Projection on a plane

The idea

The projection of a point P on a plane is the point O where the line from P perpendicular to the plane meets it; the projections of all points of an object form its projection.

A stone on a string

A drone hovers above the ground. Hang a stone on a string from it. The string hangs straight down and the stone touches the ground at one spot.

At what angle does the string meet the lines drawn on the ground?

• A different angle for each line
• 90° with every line on the ground
• 90° with only one of the lines

Find the foot of P

The point P hovers above the floor. Move Q on the floor with the sliders. The three green lines QX lie on the floor. The toy measures the angles PQX. Drag the picture to look from the side.

Project a whole cube

A cube hovers above the floor. Tap each corner to drop its perpendicular to the floor. The dots on the floor are the projections of the corners. Then turn and tip the cube with the sliders.

The foot of the perpendicular

The projection of a point P on a plane is the point O where the line from P, perpendicular to the plane, meets it. The projections of all the points of an object together form the projection of the object.

OP is perpendicular to the plane when ∠POX = 90° for every line OX on the plane. One right angle is not enough.

The cube gave a square when it stood straight and a rectangle when you tipped it: the same object can have different projections.

Notes

The projection of a point P on a plane is the point O where the line from P perpendicular to the plane meets it, so ∠POX = 90° for every line OX on the plane. The projections of all the points of an object form its projection.

Check yourself

Here P is joined to Q on the floor, and the toy measured the three angles. Is Q the projection of P?

A drone P hovers 12 m above flat ground. O is the point of the ground directly below it. A rope 13 m long joins the drone to a stake X on the ground. How far is the stake from O?

Answer: 5 m

∠POX = 90°, so OX² = 13² − 12² = 169 − 144 = 25, and OX = 5 m.

A cube is held straight, with a face parallel to the floor. Its 8 corners are projected on the floor. How many different points do you get?

Answer: 4 points

The perpendicular from a top corner passes through the corner straight below it, so they land on the same spot. 8 corners give 4 points.

Ravi joins P to a point Q on the floor with a slanted line and says, "Q is the projection of P." Is he right?

• Yes. One of the angles is 90°.. The projection needs ∠PQX = 90° for every line QX on the floor, not just one of them.
• No. P has no projection on the floor.. Every point has a projection: the foot of the perpendicular from it to the plane.
• No. The three angles are not all 90°, so PQ is not perpendicular to the floor. — correct. Yes! Only one angle is 90°. For the projection, all of them must be.
• Yes. Any line from P to the floor gives the projection.. A slanted line gives a different point for every slant. Only one point is the projection.
• No. The projection is the point where the line from P perpendicular to the floor meets it. — correct. Yes! The line from P must be perpendicular to the plane.
• No. A point can only be projected on a vertical plane.. A point can be projected on any plane, for example on the floor.