosed in ( ). ______Darshan Institute of Engineering & Technology, Rajkot 4.3 Engineering Graphics (2110013) 4. Projections of Planes

Figure 4.2 (iv) Surface Inclined to HP and Perpendicular to VP The plane surface is extended to meet HP to get the HT which is a line perpendicular to XY. The plane surface is also extended to meet VP to get the VT which is a line inclined at an angle q to XY. The projections and traces obtained are drawn with references to the XY line as shown In Fig. 4.2 (iv)

4.7 Projections of a Plane with its Surface Inclined to VP and Perpendicular to HP Consider a square plane ABCD with its surface inclined at an angle f to VP and perpendicular to HP as shown in Fig. 4.2 (v).

Figure 4.2 (v) Surface Inclined to VP and Perpendicular to HP The front view a’ b’ ¢’ d’ is projected onto VP. It is smaller than the true shape and size. The top view is projected onto HP and is a line a b (c) (d) inclined at an angle to XY. The invisible corners are enclosed in ( ). ______Darshan Institute of Engineering & Technology, Rajkot 4.4 4. Projections of Planes Engineering Graphics (2110013) The plane surface is extended to meet VP to get the VT which is a line perpendicular to XY. The plane surface is also extended to meet HP to get the HT which is a line inclined at an angle to XY.

The projections and traces obtained are drawn with reference to the XY line as shown in Fig. 4.2 (v). Note: (i) When a plane is placed with its surface inclined to one plane and perpendicular to the other plane, its projections cannot be drawn directly. It is obtained in two steps and will be discussed in the examples that follow. (ii) When a plane is placed with its surface inclined to both the reference planes, its projections are obtained in three steps which will be discussed in the following examples. Problem 4.1: A 30 – 60 set square has its longer edge 55 mm on H.P. and makes an angle 45 with V.P. Draw the projection of plane if its surface makes and angle 30 to H.P. Solution:

Geometry Triangular Plane Position in Drawing

Resting on H.P. and on its longer Edge Longer Edge Left Side

Surface, plane, plate, lamina, H.P. Draw position 3 angle above “XY” diameter ,diagonal makes an Line. angle with (Position 3 angle)

Side, edge, diagonal, diameter V.P. Draw position 5 angle below “XY” makes and angle with Line. (position 5 angle)

Figure 4.3 Problem No. 4.1 ______Darshan Institute of Engineering & Technology, Rajkot 4.5 Engineering Graphics (2110013) 4. Projections of Planes Problem 4.2: A pentagonal plane resting on V.P. on it's side and it's surface makes an angle 35 to V.P. and it's side on which it's resting on makes an angle 40 to H.P. Draw it's Projections. Solution:

Geometry Pentagon Plane Position in Drawing

Resting on V.P. and on its one of the side Side Left Side

Surface, plane, plate, lamina, V.P. Draw position 3 angle below “XY” diameter ,diagonal makes an Line. angle with (Position 3 angle)

Side, edge, diagonal, H.P. Draw position 5 angle above “XY” diameter makes and angle Line. with (position 5 angle)

Figure 4.4 Problem No. 4.2

______Darshan Institute of Engineering & Technology, Rajkot 4.6 4. Projections of Planes Engineering Graphics (2110013) Problem 4.3: A circular plane of diameter 50 mm resting on H.P. on one of the point of its periphery A. Plane makes an angle 45 to H.P. and (1) top view of it's diameter “ae” makes an angle 30 to V.P. (2) it's diameter “ae” makes an angle 30 to V.P. Draw it's projections. Solution:

Geometry Circular Plane Posotion in Drawing

Resting on H.P. on it's point of Draw Circle Below “XY” Line. periphery.

Surface, plane, plate, lamina, H.P. Draw positin 3 angle above “XY” diameter ,diagonal makes an Line. angle with (Position 3 angle)

Side, edge, diagonal, V.P. Draw position 5 angle below “XY” diameter makes and angle Line. with (position 5 angle)

In case of apparent angle (beta, alpha) If true length equals to plane length or elevation length no need to draw angle (alpha or beta) If top view or front view of any (side, edge, diagonal, diameter) makes and angle is given then no need to draw angle (alpha, Beta). If True Length does not equal to plane length or elevation length in that case angle (alpha or beta) must to be drawn.

Figure 4.5 Problem No. 4.3

______Darshan Institute of Engineering & Technology, Rajkot 4.7 Engineering Graphics (2110013) 4. Projections of Planes Problem No. 4.4 A semi-circular lamina 60 mm diameter is on H.P. and it's surface makes an angle 30 to H.P. and it's diameter makes an angle 45 to V.P. Draw it's position. Solution:

Geometry Circular Plane Position in Drawing

Resting on H.P. on it's diameter. Draw half Circle Below “XY” Line diameter left side.

