How unit 2 is examined
Covers the principle of orthographic projection with first and third angle conventions, projections of lines inclined to both planes, and projections of inclined plane figures; all three topics are high weight and every past question is a 7-mark drawing, comparison or fill-in.
Principles of Orthographic Projections- Conventions
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Definition. <mark>Orthographic projection is the method of representing a solid on a plane by drawing the feet of perpendiculars (projectors) from every point of the object to the plane, the projectors being parallel to each other and perpendicular to the plane of projection.</mark>
Key points.
- Projection means the image formed on a plane by lines (projectors) drawn from the observer's eye through points of the object; in orthographic projection the projectors are parallel and perpendicular to the plane, so the observer is imagined at infinity.
- Two principal planes are used: the Horizontal Plane (HP) and the Vertical Plane (VP), meeting at the reference line $xy$; a third, the profile plane (PP), is perpendicular to both and is used for side views.
- HP and VP divide space into four quadrants (dihedral angles); the first and third angles are used in practice because in the second and fourth the two views overlap.
- The view on VP is the front view (elevation), the view on HP is the top view (plan), and the view on PP is the side view; each shows the object from one direction only, so at least two views are needed to describe it.
- Rotating HP downward into the plane of VP about $xy$ brings the plan and elevation into one drawing sheet, with the projectors joining them perpendicular to $xy$.
- In first angle projection the object lies in the first quadrant, between the observer and the plane, and the planes are opaque.
- In third angle projection the object lies in the third quadrant, the plane lies between the observer and the object, and the planes are transparent, so each view is placed on the side of the object it represents.
Diagram. Draw the two symbols side by side: a frustum of a cone with its circles drawn to the right of the front view (first angle) and to the left of the front view (third angle).
| Point | First angle | Third angle |
|---|---|---|
| Object position | First quadrant, between observer and plane | Third quadrant, plane between observer and object |
| Nature of plane | Opaque | Transparent |
| Plan position | Below the elevation | Above the elevation |
| Right side view | Drawn on the left of the elevation | Drawn on the right of the elevation |
| Order of object, plane, observer | Observer, object, plane | Observer, plane, object |
| Symbol | Frustum with the circles on the right | Frustum with the circles on the left |
| Used in | India, UK, Europe | USA, Canada |
Fill in the blanks (Q2). (i) six (three pairs of mutually perpendicular planes); (ii) perpendicular; (iii) first angle: in the first quadrant, between the observer and the plane; (iv) third angle; (v) one-fourth (one quarter); (vi) internal (hidden) details; (vii) plotter (printer); hatching lines at $45^\circ$; mass properties give volume, centre of gravity and moment of inertia; third angle: object in the third quadrant.
Answer frame. Open with the definition of projection and the principle of orthographic projection; draw the two symbols; then the 6-7 row table in the order object position, plane nature, plan position, side view; close with the country of use. For fill-in questions, write only the word or phrase for each blank.
Pitfall: Writing the symbols the wrong way round; first angle has the circles on the right, third angle on the left.
Asked: [7 marks] (Jun 2022, Dec 2024) Differentiate between the first angles and third angles projection. Asked: [7 marks] (Dec 2023, Jun 2023) Fill in the blanks (planes surrounding an object, projectors, first angle, third angle, half sectional view, sectional views, hard copy, hatching angle, mass prop). Asked: [7 marks] (Dec 2023, Jun 2025) What is meant by projection? Explain the principle of projection and differentiate between first angle and third angle projection.
Projections of Points and lines inclined to both planes
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Definition. <mark>A line inclined to both planes has its front view and top view both shorter than its true length, and both inclined to $xy$ at angles (apparent angles) larger than the true inclinations $\theta$ (to HP) and $\phi$ (to VP).</mark>
Key points.
- A point above HP and in front of VP has its elevation $a'$ above $xy$ by its height and its plan $a$ below $xy$ by its distance from VP; the projectors join $a$ and $a'$ perpendicular to $xy$.
- A line inclined to HP by $\theta$ and to VP by $\phi$ has plan length $= TL\cos\theta$ and elevation length $= TL\cos\phi$.
- Height difference of the ends $= TL\sin\theta$ and depth difference $= TL\sin\phi$.
- The plan length is the same in the elevation-rotation as in the original, so the length of the plan stays $TL\cos\theta$ when the line is turned about a vertical axis; likewise the elevation stays $TL\cos\phi$ when it is turned about a horizontal axis.
- The sum $\theta+\phi$ cannot exceed $90^\circ$; if it equals $90^\circ$ the line lies in a profile plane.
