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BT-105 · Engineering Graphics/Quick Revision Short Notes

Engineering Graphics (BT-105) - Unit 1 Short Notes

How unit 1 is examined

This unit covers drawing basics, instruments and lettering, then the constructions that carry the marks: scales (diagonal and vernier) and cycloidal curves (both high weight), with conics at medium weight.

Principles of Engineering Graphics and their significance

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Definition. <mark>Engineering graphics is the universal language of engineers, in which shape, size and construction of an object are communicated exactly through drawings.</mark>

Key points.

  1. A drawing states shape, size, material and finish without ambiguity, so anyone can make the part.
  2. It follows standard conventions (BIS/ISO) for line types, dimensioning, projection and symbols.
  3. It is independent of spoken language, so it works across countries and trades.
  4. Its principles are accuracy, clarity, uniformity and neatness.

Usage of drawing instruments

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Definition. <mark>Drawing instruments are the tools used to draw lines, circles, angles and curves accurately on a sheet.</mark>

Key points.

  1. The drawing board and T-square (or mini-drafter) give the horizontal reference, and set squares of 45 degrees and 30-60 degrees slide on it to draw verticals and inclined lines.
  2. The compass draws circles and arcs, and the divider transfers or divides lengths without marking the sheet.
  3. The protractor measures angles, and the scale (ruler) measures lengths.
  4. Pencils are graded: H or 2H for construction and thin lines, HB for lettering, and B for thick outlines.
  5. French curves draw smooth curves through plotted points.

Lettering

<span style="display:inline-block;padding:.16em .6em;border:1.5px solid currentColor;border-radius:999px;font-size:.68em;font-weight:700;letter-spacing:.06em;text-transform:uppercase;opacity:.75">Not asked since 2022</span>

Definition. <mark>Lettering is the writing of titles, notes and dimensions on a drawing in a uniform, legible single-stroke style.</mark>

Key points.

  1. Letters are single-stroke, either vertical or inclined at 75 degrees to the horizontal, as per BIS.
  2. Height of capitals is 2.5, 3.5, 5, 7, 10, 14 or 20 mm, and stroke thickness is about one-tenth of the height.
  3. Spacing between letters is about one-fifth of the height and between words about the letter height.
  4. Guide lines are drawn lightly with a hard pencil so that the height stays uniform.

Conic sections including the Rectangular Hyperbola (General method only)

<span style="display:inline-block;padding:.16em .6em;border:1.5px solid currentColor;border-radius:999px;font-size:.68em;font-weight:700;letter-spacing:.06em;text-transform:uppercase;opacity:.75">Medium weight</span>

Definition. <mark>A conic is the locus of a point that moves so that the ratio of its distance from a fixed point (focus) to its distance from a fixed line (directrix) is constant; this constant is the eccentricity $e$.</mark>

Key points.

  1. If $e<1$ the conic is an ellipse, if $e=1$ a parabola, and if $e>1$ a hyperbola; a rectangular hyperbola has $e=\sqrt2$.
  2. The vertex $V$ lies on the axis, with $VF/VC=e$, where $C$ is the point where the axis meets the directrix.
  3. Since $CF=d$, we get $VF=\dfrac{e\,d}{1+e}$ and $VC=\dfrac{d}{1+e}$; for a parabola $V$ is the midpoint of $CF$.
  4. Any point $P$ on the curve satisfies $PF/PD=e$, where $PD$ is its perpendicular distance from the directrix.
  5. The ellipse has a major axis $2a=\dfrac{2ed}{1-e^2}$, a minor axis $2b=2a\sqrt{1-e^2}$ and foci $2ae$ apart.
  6. Tangent at $P$: join $PF$, draw a line through $F$ perpendicular to $PF$ to meet the directrix at $T$; then $TP$ is the tangent and the normal is perpendicular to it at $P$.

Steps.