Surface, plane, plate, lamina, H.P. Draw position 3 angle above “XY” diameter ,diagonal makes an Line. angle with (Position 3 angle)

Side, edge, diagonal, diameter V.P. Draw position 5 angle below “XY” makes and angle with (position 5 Line. angle)

Figure 4.6 Problem No. 4.4

______Darshan Institute of Engineering & Technology, Rajkot 4.8 4. Projections of Planes Engineering Graphics (2110013) Problem No. 4.5 A rhombus longer diagonal 50 mm and shorter diagonal 35 mm is resting on H.P. on it's corner such a way that it's plan appears square draw the projections when (1) top view of diagonal makes and angle 20 to V.P. (2) Diagonal makes an angle 20 to V.P. anf find the inclination angle of surface with H.P. Solution:

Geometry Circular Plane Position in Drawing

Resting on H.P. on it's corner. Draw rhombus on corner Below “XY” Line.

Surface, plane, plate, lamina, H.P. Draw position 3 angle above “XY” diameter ,diagonal makes an Line. angle with (Position 3 angle)

Side, edge, diagonal, diameter V.P. Draw position 5 angle below “XY” makes and angle with (position Line. 5 angle)

In case of apparent angle (beta, alpha) If true length equals to plane length or elevation length no need to draw angle (alpha or beta) If top view or front view of any (side, edge, diagonal, diameter) makes and angle is given then no need to draw angle (alpha, beta). If True Length does not equal to plane length or elevation length in that case angle (alpha or beta) must to be drawn.

Figure 4.7 Problem No. 4.5

______Darshan Institute of Engineering & Technology, Rajkot 4.9

5. Projection & Section of Solid

Projection of Solid 5.1 Introduction A solid is a 3-D object having length, breadth and thickness and bounded by surfaces which may be either plane or curved, or combination of the two. Solids are classified under two main headings: 1. Polyhedron 2. Solids of revolution

A regular polyhedron is solid bounded only by plane surfaces (faces). Its faces are formed by regular polygons of same size and all dihedral angles are equal to one another. For example tetrahedron, cube etc. When faces of a polyhedron are not formed by equal identical faces, they may be classified into prisms and pyramids.

Figure 5.1 Types of solids Prism – a polyhedron formed by two equal parallel regular polygons, end faces connected by side faces which are either rectangles or parallelograms. Some definitions regarding prisms Base and lateral faces: When the prism is placed vertically on one of its end faces, the end face on which the prism rests is called the base. The vertical side faces are the lateral faces, as shown in Figure 5.2.

______Darshan Institute of Engineering & Technology, Rajkot 5.1 Engineering Graphics (2110013) 5. Projection & Section of Solid Base edge/Shorter edge: These are the sides of the end faces, as shown in figure 5.2. Axis: it is the imaginary line connecting the end faces is called axis and is shown in figure 5.2. Longer edge/lateral edges: These are the edges connecting the respective corners of the two end faces. The longer edge of a square prism is illustrated in figure 5.2. Right prism: A prism whose axis is perpendicular to its end face is called as a right prism .Prisms are named according to the shape of their end faces, i.e, if end faces are triangular, prism is called a triangular prism. Oblique prism: It is the prism in which the axis is inclined to its base.

Figure 5.2 Parameters of solids Pyramids Pyramid is a polyhedron formed by a plane surface as its base and a number of triangles as its side faces, all meeting at a point, called vertex or apex. Axis: The imaginary line connecting the apex and the center of the base. Inclined/slant faces: Inclined triangular side faces Inclined/slant/longer edges: The edges which connect the apex and the base corners. Right pyramid: When the axis of the pyramid is perpendicular to its base. Oblique pyramid: When the axis of the pyramid is inclined to its base.

5.2 Solids of Revolution When some of the plane figures are revolved about one of their sides – solids of revolution is generated some of the solids of revolution are: Cylinder: when a rectangle is revolved about one of its sides, the other parallel side generates a cylinder. Cone: when a right triangle is revolved about one of its sides, the hypotenuse of the right triangle generates a cone. Oblique cylinder: when a parallelogram is revolved about one of its sides, the other parallel side generates a cylinder. ______Darshan Institute of Engineering & Technology, Rajkot 5.2 5. Projection & Section of Solid Engineering Graphics (2110013) Sphere: when a semi-circle is revolved about one of its diameter, a sphere is generated. Frustums of solids: When prisms, pyramids, cylinders are cut by cutting planes parallel to the base the lower portion of the solids (without their top portions) are called frustum of these solids. Some examples are shown in figure 5.3.

Figure 5.3 Frustums of Solids

5.3 Projections of Solids Placed in Different Positions The solids may be placed on HP or VP in various positions. Axes of solid perpendicular to HP: A solid when placed on HP with its axis perpendicular to it, then it will have its base on HP. When the solid is placed with the base on HP position, in the top view, the base will be projected in its true shape. Hence, when the base of the solid is on HP, the top view is drawn first and then the front view and the side views are projected from it. There is only one position in which a cylinder or a cone may be placed with its base on HP.

