Parallel and Orthogonal Projections
Senior 2 · Mathematics
Parallel and Orthogonal Projections
Lesson 4 of 4
18 min
Surveyors, builders and engineers work with projections every day. A house plan is an orthogonal projection onto the ground. A topographic map shows horizontal distances, so a road climbing towards Musanze is longer than it looks on the map. And shadows, the oldest projections of all, let you measure a tree without climbing it.
At one moment, the sun's rays are parallel, so all shadows are made by the same parallel projection onto the ground. Vertical poles and trees are parallel segments, and a parallel projection keeps the ratio of parallel segments. So, at the same moment, height ÷ shadow is the same for every vertical object.
Figure 4. At the same moment, the student and the tree cast shadows along parallel rays, so height ÷ shadow is the same for both. Diagram: Ganzaa (not to scale).
Worked example 1
At 3 p.m. a student 1.5 m tall casts a shadow 2 m long. At the same moment a eucalyptus tree casts a shadow 16 m long. How tall is the tree?
h ÷ 16 = 1.5 ÷ 2, so h = 16 × 1.5 ÷ 2 = 12 m.
Engineers draw an object as orthogonal projections: the plan (the view from directly above, projected onto the ground) and the elevations (views from the front and side, projected onto upright walls). Because the projection is orthogonal, true sizes can be measured on the drawing for every line that is parallel to the drawing, which is why working drawings use it.
But sloping lines look shorter on a plan. Worked example 2. A roof face slopes 5 m from the ridge to the edge and drops 3 m. It is 10 m long.
On the plan its width is √(5² − 3²) = √16 = 4 m, so the plan area is 4 × 10 = 40 m². The iron sheets must cover the real slope: 5 × 10 = 50 m². Ordering sheets from the plan area alone would leave 10 m² of roof uncovered.
A map shows the orthogonal projection of the land onto a horizontal plane. Worked example 3. A footpath measures 400 m on a map and climbs 90 m. The real path length is √(400² + 90²) = √168 100 = 410 m.
A photograph taken from the stands of Amahoro Stadium shows the far goal smaller than the near one: the camera's rays all pass through one point (the lens), so they are not parallel. That is a perspective projection, and it does not keep ratios. Only a parallel projection does.
Tip
Plan = view from directly above = orthogonal projection onto the ground. Anything that slopes is shorter on the plan than in real life, so use Pythagoras to get the true length before you order materials.
A 2 m pole casts a 3 m shadow. At the same moment a school building casts an 18 m shadow. How tall is the building?
Solution: h = 18 × 2 ÷ 3 = 12 m.
A roof face is 6.5 m along the slope, and it is 6 m wide on the plan. How far does it drop?
Solution: drop² = 6.5² − 6² = 42.25 − 36 = 6.25, so the drop is 2.5 m.
A road is 240 m long on a map and climbs 70 m. How long is the road itself?
Solution: √(240² + 70²) = √(57 600 + 4 900) = √62 500 = 250 m.
Why can you measure the height of a tree from its shadow, but not from a photograph taken close to the tree?
Solution: the sun's rays are parallel, so the shadow is a parallel projection and keeps the ratio height ÷ shadow. A camera makes a perspective projection, which does not keep ratios.
Remember
Sources and credits
The Rwandan examples, practice items and exercises are written by Ganzaa.
This lesson adapts CC BY-SA material and is shared under CC BY-SA 4.0.