- When a midpoint is given, the midpoint's projections are fixed first and the end points lie half the difference above and below it (half of the height difference, half of the depth difference).
- Methods to find true length and inclinations: rotating line (revolution) method, rotating trapezoid method, and auxiliary plane method.
Steps (given $TL$, $\theta$, $\phi$ and end A).
Step 1: Draw xy; mark a' at the height of A above xy and a at its distance below xy.
Step 2: From a' draw a'b1' = TL at angle theta to xy; drop b1' to a horizontal locus of b (through a).
Step 3: From a draw ab2 = TL at angle phi to xy; project b2 up to a horizontal locus of b' (through a').
Step 4: Project b1' vertically down to the locus of b; that point is b; join ab for the top view.
Step 5: Project b2 up to the locus of b'; that point is b'; join a'b' for the front view.
With given lengths of the views: draw the true-length line, find $\theta=\cos^{-1}(TV/TL)$ and $\phi=\cos^{-1}(FV/TL)$ or use the height and depth differences.
Example (Jun 2024, Q10). TL $=100$, TV $=80$, FV $=70$, midpoint $M$: $36$ above HP, $46$ in front of VP.
| Item | Working | Value |
|---|---|---|
| $\theta$ | $\cos\theta=80/100$ | $36.87^\circ$ |
| $\phi$ | $\cos\phi=70/100$ | $45.57^\circ$ |
| Height diff | $\sqrt{100^2-80^2}$ | $60$ mm |
| Depth diff | $\sqrt{100^2-70^2}$ | $71.4$ mm |
Plot $m'$ at $36$ and $m$ at $46$; $a'b'=70$ through $m'$ with ends $30$ above and below $m'$; $ab=80$ through $m$ with ends $35.7$ in front of and behind $m$ in depth. Result: $\theta=36.87^\circ$, $\phi=45.57^\circ$.
Other given data (checked).
| Question | Result |
|---|---|
| Dec 2024 (120 mm, $45^\circ$, $30^\circ$) | Half height $=42.43$, half depth $=30$; B is $62.43$ above HP and $30$ in front; A is $22.43$ below HP and $30$ behind VP; TV $=84.85$, FV $=103.92$ |
| Jun 2022 (70; FV 50, TV 60) | $\theta=31.0^\circ$, $\phi=44.4^\circ$ |
| Jun 2025 (70, $\theta=30^\circ$, FV 50) | Height diff $35$; TV $=60.6$; depth diff $=49.0$; $\phi=44.4^\circ$ |
| Nov 2022 (CD parallel to HP, FV 50, $30^\circ$ to VP) | $TL=50/\cos30^\circ=57.74$ mm; $cd$ at $30^\circ$ to $xy$ |
| Dec 2023 (75; plan 54) | $\theta=43.95^\circ$; depth difference $52$ so $\phi=\sin^{-1}(52/75)=43.9^\circ$; HT and VT where the extended views meet $xy$ |
| Jun 2023 (FV 65, TV 50) | Midpoint: $m'$ at $30$, $m$ at $38$; $b$ at $2\times38-10=66$ from $xy$ |
Traces. The HT (horizontal trace) is where the line meets HP: extend the elevation to $xy$, project down to the extended plan. The VT is where it meets VP: extend the plan to $xy$, project up to the extended elevation.
Answer frame. Open with the given data and the relations $TV=TL\cos\theta$, $FV=TL\cos\phi$; draw $xy$, the two views with loci and the true-length arcs; then follow the numbered steps; close by stating the measured $\theta$, $\phi$ and true length. For the "methods" question, name the three methods with one line each: rotating line (turn the line parallel to a plane), trapezoid (draw the trapezium of the line and its projections), auxiliary plane (view from a plane parallel to the line).
Pitfall: Rotating the line about the wrong view; the elevation is drawn from the top-view loci, and the top view from the elevation loci. Also, the Jun 2023 data give a depth difference of $56$ against a plan of $50$, so follow the examiner's midpoint method and state the values you take.