Figure 5.4 Cylinder and cone resting on HP on its base For prism and pyramid initial position of solid depends upon whether it is resting on its side of the base or corner of the base on HP. Axes of solid perpendicular to VP: When a solid is placed with its axis perpendicular to VP, the base of the solid will always be perpendicular to HP and parallel to VP.

______Darshan Institute of Engineering & Technology, Rajkot 5.3 Engineering Graphics (2110013) 5. Projection & Section of Solid Hence in the front view, base will be projected in true shape. Therefore, when the axis of the solid is perpendicular to VP, the front view is drawn first and then the top and side views are drawn from it. There is only one position in which a cylinder or a cone may be placed with its base on VP.

Figure 5.5 Cylinder and cone resting on VP on its base For prism and pyramid initial position of solid depends upon whether it is resting on its side of the base or corner of the base on VP. Case 1: When the solid lies with an edge of base on HP: If the solid is required to be placed with an edge of the base on HP, then initially the solid has to be placed with its base on HP such that an edge of the base is perpendicular to VP, i.e., to XY line in top view preferably to lie on the right side.

______Darshan Institute of Engineering & Technology, Rajkot 5.4 5. Projection & Section of Solid Engineering Graphics (2110013)

Figure 5.6 Solid resting on HP on its base edge Case 2 : When the solid lies on one of its corners of the base on HP When a solid lies on one of its corners of the base on HP, then the two edges of the base containing the corner on which it lies make either equal inclinations or different inclination with HP. Corner of the base on HP with two base edges containing the corner on which it rests make equal inclinations with HP Initially the solid should be placed with its base on HP such that an imaginary line connecting the center of the base and one of its corners is parallel to VP, i.e. to XY line in the top view, and preferably to lie on the right side.

______Darshan Institute of Engineering & Technology, Rajkot 5.5 Engineering Graphics (2110013) 5. Projection & Section of Solid

Figure 5.7 Solid resting on HP on its base corner Case 3: When a pyramid lies on one of its triangular faces on HP If a pyramid has to be placed on one of its triangular faces on HP, then initially let the pyramid be placed with its base on HP such that the base edge containing that face is perpendicular to VP. (i.e. perpendicular to XY line).

______Darshan Institute of Engineering & Technology, Rajkot 5.6 5. Projection & Section of Solid Engineering Graphics (2110013)

Figure 5.8 Solid resting on one of its triangular face on HP CASE-4: When a pyramid lies on one of its slant edges on HP When a pyramid lies with one of its slant edges on HP, then two triangular faces containing the slant edge on which it rests make either equal inclinations or different inclinations with HP.

______Darshan Institute of Engineering & Technology, Rajkot 5.7 Engineering Graphics (2110013) 5. Projection & Section of Solid

Figure 5.9 Solid resting on one of its slant edge on HP

5.4 Methods of drawing the projections of solids Change of position method: the solids are placed first in the simple position and then tilted successively in two or three stages to obtain the final position. Steps to solve problems in solids: Step 1: assume solid standing on the plane with which it is making inclination. (If it is inclined to HP, assume it standing on HP) (If it is inclined to VP, assume it standing on VP) If standing on HP - It’s tv will be true shape of it’s base or top: If standing on VP - it’s fv will be true shape of it’s base or top. Begin with this view: It’s other view will be a rectangle (If solid is cylinder or one of the prisms): It’s other view will be a triangle (If solid is cone or one of the pyramids): Draw fv & tv of that solid in standing position: Step 2: considering solid’s inclination (axis position) draw it’s fv & tv. Step 3: in last step, considering remaining inclination, draw it’s final fv & tv.

______Darshan Institute of Engineering & Technology, Rajkot 5.8 5. Projection & Section of Solid Engineering Graphics (2110013) Problem 1: A right circular cone, 40 mm base diameter and 60 mm long axis is resting on HP on one point of base circle such that it’s axis makes 450 inclination with Hp and 400 inclination with VP. Draw it’s projections.

Figure 5.10 Solution of problem 1 Problem 2: A pentagonal pyramid of 35 mm base edge and 70 mm height is resting on the Horizontal Plane with one of its triangular surfaces perpendicular to the Horizontal Plane and parallel and nearer to Vertical Plane. Draw its projections.

Figure 5.11 Solution of problem 2

______Darshan Institute of Engineering & Technology, Rajkot 5.9 Engineering Graphics (2110013) 5. Projection & Section of Solid Problem 3: ABCD is tetrahedron of 60 mm long edge. The edge AB is in the H.P. The edge CD is inclined at an angle of 300 to the H.P. and 450 to the V.P. Draw the projections of the tetrahedron.

Figure 5.12 Solution of problem 3 Problem 4: A cube of 50 mm long edges is so placed on HP on one corner that a body diagonal is parallel to Hp and 450 inclined to VP. Draw it’s projections.

