Projection of Points and Straight Lines
Draw as you read. This topic cannot be absorbed by reading. Have paper and a set square beside you and construct each figure as it is described. The procedures below are written so you can follow them line by line.
The two planes
Two planes at right angles:
- HP — horizontal plane. The top view (plan) lies on it.
- VP — vertical plane. The front view (elevation) lies on it.
Their intersection is the reference line XY. The HP is then rotated 90° clockwise until both planes are flat on the paper, which is why the top view ends up below XY and the front view above it.
The two views are always vertically aligned — the front and top view of the same point lie on one vertical projector. If your two views drift out of vertical alignment, everything after that is wrong, and it is the single most common cause of a failed construction.
Notation
Use it consistently or the work becomes unreadable:
- Point in space: capital, A
- Front view: lowercase with prime, a′
- Top view: lowercase, a
- Side view: lowercase with double prime, a″
Projection of points, by quadrant
The two planes divide space into four quadrants. Which one a point occupies determines where its views fall relative to XY:
| Quadrant | Position | Front view a′ | Top view a |
|---|---|---|---|
| I | above HP, in front of VP | above XY | below XY |
| II | above HP, behind VP | above XY | above XY |
| III | below HP, behind VP | below XY | above XY |
| IV | below HP, in front of VP | below XY | below XY |
The two rules that generate the whole table:
- Above HP → front view above XY. Below HP → front view below XY.
- In front of VP → top view below XY. Behind VP → top view above XY.
Learn those two sentences rather than the table; the table then writes itself, and you cannot misremember a row.
Distances are preserved: the distance of from XY is the height above HP, and the distance of from XY is the distance in front of VP.
Almost all exam work is in the first quadrant, which is why first angle projection is the default.
Line parallel to one plane
If a line is parallel to VP, its front view shows true length and its true inclination to the HP. If it is parallel to HP, its top view shows true length.
The general principle, and the one worth memorising: a line shows its true length in the view on the plane it is parallel to.
Line inclined to one plane
Say the line is inclined at θ to the HP and parallel to the VP:
- Front view: true length, inclined at θ to XY.
- Top view: shorter than true length, parallel to XY.
The top view is a foreshortened projection. The word matters: a projection can never be longer than the true length, only equal or shorter. If a computed true length comes out shorter than a view, the work is wrong.
Line inclined to both planes — the core procedure
Neither view now shows true length, and neither shows a true inclination. This is the standard exam problem and it has a fixed four-stage procedure.
Let the true length be TL, true inclinations θ to HP and φ to VP, and apparent inclinations α and β.
Stage 1 — put the line parallel to VP, inclined at θ to HP. Draw the front view at true length, at angle θ to XY. Project down to get the top view: a horizontal line, shorter than TL. Record that top view length.
Stage 2 — put the line parallel to HP, inclined at φ to VP. Draw the top view at true length, at angle φ to XY. Project up to get the front view: a horizontal line, shorter than TL. Record that front view length.
Stage 3 — combine. The final views must simultaneously have:
- top view of the length found in Stage 1, and
- front view of the length found in Stage 2.
Swing arcs from the fixed end to lay off those lengths, and complete both views keeping the ends vertically aligned.
Stage 4 — read off the apparent inclinations α and β from the finished views.
Why this works, in one sentence: tilting a line about one axis does not change its projected length on the plane containing that axis, so each stage fixes one projected length, and the final position is the one that satisfies both at once.
The reverse problem — given the two views, find TL and the true inclinations — is solved by rotation: rotate each view until it is parallel to XY, project, and the resulting hypotenuse is the true length, with the angle it makes giving the true inclination.
Traces
The trace is where the line, extended if necessary, pierces a plane.
- HT — horizontal trace, where the line meets the HP.
- VT — vertical trace, where the line meets the VP.
Procedure for HT: extend the front view to cut XY; from that intersection drop a projector; where it meets the extended top view is the HT.
Procedure for VT: extend the top view to cut XY; from that intersection raise a projector; where it meets the extended front view is the VT.
Note the crossover — you extend the front view to find the horizontal trace. Getting this the wrong way round is the standard error, and it is worth writing the two lines above onto your formula sheet verbatim.
Special cases worth recognising instantly: a line parallel to a plane has no trace on it, since it never meets it; a line perpendicular to a plane has its trace on the axis; and a line in a plane has its trace at the line itself.