Projection of Points, Lines & Planes Notes

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.

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