Figure 5.13 Solution of problem 4 FREELY SUSPENDED SOLIDS:

______Darshan Institute of Engineering & Technology, Rajkot 5.10 5. Projection & Section of Solid Engineering Graphics (2110013) Positions of CG, on axis, from base, for different solids are shown below.

Figure 5.14 Position of CG for prism and pyramid Problem 5: A square pyramid, side of base 50 mm and height 64 mm, is freely suspended from one of the corners of the base. Draw its projection when vertical plane containing the axis makes an angle of 45o to VP.

Figure 5.15 Solution of problem 5 Section of solid 5.5 Introduction An object (here a solid ) is cut by some imaginary cutting plane to understand internal details of that object.

______Darshan Institute of Engineering & Technology, Rajkot 5.11 Engineering Graphics (2110013) 5. Projection & Section of Solid The action of cutting is called sectioning a solid. The plane of cutting is called section plane.

5.6 Types of Cutting Plane and its Position Horizontal Cutting Plane This type of Cutting plane is perpendicular to Vertical Plane and parallel to the horizontal plane. Cutting plane cuts – Elevation/Front View in First Angle Position Cutting Line is Drawn Parallel to XY Line. Section appears in plan/top view and sectional plan shows true shape. Probable conditions to set cutting plane from problem data: 3. Distance on axis either from apex (pyramid/cylinder) or from top surface. (prism/pyramid) 4. Distance on axis from base.

Figure 5.3(i) Horizontal Cutting Plane Vertical Cutting Plane This type of Cutting plane is perpendicular to Horizontal Plane and parallel to the vertical plane. Cutting plane cuts – plan/top View in First Angle Position. Cutting Line is Drawn Parallel to XY Line. Section appears in Elevation/Front view and sectional Elevation shows true shape.

Probable Conditions to set Cutting plane From Problem Data: 1. Distance in front of axis from apex (pyramid) or from top surface (prism).

Figure 5.3 (ii) Vertical Cutting Plane

______Darshan Institute of Engineering & Technology, Rajkot 5.12 5. Projection & Section of Solid Engineering Graphics (2110013) Cutting plane perpendicular to both H.P/V.P and parallel to profile plane This type of Cutting plane is perpendicular to both H.P & V.P and parallel to Profile Plane. Cutting plane cuts – plan/top View and Elevation/front view both in First Angle Position. Cutting Line is Drawn Perpendicular to XY Line. Section appears in Side view and sectional side view shows true shape. Probable Conditions To set Cutting plane From Problem Data: 1. Distance on base from axis. 2. Distance on base from any one base corner in prism or pyramid.

Figure 5.3 (iii) Cutting Plane Perpendicular to both H.P. and V.P. Auxiliary Inclined Cutting Plane. (A.I.P) This type of Cutting plane is Inclined to H.P. and perpendicular to vertical plane. Cutting plane cuts – Elevation/ Front View Cutting Line is Drawn Inclined and above to XY Line. Section appears in Plan/Top View. True Shape is drawn separately parallel to cutting plane line. Probable Conditions To set Cutting plane From Problem Data: 1. Angle with H.P. and bisecting the axis. 2. Angle with H.P and Cutting plane passing distance on axis either from base or from apex or top surface. 3. Cutting plane passing through any corner of the base. Figure 2.5 Auxiliary Inclined Cutting Plane AUXILIARY VERTICAL CUTTING PLANE (A.V.P) This type of Cutting plane is Inclined to V.P. and perpendicular to H.P. ______Darshan Institute of Engineering & Technology, Rajkot 5.13 Engineering Graphics (2110013) 5. Projection & Section of Solid Cutting plane cuts – Plan/ Top View Cutting Line is Drawn Inclined and below to XY Line. Section appears in Elevation/Front View. True Shape is drawn separately parallel to cutting plane line. Probable Conditions To set Cutting plane From Problem Data: 1. Angle with V.P. Distance from axis in top view. 2. Passing through any corner in top view.

Figure 5.3 (iv) Auxiliary Vertical Cutting Plane Problem 5.1: A Hexagonal prism with 30 mm side and 60 mm height resting on its base , on its corner on H.P. A cutting plane cuts the prism by making 45 with H.P. and passing 40 mm from base. Draw front view, sectional top view, sectional side view and true shape of the section.

______Darshan Institute of Engineering & Technology, Rajkot 5.14 5. Projection & Section of Solid Engineering Graphics (2110013) Figure 5.4 Hexagonal Prism cut by (A.I.P) Solution: 1. Draw the line XY. 2. Draw the top view as a Hexagon on corner and name its corners. 3. Draw projectors from each corner of the top view through XY. 4. Draw the front view as shown in the figure and name its corners on base as well as top. 5. Draw the section plane in the front view at 40 mm from the base and name the sectional points. 6. Draw projectors from each sectional point in front view so that they cut the corresponding edges or corner in the top view. 7. Name these points and join them. 8. Draw the hatching lines to get sectional top view. 9. For side view take projection of corner points from top view up to xy line then draw perpendicular lines to XY from projected points and complete the side view as shown in figure. 10. Take projection from cutting points in front view and extend where it’s intersect to corresponding points in side view as shown in figure and give the points. 11. Joint all the points and draw hatching line in it.

