What is a projection?
A projection is a way to draw a 3D object on a flat surface. We need three things:
- the object (for example a box),
- the projection plane or screen (the paper, a wall, the ground),
- projecting rays: straight lines through each point of the object.
Where a ray meets the plane we get the image (projection) of that point. Join the images of the corners and you get the image of the whole object. A shadow is a projection made by light.
Every system of drawing is decided by one question: where do the rays come from?
Central projection
In a central projection all rays start from one point, the centre of projection (a lamp, a candle, your eye).
- Bring the centre closer to the object → the image becomes bigger.
- Move the object closer to the screen → the image becomes smaller and closer to true size.
- Parallel lines in the object may not stay parallel in the image; they can meet at a point.
Size works like similar triangles: if the lamp is 2 m from a stick and 6 m from the wall, the shadow is 6 ÷ 2 = 3 times as tall as the stick.
Parallel and orthogonal projection
In a parallel projection all rays are parallel. Sunlight is the everyday example because the Sun is so far away.
- The image size does not depend on how far the object is from the light.
- Parallel lines stay parallel. Midpoints stay midpoints.
If the rays meet the plane at an angle it is an oblique parallel projection: the shape is stretched or squashed. If the rays are perpendicular (90°) to the plane it is an orthogonal projection: any face parallel to the plane keeps its true shape and size. Front, top and side views of a solid are orthogonal projections.
Conical perspective: drawing what the eye sees
Conical (linear) perspective is a central projection where the centre is the viewer's eye and the plane is a picture plane between the eye and the object. The rays form a cone from the eye, hence the name.
- Horizon line: the line at eye height.
- Vanishing point: parallel lines going away from us seem to meet here.
- Frontal (one-point) perspective: one face of the object is parallel to the picture plane; lines going into the distance meet at one vanishing point. Good for rooms, roads and corridors.
- Oblique (two-point) perspective: the object is turned; its two sets of horizontal edges go to two vanishing points on the horizon. Good for buildings seen from a corner.
Comic artists move the horizon high (looking down, the hero looks small) or low (looking up, the hero looks powerful).
Which system for which job?
| System | Rays | Best for |
|---|---|---|
| Orthogonal views (multiview) | parallel, 90° | exact sizes: plans, machine parts, building drawings |
| Axonometric (isometric) | parallel, 90°, object tilted | one 3D-looking view that still can be measured |
| Oblique (cavalier, cabinet) | parallel, slanted | quick sketches with a true front face |
| Conical perspective | from one point | realistic pictures: architecture, art, comics, games |
Try it: stand a pencil on a table at night under one bulb and then in sunlight. Measure both shadows as you move the pencil closer to and farther from the light.
Key formulas and definitions
- Central projection size: image height ÷ object height = (centre-to-screen distance) ÷ (centre-to-object distance)
- Parallel projection: image size does not depend on distance
- Orthogonal projection: rays ⟂ plane; a face parallel to the plane is shown in true size
- A segment of length L at angle θ to the plane has orthogonal image length L·cos θ
- Perspective: horizon line = eye height; parallel receding lines meet at a vanishing point
Worked examples
1. A lamp is 1.5 m from a 20 cm toy and 4.5 m from the wall. How tall is the shadow on the wall?
Central projection: ratio = 4.5 ÷ 1.5 = 3. Shadow = 3 × 20 cm = 60 cm.
2. A 2 m pole stands in sunlight and casts a 3 m shadow. A tree nearby casts a 12 m shadow at the same time. How tall is the tree?
Sun rays are parallel, so heights and shadows are in the same ratio: 2/3 = h/12, h = 8 m.
3. A 10 cm stick makes an angle of 60° with a sheet of paper. How long is its orthogonal projection on the paper?
Image length = L cos θ = 10 × cos 60° = 10 × 0.5 = 5 cm.
4. Which projection keeps parallel lines parallel: central or parallel?
Parallel projection always keeps parallel lines parallel. Central projection can make them meet at a vanishing point.
Common mistakes
- Saying a central projection keeps the true size. It only does for a flat object touching the screen; otherwise size changes with distance.
- Thinking every parallel projection is orthogonal. Orthogonal is the special case where rays are at 90° to the plane.
- Putting the vanishing point anywhere. For horizontal lines it must lie on the horizon line (eye level).
- Using the wrong distances in the size ratio: use centre-to-screen ÷ centre-to-object, not object-to-screen.