Step 1: Draw directrix DD' and the axis CF perpendicular to it; mark F at distance d from C.
Step 2: Divide CF in the ratio e:1 to get V (VF:VC = e:1); parabola V = midpoint of CF.
Step 3: Draw ordinates through points 1, 2, 3 on the axis beyond V.
Step 4: Distance of a point from D'D = its axis distance from C; take radius = e x that distance, and cut the ordinate from F.
Step 5: Join the points smoothly, mark the axes, and draw the tangent and normal at P.

Example. Ellipse with $d=50$, $e=2/3$: $VF=20$, $VC=30$ mm, so major axis $=120$ mm, minor axis $\approx89.4$ mm, foci $80$ mm apart. For $d=25$, $e=3/4$: $VF\approx10.7$ mm, major $\approx85.7$ mm, minor $\approx56.7$ mm, foci $\approx64.3$ mm apart.

Answer frame. Open with the focus-directrix definition and the value of $e$ that names the curve; draw the directrix, axis, focus, vertex, plotted points, smooth curve, and the tangent and normal at $P$; develop the steps in order; close with the curve name, and axes lengths for an ellipse.

Asked: [7 marks] (Nov 2022, Dec 2023) Construct a conic when the distance of its focus from its directrix is 50 mm and eccentricity 2/3. Name the curve; mark its major and minor axes. Draw a tangent at any point P. Also: eccentricity 3/4, focus 25 mm from directrix; measure the axes and distance between the foci. Asked: [7 marks] (Jun 2024) Construct a parabola when the distance between focus and directrix is 60 mm. Draw a tangent and normal at any point.

Pitfall: Dividing CF in the wrong ratio; the vertex must satisfy $VF:VC=e:1$, so for $e=2/3$ it is 2:3, not 3:2.

Cycloid, Epicycloid, Hypocycloid and Involute

<span style="display:inline-block;padding:.16em .6em;border:1.5px solid currentColor;border-radius:999px;font-size:.68em;font-weight:700;letter-spacing:.06em;text-transform:uppercase;opacity:.75">High weight</span>

Definition. <mark>A cycloid is the locus of a point on the circumference of a circle (generating circle) that rolls without slipping along a straight line; an epicycloid rolls on the outside, and a hypocycloid on the inside, of another circle (directing circle); an involute is the locus of the free end of a taut string unwound from a circle.</mark>

Key points.

  1. In a cycloid the base line length equals the circumference $\pi D$ for one revolution, and the curve touches the base line at its start and end.
  2. In an epicycloid the generating circle rolls on the outside of the directing circle, and the centre travels on an arc of radius $R+r$.
  3. In a hypocycloid the generating circle rolls inside, and the centre travels on an arc of radius $R-r$.
  4. For both, the directing arc for one revolution subtends $\theta=\dfrac{r}{R}\times360^\circ$ at the centre $O$.
  5. Divide the generating circle and the arc $\theta$ into the same number of equal parts (8 or 12), and draw arcs of radius $r$ from the centre positions to cut the parallel locus lines.
  6. An involute of a circle: unwound length equals the arc length rolled, so the tangent at each division point is drawn with length equal to the arc from the start point.
  7. The normal at any point passes through the instantaneous contact point (cycloids) or is tangent to the circle (involute).

Steps (cycloid, $D=30$).

Step 1: Draw circle of D=30 and divide into 12 equal parts; draw the base line = pi x D = 94.25 mm.
Step 2: Divide base into 12 equal parts and erect verticals to the locus line through the centre.
Step 3: Draw horizontals from circle points 1..12 to cut these locus lines.
Step 4: Join the cut points smoothly. Result: cycloid.

Example. Bicycle wheel $D=650$ over culvert $R=1950$: $r=325$, so $\theta=\tfrac{325}{1950}\times360^\circ=60^\circ$, centre arc radius $=1950+325=2275$ mm (epicycloid; scale 1:10 or 1:20).

Answer frame. Open by naming the curve and its generation; draw the directing arc or line, centre locus arc, and the smooth curve with the tangent and normal; give $\theta$ and the divisions; close with the curve name.