True Shape of the section 1. Draw parallel line x1y1 to cutting plane line. 2. Draw projectors from each sectional point in the front view perpendicular and through X1 Y1. 3. Transfer the distances, from XY, of the sectional points in the top view to the corresponding projectors through X1Y1, measuring from X1Y1 in each case. 4. Join these points as shown and draw the hatching lines to get the true shape of the section.

______Darshan Institute of Engineering & Technology, Rajkot 5.15 Engineering Graphics (2110013) 5. Projection & Section of Solid Problem 5.2: A cone 60 mm diameter and 75 mm axis height resting on its base on H.P. A cutting plane cuts the cone making an angle 60 passing 8 mm away from its axis. Draw its top view, sectional front view and true shape of the section. Solution:

Figure 5.5 Cone cut by (A.V.P)

______Darshan Institute of Engineering & Technology, Rajkot 5.16 5. Projection & Section of Solid Engineering Graphics (2110013) Problem 5.3: A Hexagonal Prism side 25 mm and axis height 65 mm has a face on H.P. and axis parallel to V.P. It is cut by cutting plane perpendicular to H.P. and makes an angle 45 with V.P. and passing through 20 mm away from one end of the hexagonal axis. Draw sectional front view, top view and true shape of the section. Solution:

Figure 5.6 Hexagonal Prism Cut by A.V.P Combine Projection and Section of solid Problem

______Darshan Institute of Engineering & Technology, Rajkot 5.17 Engineering Graphics (2110013) 5. Projection & Section of Solid Problem 5.4: A Cylinder with 50 mm diameter and 70 mm height resting on its base, on H.P. A cutting plane cuts the cylinder by making 45 with H.P. and passing 45 mm from base. Draw front view, sectional top view, sectional side view and true shape of the section. Solution:

Figure 5.7 Cylinder Cut by A.I.P.

______Darshan Institute of Engineering & Technology, Rajkot 5.18 5. Projection & Section of Solid Engineering Graphics (2110013) Problem 5.5: A Tetrahedron resting on H.P. with one face perpendicular to V.P. such a way that true shape of section is isosceles triangle with base 40 mm (B) and altitude 28 mm (H). Determine angle which section plane makes with H.P. Also draw front view, sectional top view and true shape of the section. Solution:

Figure 5.8 Tetrahedron Reverse Section of solid Problem

______Darshan Institute of Engineering & Technology, Rajkot 5.19

6. Orthographic Projections

6.1 Projection Theory In engineering, 3-dimensional objects and structures are represented graphically on a 2- dimensional media. The act of obtaining the image of an object is termed “projection”. The image obtained by projection is known as a “view”. A simple projection system is shown in figure 6.1. All projection theory are based on two variables: 1. Line of sight 2. Plane of projection.

Line of Sight A line of sight is an imaginary ray of light between an observer’s eye and an object. Plane of Projection A plane of projection (i.e., an image or picture plane) is an imaginary flat plane upon which the image created by the line of sight is projected. The image is produced by connecting the points where the lines of sight pierce the projection plane. In effect, 3-D object is transformed into a 2-D representation, also called projections. The paper or computer screen on which a drawing is created is a plane of projection.

Figure 6.1 A Simple Projection System

6.2 Projection Methods Projection methods are very important techniques in engineering drawing. Two projection methods used are: 1. Perspective and 2. Parallel

Perspective Projection In perspective projection, all lines of sight start at a single point and is schematically shown in figure 6.2. ______Darshan Institute of Engineering & Technology, Rajkot 6.1 Engineering Graphics (2110013) 6. Orthographic Projections

Figure 6.2 A Schematic Representation of a Perspective Projection Parallel Projection In parallel projection, all lines of sight are parallel and is schematically represented in figure 6.3. The observer is assumed to be stationed at infinite distance from the object.

Figure 6.3 A Schematic Representation of a Parallel Projection ______Darshan Institute of Engineering & Technology, Rajkot 6.2 6. Orthographic Projection Engineering Graphics (2110013) 6.3 Difference between Parallel and Perspective Projection Parallel Projection Perspective Projection Distance from the observer to the object is Distance from the observer to the object is infinite. finite Projection lines are parallel. Projection lines are not parallel. Object is positioned at infinity. Object is viewed from a single point. Easier to draw. Difficult to draw.

