Shadows and projections

The idea

A shadow on a plane is similar in shape to the projection but may be scaled, stretched or distorted; light from a very distant source perpendicular to the plane, like sunlight, casts shadows indistinguishable from projections.

A shadow on the wall

Hold your hand in front of a lamp. Its shadow falls on the wall. Now carry the lamp backwards, farther and farther from your hand.

What happens to the shadow on the wall?

• It gets bigger
• It gets smaller
• It stays the same size

The lamp and the card

A house-shaped card stands between a lamp and a wall. The dark shape is its shadow. The dashed outline is its projection on the wall: where its points land when they go straight to the wall.

Shadow and projection

The shadow of an object on a plane is similar in shape to its projection, but it may be scaled up, stretched or distorted. Light from a very far source that falls perpendicular to the plane, like sunlight, casts a shadow indistinguishable from the projection.

A lamp close to the card makes its rays spread out, so the shadow is bigger. A lamp far away has almost parallel rays, so the shadow is the projection. The Sun is such a lamp.

Here the wall is 15 cm behind the card. For the middle of the card, lamp to wall ÷ lamp to card tells how many times bigger the shadow is.

• Lamp to card | Middle of shadow is
• 30 cm | 1.5 times
• 60 cm | 1.25 times
• 150 cm | 1.1 times
• 300 cm | 1.05 times
• very far (the Sun) | 1 times: the projection

Notes

The shadow of an object on a plane is similar in shape to its projection, but it may be scaled up, stretched or distorted. A very far light that falls perpendicular to the plane, like the Sun, casts shadows that are the same as projections.

Check yourself

The shadow of a card on a wall is much bigger than the card's projection on the wall. How can you make the shadow more like the projection?

A card is 12 cm tall. A lamp is 20 cm in front of it and the wall is 10 cm behind it. How tall is the shadow on the wall?

Answer: 18 cm

Lamp to wall = 20 + 10 = 30 cm and lamp to card = 20 cm, so the shadow is 30 ÷ 20 = 1.5 times as tall: 12 × 1.5 = 18 cm.

Sunlight falls straight down on flat ground. Why is a shadow on the ground almost exactly the projection of the object?

Three lamps light the same card. The dashed outline is the card's projection. Which lamp is farthest from the card?

• Move the lamp farther away from the card — correct. Yes! The farther the lamp, the more parallel its rays, and the closer the shadow gets to the projection.
• Move the lamp closer to the card. A closer lamp spreads its rays even more, so the shadow gets bigger still.
• Make the room darker. The brightness of the room does not change the shape or size of the shadow.
• The Sun is very big, so it lights everything evenly.. The size of the Sun is not the reason. The light of a lamp, big or small, spreads out if the lamp is near.
• Shadows on the ground are always as big as the object.. They are not. In the evening shadows are long, because the light is slanted and not perpendicular to the ground.
• The Sun is very far away, so its rays are almost parallel and fall straight on the ground. — correct. Yes! Far away, the rays do not spread out. A perpendicular, parallel light gives the projection.
• Lamp A. The shadow in A is still much bigger than the projection, so its lamp is nearer than the lamp in B.
• Lamp B — correct. Yes! The shadow in B is almost the same as the projection. Only a very far lamp does that.
• Lamp C. C has the biggest shadow, so its lamp is the nearest to the card.