Asked: [7 marks] (Jun 2022, Nov 2022) A line AB 60 mm long has end A 15 mm above HP and 10 mm in front of VP, inclined $45^\circ$ to HP and $30^\circ$ to VP; also line PS 65 mm, P 15 above and 15 in front, inclined $55^\circ$ and $35^\circ$. Draw its projections. Asked: [7 marks] (Jun 2022) Line AB 70 mm, A 10 above HP and 15 in front of VP; FV 50 mm and TV 60 mm. Draw the projections and determine the inclinations. Asked: [7 marks] (Nov 2022) Front view of line CD parallel to HP and inclined $30^\circ$ to VP is 50 mm; C is 15 in front of VP and 25 above HP. Draw the projections and find the true length. Asked: [7 marks] (Dec 2023) Plan of line PQ, 75 mm, measures 54 mm; midpoint 50 from VP and 15 from HP; Q is 24 from VP. Draw projections, find inclinations and locate traces. Asked: [7 marks] (Jun 2023) Projections a'b' 65 mm and ab 50 mm; midpoint 38 in front of VP and 30 above HP; A 10 in front of VP and nearer to it; B nearer to HP. Draw projections and find true length. Asked: [7 marks] (Dec 2024) Line AB 120 mm, $45^\circ$ to HP and $30^\circ$ to VP; midpoint P in VP and 20 above HP; A in third quadrant, B in first. Draw projections. Asked: [7 marks] (Jun 2024) Line 100 mm, 80 mm in plan and 70 mm in elevation; midpoint M 36 above HP and 46 in front of VP. Draw top and front views of AB. Asked: [7 marks] (Jun 2024) Name the methods used to determine the length and true inclinations of a straight line. Asked: [7 marks] (Jun 2025) Line AB 70 mm inclined $30^\circ$ to HP; A 10 above HP and 15 in front of VP; front view 50 mm. Draw the projections.
Projections of planes inclined Planes - Auxiliary Planes
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Definition. <mark>A plane figure (lamina) is projected by first placing it parallel to a reference plane, where it shows its true shape, and then tilting it stage by stage; each stage changes only one view's shape while the other view keeps its length.</mark>
Key points.
- When the lamina is parallel to HP, its top view is the true shape and its front view is a line parallel to $xy$; when parallel to VP, the front view is the true shape and the top view is a line.
- Stage 1 draws the true shape in the plane the lamina is parallel to, with the given edge or corner correctly placed and the other view projected.
- Stage 2 tilts the surface to the given angle by turning the line view (edge view) to that angle with $xy$, keeping the resting corner or edge on $xy$, then projecting the apparent-shape view.
- Stage 3 rotates the apparent-shape view (without changing its shape) so that the given edge, diagonal or median makes the stated angle with $xy$, then the final other view is found by projectors from stage 3 and horizontals from stage 2.
- The rule to choose the first stage: the surface inclination is done first and the edge or diagonal inclination last; if the edge is inclined first, the edge angle is applied in stage 1.
- A resting edge on HP is shown in the front view as a point on $xy$; a corner on HP as a point on $xy$ with the diagonal through it.
- A hole in the lamina is drawn in the true shape and projected with the same points, since it is carried through all the stages.
- A distance from VP is set on the final stage by placing the nearest corner at that distance from $xy$.
Example (Dec 2023, hexagon 30 mm). The corner on HP; surface at $45^\circ$ to HP; the plan of the diagonal through that corner at $45^\circ$ to VP.
Step 1: Draw the true hexagon (side 30) in the top view with one corner on xy; project the front view as a line on xy.
Step 2: Tilt the front view line at 45 deg to xy about that corner; project down the intermediate top view.
Step 3: Rotate the intermediate top view so the diagonal through the corner is at 45 deg to xy; keep its shape.
Step 4: Project up from the new top view and across from the tilted front view; the intersections give the final front view.
Other figures.
| Question | First stage | Second stage | Third stage |
|---|---|---|---|
| Jun 2024, Dec 2024: hexagon 25, side in HP at $60^\circ$ to VP, surface $45^\circ$ to HP | True hexagon, resting side perpendicular to VP; FV a line on $xy$ | Tilt surface $45^\circ$ to HP; new TV | Rotate TV so the resting side is at $60^\circ$ to VP; FV |
| Jun 2024, Dec 2024: pentagon 25, resting edge, $45^\circ$ to HP, perpendicular from the midpoint of the resting edge at $30^\circ$ to VP | Pentagon TV with resting edge perpendicular to $xy$ | Tilt FV $45^\circ$; new TV | Median at $30^\circ$ to $xy$; final FV |
| Jun 2024, Dec 2024: rhombus 100 by 40, long diagonal $30^\circ$ to HP, short diagonal parallel to both | Rhombus in HP, long diagonal parallel to VP | Tilt long diagonal $30^\circ$ to HP with one end on $xy$; short diagonal stays parallel to $xy$ | No rotation needed |
| Jun 2024, Dec 2024: plate 45 by 35, AC $30^\circ$ to HP, BD apparent $45^\circ$ to VP | Plate in HP, FV a line on $xy$ | Tilt AC $30^\circ$ to $xy$; new TV | Rotate TV so BD is $45^\circ$ to $xy$; FV |
| Jun 2023, Jun 2025: hexagon 40 with square hole 25, resting on a side, surface $60^\circ$ to VP, nearest corner 24 from VP | True hexagon with the hole; FV on $xy$ | Incline surface $60^\circ$; nearest corner 24 from VP | Project the final views including the hole |
| Jun 2023, Jun 2025: hexagon 30, edge in VP at $60^\circ$ to HP, surface $40^\circ$ to VP | True shape in FV, TV a line on $xy$ | Tilt TV $40^\circ$ to $xy$; project FV | Incline the edge at $60^\circ$ to HP; final TV |
Answer frame. Open with the given conditions; draw three figures in stage order with $xy$; then develop the stages 1, 2, 3 in order, projecting each new view from the previous view; close by labelling the final views and dimensioning the given angles and distances.