6.4 Orthographic Projection Orthographic projection is a parallel projection technique in which the plane of projection is perpendicular to the parallel line of sight. Orthographic projection technique can produce either pictorial drawings that show all three dimensions of an object in one view or multi-views that show only two dimensions of an object in a single view. These views are shown in figure 6.4.

Figure 6.4 Orthographic Projections of a Solid Showing Isometric, Oblique and Multi-view Drawings

6.5 Views of Objects Any object can be viewed from six mutually perpendicular directions. These are called the six principal views.

Figure 6.5 Six views of a House You can think of the six views as what an observer would see by moving around the object.

______Darshan Institute of Engineering & Technology, Rajkot 6.3 Engineering Graphics (2110013) 6. Orthographic Projections As shown in Figure 6.5, the observer can walk around a house and view its front, sides, and rear. You can imagine the top view as seen by an observer from an airplane and the bottom, or "worm's- eye view," as seen from underneath. The term "plan" may also be used for the top view. The term "elevation" is used for all views showing the height of the building. These terms are regularly used in architectural drawing and occasionally in other fields. To make drawings easier to read, the views are arranged on the paper in a standard way. The views in Figure 6.5 show the American National Standard arrangement. The top, front, and bottom views align vertically. The rear, left-side, front, and right-side views align horizontally. To draw a view out of place is a serious error and is generally regarded as one of the worst mistakes in drawing.

6.6 Multi-view Projection Planes

Figure 6.6 The principal Projection planes and quadrants used in orthographic projection In the multiview projections views are defined according to the positions of the planes of projection with respect to the object. 6.6.1 Frontal Plane of Projection

The front view of an object shows the width and height dimensions. The views in figure 6.7 are front views. The frontal plane of projection is the plane onto which the front view of a multiview drawing is projected.

______Darshan Institute of Engineering & Technology, Rajkot 6.4 6. Orthographic Projection Engineering Graphics (2110013)

Figure 6.7 Front Multiview Drawing 6.6.2 Horizontal Plane of Projection

The top view of an object shows the width and depth dimensions. The top view is projected on to the horizontal plane of projection, which is a plane suspended above and parallel to the top of the object.

Figure 6.8 Horizontal Plane of Projection 6.6.3 Profile Plane of Projection

The side view of an object shows the depth and height dimensions. In multiview drawings, the right side view is used. The right side view is projected onto the right profile plane of projection, which is a plane that is parallel to the right side of the object.

______Darshan Institute of Engineering & Technology, Rajkot 6.5 Engineering Graphics (2110013) 6. Orthographic Projections

Figure 6.9 Profile Plane of Projection

6.7 Orientation of Views from Projection Planes Multi-view drawings gives the complete description of an object. For conveying the complete information, all the three views, i.e., the Front view, Top view and side view of the object is required. To obtain all the technical information, at least two out of the three views are required. It is also necessary to position the three views in a particular order. Top view is always positioned and aligned with the front view, and side view is always positioned to the side of the Front view and aligned with the front view. The positions of each view is shown in figure 6.11. Depending on whether 1st angle or 3rd angle projection techniques are used, the top view and Front view will be interchanged. Also the position of the side view will be either towards the Right or left of the Front view.

Figure 6.10 Relative Positions and Alignment of the Views in a Multi-View Drawing

______Darshan Institute of Engineering & Technology, Rajkot 6.6 6. Orthographic Projection Engineering Graphics (2110013) 6.8 The Glass Box Approach The plane of projection can be oriented to produce an infinite number of views of an object. However, some views are more important than others. These principal views are the six mutually perpendicular views that are produced by six mutually perpendicular planes of projection. Imagine suspending an object in a glass box with major surfaces of the object positioned so that they are parallel to the sides of the box, six sides of the box become projection planes, showing the six views – front, top, left, right, bottom and rear as shown in figure 6.11.

Figure 6.11 Glass Box Producing Six Principal Views The glass box is now slowly unfolded as shown in figure 6.12. After complete unfolding of the box on to a single plane, we get the six views of the object in a single plane as shown in figure 6.13. The top, front and bottom views are all aligned vertically and share the same width dimension whereas the rear, left side, front and right side views are all aligned horizontally and share the same height dimension.

______Darshan Institute of Engineering & Technology, Rajkot 6.7 Engineering Graphics (2110013) 6. Orthographic Projections

Figure 6.12 Illustration of the Views after the Box Has Been Partially Unfolded So we can think of the system of projecting views as unfolding a glass box made from the viewing planes. There are two main systems used for projecting and unfolding the views: third-angle projection, which is used in the United States, Canada and some other countries, and first-angle projection, which is primarily used in Europe and Asia. Difficulty in interpreting the drawing and manufacturing errors can result when a first-angle drawing is confused with a third-angle drawing.

