Chapter 6 of 8

Projection of Points, Lines, Planes & Solids

In the DSSSB AE Civil syllabus under Engineering Graphics · 2 parts

Projection of Points, Lines & Planes · Projection of Solids

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📑 Contents (12 sections)

Part 1 of 2

Projection of Points, Lines & Planes

Last reviewed 16 Sept 2026 · 8 min read

Projection of points

A point is located by its distance from HP (shown in the front view from XY) and its distance from VP (shown in the top view from XY).

Notation (first-angle convention in textbooks): point A in space; a′ = front view (elevation); a = top view (plan); a″ = side view. The front and top views lie on the same projector perpendicular to XY.

Position of point	Front view (a′)	Top view (a)
First quadrant (above HP, in front of VP)	Above XY	Below XY
Second quadrant (above HP, behind VP)	Above XY	Above XY
Third quadrant (below HP, behind VP)	Below XY	Above XY
Fourth quadrant (below HP, in front of VP)	Below XY	Below XY
On HP	On XY	— (distance from VP)
On VP	— (height)	On XY

Projection of straight lines

Notation

True length $L$ ; θ = true inclination with HP; φ = true inclination with VP.
α = apparent angle of the front view with XY; β = apparent angle of the top view with XY.
$a^{'} b^{'}$ = front view (elevation); $ab$ = top view (plan).

Simple positions

Position of line	Front view	Top view
Parallel to both HP and VP	True length, parallel to XY	True length, parallel to XY
Perpendicular to HP (parallel to VP)	True length, perpendicular to XY	Point
Perpendicular to VP (parallel to HP)	Point	True length, perpendicular to XY
Parallel to VP, inclined θ to HP	True length at θ to XY	Shorter, parallel to XY
Parallel to HP, inclined φ to VP	Shorter, parallel to XY	True length at φ to XY

Line inclined to both planes

FormulaRelations for a line inclined to both HP and VP

Length of top view (plan): $L_{T V} = L cos θ$ Length of front view (elevation): $L_{F V} = L cos ϕ$

Difference in heights of ends (distance between end projectors in FV, vertical): $Δ z = L sin θ$ Difference in distances from VP (vertical separation in TV): $Δ y = L sin ϕ$

Distance between end projectors (along XY): $Δ x = L^{2} - Δ z^{2} - Δ y^{2}$

Apparent angles: $sin α = \frac{Δ z}{L _{F V}}$ , $sin β = \frac{Δ y}{L _{T V}}$

$α > θ$ and $β > ϕ$ (apparent angles exceed true angles).
$θ + ϕ \leq 90°$ ; if $θ + ϕ = 90°$ the line lies in a profile plane — both views perpendicular to XY and neither shows true length.

Finding true length and true inclinations

Rotating line (rotation) method — rotate the top view parallel to XY about one end; project to the front view — gives true length and θ. Rotating the front view parallel to XY gives true length and φ.
Trapezoidal method — construct trapezia on the views using distances from the reference planes.
Auxiliary plane method — project onto a plane parallel to the line.

Traces of a line

Horizontal trace (HT) — point where the line (produced) meets HP.
Vertical trace (VT) — point where the line (produced) meets VP.
A line parallel to a plane has no trace on that plane.
Locating HT: extend the front view to meet XY; project down to the (extended) top view. Locating VT: extend the top view to meet XY; project up to the extended front view.

Part 2 of 2

Projection of Solids

Last reviewed 16 Sept 2026 · 7 min read

Classification of solids

Polyhedra (plane faces)

Solid	Description
Regular polyhedra	All faces identical regular polygons — tetrahedron (4 equilateral triangles), cube/hexahedron (6 squares), octahedron (8), dodecahedron (12 pentagons), icosahedron (20)
Prism	Two equal and parallel polygonal bases joined by rectangular lateral faces; named by base shape (triangular, square, pentagonal, hexagonal prism)
Pyramid	Polygonal base and triangular lateral faces meeting at the apex

Solids of revolution

Solid	Generation
Cylinder	Rectangle revolving about one side
Cone	Right triangle revolving about one of its perpendicular sides
Sphere	Semicircle revolving about its diameter

Other terms

Right solid — axis perpendicular to the base; oblique solid — axis inclined to the base.
Frustum — portion of a pyramid/cone between the base and a cutting plane parallel to the base.
Truncated solid — cut by a plane inclined to the base.
Generators — straight lines on the surface of a cylinder or cone (from base to apex for a cone).
Slant height of a cone $l = r^{2} + h^{2}$ .

Positions of solids

Position	Approach
Axis perpendicular to HP	Top view shows the true shape of the base; draw it first, project the front view
Axis perpendicular to VP	Front view shows the true shape of the base; draw it first, project the top view
Axis parallel to both HP and VP	Side view shows true shape of base; draw side view first
Axis inclined to HP, parallel to VP	Two-stage method — start with axis perpendicular to HP, then tilt the front view
Axis inclined to VP, parallel to HP	Two-stage — start with axis perpendicular to VP, then rotate the top view
Axis inclined to both HP and VP	Three-stage method — tilt, then rotate

Resting conditions

Resting on its base (on HP).
Resting on a base edge or corner of the base with axis inclined.
Resting on a lateral face or lateral edge (prisms/pyramids) or a generator (cones/cylinders).
Suspended freely from a corner or point — the axis and the line joining the point of suspension to the centre of gravity are vertical (the CG of a pyramid/cone lies on the axis at 1/4 of the height from the base; of a prism/cylinder at mid-height).

Change of position method

Stage 1 (simple position): Draw the solid with its axis perpendicular to the plane to which it will be inclined — e.g. for an axis inclined to HP, draw with axis perpendicular to HP, placing the base edge/corner as required (e.g. edge perpendicular to VP).

Stage 2 (tilt): Redraw the front view tilted so that the axis (or base, face, edge) makes the given angle with HP; project the new top view using widths from stage 1 and positions from the tilted front view.

Stage 3 (rotate, if inclined to both planes): Redraw the top view rotated so that the axis (or its top view) makes the required angle with XY (φ or apparent angle β); project the final front view using heights from stage 2.

Key idea: during tilting about a line on HP, heights change but widths (from VP) of points measured perpendicular to the plane of rotation remain; during rotation about a vertical axis, heights remain unchanged.

Visibility

In the front view, points nearest to the observer (farthest in front of VP) are visible.
In the top view, points highest (farthest above HP) are visible.
Edges hidden behind surfaces are drawn dashed; the outline of any view is always visible.
For pyramids with apex up, all lateral edges are visible in the top view.

Useful geometric relations

FormulaDimensions of common solids

Regular tetrahedron of edge $a$ : height $h = a \frac{2}{3} \approx 0.816 a$
Square pyramid (base side $b$ , height $h$ ): slant edge $= h^{2} + \frac{b ^{2}}{2}$ ; slant height of a face $= h^{2} + \frac{b ^{2}}{4}$
Hexagonal base of side $a$ : across corners $= 2 a$ ; across flats $= 3 a$
Cone: slant height $l = r^{2} + h^{2}$ ; semi-apex angle $= tan^{- 1} (r / h)$
Angle of a pyramid face with the base $= tan^{- 1} (\frac{h}{b /2})$ (square pyramid)
Cube of side $a$ : face diagonal $a 2$ ; body diagonal $a 3$

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