Asked: [14 marks] (Dec 2023, Dec 2024) Bicycle wheel diameter 650 mm passes over a segmental culvert of radius 1950 mm; draw locus of P for one revolution. Also: epicycloid with $r=20$ mm, $R=72$ mm ($\theta=100^\circ$), normal and tangent at a point. Asked: [7 marks] (Jun 2023, Dec 2024) String unwound 122 mm from a drum of diameter 30 mm ($122>\pi D=94.25$, about 1.3 turns, so 466 degrees); locus of the free end. Also: thread unwinding from a drum of radius 60 mm through $180^\circ$ (length $\pi r=188.5$ mm). Asked: [7 marks] (Jun 2022) Hypocycloid, rolling circle 50 mm, directing circle 175 mm ($\theta=\tfrac{25}{87.5}\times360^\circ=102.86^\circ$); tangent at a point 50 mm from centre. Asked: [7 marks] (Jun 2025) A coin of 30 mm diameter rolls on a straight line; plot and name the locus for one revolution (cycloid). Asked: [14 marks] (Jun 2025) Short notes on any three: Epicycloid; Traces of a line; Tool bar; View ports.

Pitfall: Using diameter instead of radius in $\theta=\tfrac{r}{R}\times360^\circ$; 175 mm is a diameter, so $R=87.5$.

Scales – Plain, Diagonal and Vernier Scales

<span style="display:inline-block;padding:.16em .6em;border:1.5px solid currentColor;border-radius:999px;font-size:.68em;font-weight:700;letter-spacing:.06em;text-transform:uppercase;opacity:.75">High weight</span>

Definition. ==A scale is a ruler-like graduated line used to represent an actual length on a drawing at a reduced, full or enlarged size; the Representative Fraction is $RF=\dfrac{\text{drawing length}}{\text{actual length}}$ in the same units.==

Key points.

  1. Scales are full size ($RF=1$), reducing ($RF<1$) or enlarging ($RF>1$).
  2. Plain scale reads two units (e.g. m and dm); diagonal scale reads three (e.g. km, hm, dam) using diagonals; vernier scale reads three units with a sliding vernier, and has smaller least count.
  3. Length of scale $LOS=RF\times\text{maximum length}$.
  4. For an area ratio, $RF=\sqrt{\text{map area}/\text{actual area}}$.
  5. Diagonal scale: least count $=\dfrac{\text{main division}}{10}\times\dfrac1{10}$.
  6. Vernier: $LC=1\,\text{main division}-1\,\text{vernier division}$; a forward vernier has 10 divisions equal to 9 main divisions.
  7. Scale of chords draws angles: chord of $\theta=2R\sin(\theta/2)$.

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Point Diagonal scale Vernier scale
Principle Similar triangles on diagonals Difference of main and vernier division
Construction Rectangle grid with 10 vertical parts Main scale plus sliding vernier
Least count Small, mm-level Smaller, finest
Use Map distances Precision instruments

Worked results.

  • Map 1 (RF $\tfrac1{40000}$, 6 km): $LOS=15$ cm, 6 divisions of 2.5 cm, LC 0.01 km; mark 3.34 and 0.59 km.
  • Map 2 ($\tfrac{2.5}{100}$, 6 m): $RF=\tfrac1{40}$, $LOS=15$ cm; show 4.33 m.
  • Tank: $\sqrt[3]{216/27}=2$, so 6 cm : 3 m, $RF=\tfrac1{50}$; 5 m $=10$ cm; 3.95 and 0.042 m.
  • $144\text{ cm}^2 : 36\text{ km}^2$: $RF=\tfrac{12}{6\times10^5}=\tfrac1{50000}$, $LOS=20$ cm, 10 parts of 2 cm; 7.56 km.
  • 352 km : 70.4 mm: $RF=\tfrac1{5\times10^6}$, $LOS=18$ cm, 9 parts of 100 km; 649 km.
  • 100 km : 30 cm: $RF=\tfrac1{333333}$; 30 cm, 10 parts of 10 km; 46.8 and 32.4 km.
  • 0.45 ha : 5 cm²: $RF=\sqrt{5/4.5\times10^7}=\tfrac1{3000}$, $LOS=13.33$ cm, 4 parts of 100 m.