______Darshan Institute of Engineering & Technology, Rajkot 6.8 6. Orthographic Projection Engineering Graphics (2110013)

Figure 6.13 The Views of the Object with Their Relative Positions after the Box Has Been Unfolded Completely on to a Single Plane

6.9 Third Angle Projection Figure 6.14 shows the concept of third-angle orthographic projection. If the object is placed below the horizontal plane and behind the vertical plane, the object is in the third angle. Placing the object in the third quadrant puts the projection planes between the viewer and the object.

Figure 6.14 Third Angle Projection Technique ______Darshan Institute of Engineering & Technology, Rajkot 6.9 Engineering Graphics (2110013) 6. Orthographic Projections 6.10 First Angle Projection If the object is placed above the horizontal plane and in front of the vertical plane, the object is in the first angle. In first-angle projection the observer looks through the object to the planes of projection. The right- side view is still obtained by looking toward the right side of the object, the front by looking toward the front, and the top by looking down toward the top; but the views are projected from the object onto a plane in each case.

Figure 6.15 First Angle Projection Technique The biggest difference between third-angle projection and first-angle projection is in how the planes of the glass box are unfolded, as shown in Figure 6.15. In first-angle projection, the right-side view is to the left of the front view, and the top view is below the front view, as shown.

6.11 Symbolic Representation of First Angle and Third Angle Projection System

Figure 6.16 Symbols of Third and First Angle Projection System ______Darshan Institute of Engineering & Technology, Rajkot 6.10 6. Orthographic Projection Engineering Graphics (2110013) To avoid misunderstanding, international projection symbols have been developed to distinguish between first-angle and third-angle projections on drawings. The symbol in Figure 6.16 shows two views of a truncated cone. You can examine the arrangement of the views in the symbol to determine whether first-or third-angle projection was used. On international drawings you should be sure to include this symbol.

6.12 Precedence of Line Visible lines, hidden lines, and centerlines (which are used to show the axis of symmetry for contoured shapes, like holes) often coincide on a drawing. There are rules for deciding which line to show. A visible line always takes precedence over and covers up a centerline or a hidden line when they coincide in a view. A hidden line takes precedence over a centerline.

Figure 6.17 Precedence of Line

______Darshan Institute of Engineering & Technology, Rajkot 6.11

7. Isometric Projections

7.1 Introduction In Multi-View (or multiple plane) Orthographic Projections, a minimum of two views are necessary to give information about the three dimensions of an object. For that minimum two Reference planes are required to draw two views. When multi-view orthographic projections are given, a good imagination is needed to visualize the object in 3D space. Therefore, it is useful only for technically trained person. Single-View Orthographic Projections (or axonometric projections) can easily be understood even by person without technical training because such a drawing shows the several faces of an object in a single view. But, Single-View Orthographic Projections shows the appearance of an object but not the true shape and size of its faces. If Axonometric Plane is equally inclined to all the three principal planes then that plane is known as the isometric plan and orthographic projection on it is known as isometric projection.

Figure 7.1 Illustrates the four principle projection techniques

______Darshan Institute of Engineering & Technology, Rajkot 7.1 Engineering Graphics (2110013) 7. Isometric Projections 7.2 Axonometric Projections In axonometric projections, only one view showing all the three dimensions of an object is drawn on a POP (plane of Projection). The orientation of the object is kept in such a way that its three principle edges (mutually perpendicular edges) will remain inclined to the POP (Projection of Plane). Depending upon the inclinations of principle edges of the object with the POP, axonometric projections are classified as: 1. Isometric  If the object is so placed that its mutually perpendicular edges makes equal angles with the plane of projection that the edges OA = OB = OC and axonometric angles α, β and  are equal then the projection method is known as isometric projection method. 2. Diametric  If two mutually perpendicular edges of the object are equally inclined and third edge is inclined at other angle to the plane of projection means OA = OB and axonometric angle between this two edges is also equal, then the projection method is known as Diametric Projection method. 3. Trimetric.  If the object is place such that its perpendicular edges have not equal angle with the plane of projection that means OA ≠ OB ≠ OC then the projection method is known as Trimetric Projection method.

Figure 7.2 axonometric projections 7.3 TERMINOLOGY 1. Isometric axes:  The three lines CB, CD and CC, meeting at point C and inclined at an angle of 120° with each other are called isometric axes. 2. Isometric lines:  The lines parallel to the isometric axes are called isometric lines. Here lines AB, BF, FG, GH, DH and AD are isometric lines. 3. Non-isometric lines:  The lines which are not parallel to isometric axes are known as non-isometric lines. For example, diagonals BD, AC, CF, BG, etc., are non-isometric lines. 4. Isometric plane:  A plane representing any face of the cube as well as other plane parallel to it is called an isometric plane. For example, ABCD, BCGF, CGHD, etc., are isometric planes. ______Darshan Institute of Engineering & Technology, Rajkot 7.2 7. Isometric Projection Engineering Graphics (2110013) 5. Isometric scale:  It is the scale which is used to convert the true length into isometric length. Mathematically, Isometric length = 0.816 x True length.