Pitfall: Rotating the wrong view in stage 3; the view that is turned is the one that was obtained in stage 2 and its shape must not change.
Asked: [7 marks] (Dec 2023) Regular hexagonal lamina 30 mm side with a corner on HP, surface $45^\circ$ to HP, plan of the diagonal through the corner at $45^\circ$ to VP. Draw its projections. Asked: [7 marks] (Jun 2023, Jun 2025) Hexagonal lamina 40 mm side with a centrally cut square hole of 25 mm, resting on a side on HP, surface $60^\circ$ to VP, nearest corner 24 mm from VP; also hexagonal lamina 30 mm with an edge in VP inclined $60^\circ$ to HP and surface $40^\circ$ to VP. Asked: [7 marks] (Jun 2024, Dec 2024) Regular hexagon 25 mm side with one side in HP inclined $60^\circ$ to VP and surface $45^\circ$ to HP; also pentagonal plane 25 mm resting on an edge, $45^\circ$ to HP, perpendicular from the midpoint of the resting edge $30^\circ$ to VP. Asked: [7 marks] (Jun 2024, Dec 2024) Rhombus with 100 mm and 40 mm diagonals, bigger diagonal $30^\circ$ to HP with one end in HP, smaller diagonal parallel to both planes; also rectangular plate ABCD $45\times35$ mm with AC at $30^\circ$ to HP and BD at an apparent angle of $45^\circ$ to VP.
Last-minute revision
- Orthographic projection: parallel projectors perpendicular to the plane of projection.
- Planes surrounding an object: six (three pairs); HP and VP make four quadrants.
- First angle: object between observer and plane, plan below elevation; third angle: plane between, plan above elevation.
- First angle symbol has the circles on the right of the front view; third angle on the left.
- Line: $TV=TL\cos\theta$, $FV=TL\cos\phi$; height diff $=TL\sin\theta$, depth diff $=TL\sin\phi$.
- $\theta+\phi\le90^\circ$; equal to $90^\circ$ means the line lies in a profile plane.
- Jun 2024 line: $\theta=36.87^\circ$, $\phi=45.57^\circ$; Dec 2023: $\theta=43.95^\circ$.
- Nov 2022 line CD: $TL=50/\cos30^\circ=57.74$ mm.
- HT is where the line meets HP; VT where it meets VP.
- Planes: stage 1 true shape, stage 2 surface tilt, stage 3 rotate for the edge or diagonal angle.
- Hatching lines at $45^\circ$.
Memory hooks
- "TV cos theta, FV cos phi": cosines of the angle with the plane the view is on.
- First angle: "object in FRONT of the plane, plan BELOW"; third: the opposite.
- Sine gives the difference of ends: height with theta, depth with phi.
- Planes: "Tilt first (surface), turn last (edge)".
Coverage checklist
- Principles of Orthographic Projections- Conventions: first and third angle comparison (Jun 2022, Dec 2024), fill in the blanks (Dec 2023, Jun 2023), projection and principle (Dec 2023, Jun 2025).
- Projections of Points and lines inclined to both planes: all nine line drawings (Jun 2022, Nov 2022, Dec 2023, Jun 2023, Dec 2024, Jun 2024, Jun 2025) and the methods question (Jun 2024).
- Projections of planes inclined Planes - Auxiliary Planes: hexagon, hole hexagon, pentagon, rhombus, plate (Dec 2023, Jun 2023, Jun 2025, Jun 2024, Dec 2024).