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Introduction to Three Dimensional Geometry
Introduction
Coordinate Axes and Coordinate Planes in Three Dimensional Space
Coordinates of a Point in Space
Practice: Points and Octants in Space
Distance between Two Points
Practice: Distance in Three Dimensions
Miscellaneous Examples
Chapter Review: Mixed Practice
Summary
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Introduction to Three Dimensional GeometryIntroduction

Reading time: ~15 min

Riya has just moved into a new flat, and tomorrow an electrician is coming to change the bulb that hangs from her bedroom ceiling. He asks her on the phone exactly where the bulb is, so that he can bring a ladder of the right height.

On a sheet of graph paper, two numbers are enough to mark any spot — that is what x and y have done for you since Class 9. But the bulb is not lying on the floor. Are two numbers still enough?

Find out with Riya's toy drone. The picture is one corner of her room: the floor and the two walls behind it. The three dashed lines are the drone's straight-line distances to the back wall, the left wall and the floor. Drag the picture sideways to turn the room, then use the sliders below it to complete both missions.

Fly the drone with the three sliders and complete both missions.

Distance from the back wall: ${a} m

Distance from the left wall: ${b} m

Height above the floor: ${h} m

Two numbers took the drone to the right spot on the floor — the bulb's shadow — but above that spot lies a whole vertical line of positions. Only the third number, the height, picked out the bulb itself. So the bulb's position is (4, 5, 3): 4 m from the back wall, 5 m from the left wall and 3 m above the floor.

That is the idea behind this whole chapter. A point on a plane needs numbers, but a point in space needs : its distances from three planes that meet at right angles, like the floor and two walls of a room. These three numbers are the coordinates of the point with respect to the three coordinate planes. Your phone does exactly this when it fixes your latitude, longitude and height above sea level, and so does an air-traffic controller following a plane, or a video game placing a character in its world.

A Fly on the Ceiling

A famous story — probably more legend than fact — says that René Descartes, lying in bed one morning, watched a fly crawl across his ceiling and realised that two numbers could say exactly where it was: its distances from two of the walls.

That is the coordinate geometry of the plane. But the moment the fly took off into the room, two numbers were no longer enough. A fly in the air needs numbers.

Descartes (1596–1650) published his geometry in 1637. He, his co-inventor Pierre de Fermat (1601–1665) and, later, Philippe de La Hire (1640–1718) worked almost entirely in the plane; their writings hint at a third coordinate but never develop it. In 1700 Antoine Parent (1666–1716) presented the first systematic geometry of solids with coordinates to the French Academy. In a letter of 1715 to Leibniz, Johann Bernoulli (1667–1748) described the three coordinate planes we still use today. And in 1748 Leonhard Euler (1707–1783) set out three-dimensional coordinate geometry in full, in an appendix to the second volume of his Introductio.

Drag these milestones into the order in which they happened, earliest at the top.

Johann Bernoulli describes the three coordinate planes in a letter to Leibniz
Descartes publishes his coordinate geometry of the plane
Einstein's relativity joins time to space as a fourth dimension
Antoine Parent presents a systematic geometry of solids with coordinates
Euler sets out three-dimensional coordinate geometry in his Introductio

Geometry in more than three dimensions arrived only in the middle of the nineteenth century, and its most famous use is in Albert Einstein's theory of relativity, where time and the three dimensions of space form one four-dimensional "space-time". In this chapter we stay with three: axes, planes, coordinates, and the distance between two points.

Sina