Figure 7.3 Principle of Isometric Projection

7.4 Isometric Scale: For isometric projection an isometric scale is required. In isometric projection, the length of each isometric line is 0.815 times the normal line which is given in orthographic projection. To draw the isometric scale use the following procedure. . Draw the horizontal line OA. . Draw the line OC-100 mm at 45° from O. . Draw another line OB at 30° from O to measure the isometric scale. . Divide line OC in 10 equal parts. Mark them 10, 20, 30…. 100. . From each point draw the vertical line to cut the line OB. Mark them 10', 20', 30'….100'. . The scale obtained on OC is Normal Scale. . The scale obtained on OB is Isometric Scale.

Figure 7.4 Isometric scale

______Darshan Institute of Engineering & Technology, Rajkot 7.3 Engineering Graphics (2110013) 7. Isometric Projections OC = Normal Length OB = Isometric Length OA cos 45° = ∴ OA = cos 45° × OC … …. (1) OC OA cos 30° = ∴ OA = cos 30° × OB … … . (2) OB Compare the equation (1) & (2)…. cos 45° × OC = cos 30° × OB cos 45° 9 OB = OC × = × OC = 0.816 × OC cos 30° 11  Isometric length = 0.816 x Normal length

7.5 Isometric View or Isometric Drawing The pictorial view drawn with normal scale is called the Isometric View or Isometric Drawing.

7.6 Isometric Projection The pictorial view drawn with the help of isometric scale is called the Isometric Projection.

Figure 7.5 Comparison between Isometric View and Projection

7.7 Isometric Views of Standard Shapes

1. Square:

______Darshan Institute of Engineering & Technology, Rajkot 7.4 7. Isometric Projection Engineering Graphics (2110013) 2. Rectangle:

3. Triangle:

4. Hexagon:

5. Angle in Isometric: Any angle in orthographic view is never seen as it is in isometric. It is obtained in isometric by locating and joining the end points of the two lines making the angle.

______Darshan Institute of Engineering & Technology, Rajkot 7.5 Engineering Graphics (2110013) 7. Isometric Projections 6. Circle: a) Method of Four Centres:

b) Method of Points:

7. Semicircle:

8. Quarter of a Circle:

______Darshan Institute of Engineering & Technology, Rajkot 7.6 7. Isometric Projection Engineering Graphics (2110013) 7.8 Isometric Solids 1. Cylinder

Figure 7.6 Isometric View of cylinder 2. Sphere:

Figure 7.7 Isometric Projection & View of Sphere

The isometric projection of a sphere appears same in size as that of the actual sphere while the isometric view of a sphere is seen larger in size than the actual size of the sphere.

______Darshan Institute of Engineering & Technology, Rajkot 7.7 Engineering Graphics (2110013) 7. Isometric Projections Problem 7.1 PROJECTIONS OF FRUSTOM OF PENTAGONAL PYRAMID ARE GIVEN.DRAW IT’S ISOMETRIC VIEW.

Figure 7.8 Problem 7.1

First draw isometric of its base. Then draw same shape as top, 60 mm above the base pentagon center. Then reduce the top to 20 mm sides and join with the proper base corners.

______Darshan Institute of Engineering & Technology, Rajkot 7.8 7. Isometric Projection Engineering Graphics (2110013)

Problem 7.2 Draw the isometric view from the following orthographic views.

Figure 7.9 Problem 7.2 Step 1. Determine the desired view of the object. Here the object will be viewed from above (regular isometric). The isometric axes is then drawn as shown in step-1. Step 2. Construct the front isometric plane using W and H dimensions. Width dimensions are drawn along 30 lines from the horizontal. Height dimensions are drawn as vertical lines. Step 3. Construct the top isometric plane using the W and D dimensions. Both W and D dimensions are drawn along 30 lines from the horizontal. Step 4. Construct the right side isometric plane using D and H dimensions. Depth dimensions are drawn along 30 lines and height dimensions are drawn as vertical lines. Step 5. Transfer some distances for the various features from the multi-view drawing to the isometric lines that make up the isometric rectangle on the front and top planes of the isometric box. e.g. distance A is measured from the multi-view drawing./ It is then transferred along the width line in the front plane of the isometric rectangle. Draw the details of the block by drawing isometric lines between the points transferred from the multi-view drawing. e.g., the notch is taken out of the block by locating its position on the front and the top planes of the isometric box.

______Darshan Institute of Engineering & Technology, Rajkot 7.9 Engineering Graphics (2110013) 7. Isometric Projections Step 6. Transfer the remaining features from the multi-view drawing to the isometric drawing. Block in the details by connecting endpoints of the measurements taken from the multi-view drawing. Step 7. Darken all visible lines and erase or lighten the construction lines Problem 7.3 Draw the isometric view from the following orthographic views.

Figure 7.10 Problem 7.3

______Darshan Institute of Engineering & Technology, Rajkot 7.10