Answer frame. Open with the definition and RF; find RF and LOS; draw the scale with main divisions, subdivisions and the height divided into 10; show the required distance with a bold line; close with the RF written under the scale.

Asked: [7 marks] (Jun 2022, Jun 2024, Jun 2025) Forward reading vernier scale, map 1:40000, up to 6 km, mark 3.34 km and 0.59 km. Also: 1 m = 2.5 cm, up to 6 m, show 4.33 m. Also: tank 27 m³ shown as 216 cm³, up to 5 m, show 3.95 m and 0.042 m. Asked: [7 marks] (Nov 2022, Dec 2023, Dec 2024) 144 cm² map = 36 km² field; find RF, diagonal scale to 10 km, indicate 7 km 5 hm 6 dam. Also: 352 km = 70.4 mm, up to 900 km, show 649 km. Also: 100 km = 30 cm, show 46.8 km and 32.4 km. Asked: [7 marks] (Jun 2023) Plot 0.45 hectare shown by 5 cm²; find RF; scale to single metre, up to 400 m. Asked: [7 marks] (Jun 2023, Jun 2024) What is a scale? Classify its types; how is RF calculated; main uses. Asked: [7 marks] (Jun 2025) Compare diagonal scale and Vernier scale in construction and precision; give a use of each. Asked: [7 marks] (Dec 2023) Draw angles of $45^\circ$ and $135^\circ$ with the scale of chords. Asked: [14 marks] (Jun 2022) Short notes: Types of scales; Editing commands in CAD; Orthographic projection.

Pitfall: Mixing units when finding RF; convert both lengths to the same unit first (1 km = $10^5$ cm).

Last-minute revision

  • $RF=\text{drawing size}/\text{actual size}$, same units; $LOS=RF\times\text{maximum length}$.
  • Area-based RF: take the square root of the area ratio; volume-based: the cube root.
  • Diagonal scale: 10 main divisions, first divided into 10, height into 10.
  • Vernier: 10 vernier divisions equal 9 main divisions.
  • $\theta=\tfrac{r}{R}\times360^\circ$ for epicycloid and hypocycloid; convert diameter to radius first.
  • Epicycloid centre arc: $R+r$; hypocycloid: $R-r$.
  • Cycloid base line length is $\pi D$.
  • Involute: tangent length equals the arc unwound.
  • Conic: $e<1$ ellipse, $e=1$ parabola, $e>1$ hyperbola; $VF:VC=e:1$.
  • Tangent to a conic: perpendicular to $PF$ at $F$ meets the directrix at $T$; join $TP$.
  • Chord of $\theta=2R\sin(\theta/2)$; chord of $60^\circ=R$.

Memory hooks

  • Epi = outside (epidermis), Hypo = inside (hypodermic needle).
  • "Eccentricity less than one is an Ellipse; equal one is Parabola; more is Hyperbola" (E-P-H order).
  • Diagonal scale: 10-10-10, one grid gives three units.
  • Vernier: 10 on vernier equal 9 on main, so least count is 1/10 of a division.

Coverage checklist

  • Principles of Engineering Graphics and their significance: no past questions.
  • usage of Drawing instruments: no past questions.
  • lettering: no past questions.
  • Conic sections including the Rectangular Hyperbola (General method only): ellipse e=2/3 and 3/4, parabola 60 mm.
  • Cycloid, Epicycloid, Hypocycloid and Involute: culvert, epicycloid 20/72, involute 122 mm and 180 degrees, hypocycloid 50/175, coin cycloid, short notes epicycloid.
  • Scales – Plain, Diagonal and Vernier Scales: vernier 6 km, 6 m, tank; diagonal 10 km, 900 km, 100 km, 0.45 ha; types and RF; compare diagonal and vernier; scale of chords; short notes types of scales.
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