Exploring Some Geometric Themes · Lesson 9 of 13
Foundations of Projection
“Project points, segments, and shapes onto a plane and identify which information a view preserves or loses.”
Locate the perpendicular projection of a point on a plane. Explain why a segment’s projected length cannot exceed its actual length. Recognise preserved parallel directions and possible collapsed views. Use projections to compare different orientations of a solid.
A drawing lies on a flat page, while a building, machine, or box occupies space. To communicate a solid accurately, we need a rule for deciding where each point should appear on the page. Projection provides that rule.
Artists may use many drawing methods, but measured geometric views require a consistent construction. We will begin with a single point and build up to segments, polygons, and solids. Small ideas about one point will explain larger patterns in an entire drawing.
Projecting a Point onto a Plane
A plane is an ideal flat surface extending in every direction. Imagine a tabletop extended beyond its edges. To project a point onto it, draw a perpendicular from the point to the plane. The place where that line meets the plane is the point’s projection.
If M is the original point and P is its projection, MP is perpendicular to the plane. A point already on the plane projects to itself. Two points lying above one another on the same perpendicular line have the same projection, so a projection can hide their separation.
A mapping to a plane in which each point is carried along a line perpendicular to that plane.
Problem
Two points are 3 cm and 8 cm directly above the same point P on a horizontal plane. Where do their top projections fall?
- 1.The viewing lines for a top projection are perpendicular to the horizontal plane.
- 2.Both points lie on the same perpendicular through P.
- 3.Therefore both project to P. Their 5 cm separation in height disappears in this view.
Projecting a Line Segment
Project both endpoints of a segment and join their projected points. This gives its projected segment, unless both endpoints project to the same point. The projection records the part of the segment’s direction that runs along the plane; motion directly towards or away from the plane is lost.
A segment parallel to the plane retains its full length in the projection. A tilted segment appears shorter. A segment perpendicular to the plane projects to a single point. These are different orientations of the same possible actual length.
| Orientation to projection plane | Projected result |
|---|---|
| Parallel | A segment with the same length |
| Tilted, neither parallel nor perpendicular | A shorter segment |
| Perpendicular | A point; projected length zero |
Let L be the actual segment length, p its projected length, and d the difference between its endpoint heights above the plane. These form a right triangle with L as the hypotenuse. Thus L² = p² + d². Since d² cannot be negative, p cannot exceed L. Equality occurs when d = 0, meaning the segment is parallel to the plane.
Problem
A segment is 13 cm long. Its endpoints are 2 cm and 7 cm above a horizontal plane. Find its top-projected length.
- 1.The endpoint height difference is 7 − 2 = 5 cm, so d = 5 cm.
- 2.Use p² = L² − d² = 13² − 5² = 169 − 25 = 144.
- 3.Therefore p = 12 cm.
- 4.Using 7 cm in place of the difference would be wrong: both endpoints are already above the plane.
A projected length may be shorter than the actual length. Do not measure a slanting drawing and assume the result is the true physical length. First identify the projection direction and the segment’s orientation.
Projecting Polygons
Project every vertex of a polygon and join corresponding images in the original order. A square tilted relative to the projection plane generally appears as a parallelogram. Its opposite sides remain parallel, but its angles and lengths need not remain equal to the original ones.
Parallel segments have the same direction in space. Projecting them along one common perpendicular direction preserves that parallel direction whenever they do not collapse to points. If an entire flat polygon is viewed exactly edge-on, it collapses to a segment instead of retaining a visible polygonal interior.
Problem
Describe a square card’s projection when it is parallel to a screen, tilted, and finally edge-on.
- 1.Parallel to the screen, its projection is a square of the same size.
- 2.When tilted, its shape is generally a parallelogram; a tilt about one side can give a rectangle with one shortened dimension.
- 3.Exactly edge-on, the card’s thickness is ignored in the ideal model and the square projects to a segment.
- 4.The object remains a square card throughout; the projection changes because its orientation changes.
The caveat about collapse matters. It would be inaccurate to promise that every polygon always projects to a polygon with the same number of visible sides. Similarly, a right angle on an object may look oblique in a projection. The drawing communicates a view, not every measurement at once.
Projecting Solids
For a solid, project its boundary onto the plane. Some boundary points lie behind others along the viewing direction. A usual visible view shows the nearest surfaces and the outer profile; hidden lines may be added when they are useful for understanding the structure.
A cube viewed face-on gives a square. Viewed along a suitable edge direction it can give a rectangular outline. Viewed along a body diagonal it gives a regular hexagonal outline. The hexagon does not mean the cube has a hexagonal face; it is the combined outline of several projected faces.
A cone viewed along its axis gives a circular outline, while a side view gives a triangle. A cylinder similarly changes from a circle to a rectangle. These familiar examples help distinguish the shape of one face from the shape of the entire projected outline.
Find What a View Loses
Use a square card and a small box. Choose a fixed imaginary screen in front of you and change the object’s orientation slowly.
- Hold the card parallel to the screen and describe its visible shape.
- Tilt it about a side and follow how one dimension shortens.
- Turn it edge-on and explain why the area of the projection disappears in the ideal flat-card model.
- Repeat with the box and identify an edge whose projected length changes as you rotate it.
The exercise shows why several coordinated views are useful in design. A single projection may hide depth, shorten lengths, merge different points, or conceal a face. A second direction can reveal information lost in the first.
Quiz
How is a point’s perpendicular projection located?
When does a segment retain its full length in perpendicular projection?
What is the projection of a segment perpendicular to the plane?
A 10 cm segment has endpoint height difference 6 cm. What is its projected length?
What happens to a flat square viewed exactly edge-on in the ideal model?
Practice Problems
- Find a 17 cm segment’s projected length if its endpoint height difference is 8 cm.
- A segment projects to 12 cm and has endpoint height difference 9 cm. Find its actual length.
- Two points at heights 5 cm and 11 cm lie on the same perpendicular. Describe their projection.
- A learner says every projected square must be a square. Correct the statement.
- Explain why a hexagonal outline of a cube does not imply that the cube has a hexagonal face.
1. Use p² = 17² − 8² = 289 − 64 = 225. 2. The projected length is 15 cm.
Key Takeaways
Perpendicular projection sends points to a plane along perpendicular lines. Several points can share one projected point, so depth can be lost. A segment’s projected length is at most its actual length. Parallel directions are preserved when their projections do not collapse. The outline of a projected solid is not necessarily the shape of one of its faces.