The point

Think of a treasure map: an X marks a spot, and a dotted path joins one spot to the next. Geometry begins in exactly the same way — with points that mark exact places and lines that join them up.

Geometry is built from a few plain ideas, and the simplest is the point. A point marks an exact position — a single spot — and nothing more. It has no size at all: no width, no height, no thickness. The dot we draw is only a marker; the true point is the precise place at its centre.

We name a point with a single capital letter, so we can talk about it: point A, point B, and so on.

Look up on a clear night. Each star looks like a tiny pin-prick of light — a single position in the sky, with no size you can measure by eye. Mapmakers of the sky give each bright star a name and treat it as a point, then join the points up to make the shapes we call constellations. That is exactly how geometry uses points: mark the spots first, then join them into lines and shapes.

Lines, segments and rays

A line is perfectly straight and goes on forever in both directions — it never bends and never stops. We draw little arrowheads on each end to remind us that it keeps going past the edge of the page. We can name a line by any two points on it: the line through A and B is called line AB.

Two related shapes are just parts of a line:

• a line segment is the piece between two points — it has two endpoints, like the segment from A to B, written segment AB. A segment is the only one of the three you could measure with a ruler;
• a ray starts at one endpoint and goes on forever in one direction — one end, one way, like a beam of light from a torch. We name a ray with its endpoint first: ray CD starts at C and shoots out through D.

There is one more quietly important fact: through any two distinct points there passes exactly one straight line. Pick two spots and there is one — and only one — perfectly straight line that joins them.

When sunlight streams through a gap in the clouds, each beam starts at the Sun and travels out in one straight direction — it has a clear starting end and then just keeps going. That is a perfect picture of a ray: one endpoint, then on and on forever in a single direction. The Sun is the endpoint; the beam is the ray.

Parallel and intersecting lines

When two straight lines share the same flat surface, exactly one of two things happens:

• they intersect — they cross at exactly one point (two straight lines can never cross more than once); or
• they are parallel — they run in the same direction, always the same distance apart, and never meet, no matter how far you follow them.

We mark parallel lines with matching little arrowheads, and we can write AB \parallel CD to say "line AB is parallel to line CD".

The two rails of a railway track are the most famous parallel lines in the world. They stay exactly the same distance apart for mile after mile — they have to, or the train would fall off! Stand between the rails and look into the distance and they seem to meet at the horizon, but that is a trick of your eyes. True parallel lines never actually touch.

Naming them — three worked examples

Geometry is careful about names. Here is how the three line-objects get theirs.

Example 1 — a segment. Two points P and Q mark the ends of a short straight piece. It is the segment PQ. Because a segment has two equal ends, the order does not matter: PQ and QP are the same segment.

Example 2 — a ray. A torch sits at point S and its beam passes through point T. This is ray ST — endpoint first. Here the order does matter: ray ST (starting at S) points the opposite way to ray TS (starting at T).

Example 3 — parallel or intersecting? Line AB runs east–west and line CD also runs east–west, one above the other. They keep the same gap forever, so they are parallel: AB \parallel CD. But line EF, which runs north–south, must cut across both of them — so EF intersects each of them at exactly one point.

The plane

A plane is a perfectly flat surface that extends forever in every direction — imagine a tabletop that never reaches an edge. Points and lines live in planes: a plane is the endless flat sheet on which we draw our geometry.

Points that all lie on one straight line have a special name — they are collinear. Step through the figure to watch a point grow into a line, a segment and a ray, and finally a whole plane.

Try it: join the points

Here are six random points, two on each row, each named with a capital letter. The top pair is joined into a segment (two ends), the middle pair into a ray (one arrowhead — it starts at one point and shoots past the other), and the bottom pair into a line (arrowheads at both ends — it runs on forever). Press Refresh to scatter fresh points and watch the same three shapes redraw.