Plane Geometry, Points and Lines

Point, Line, Segment, Ray, Plane

A point is actually a location, rather than a thing. It might be the point 7 inches from the left edge of a ruler, or it might be the point on your paper that is 3 inches from the top and 4 inches from the left, or it might be a point in the middle of the room, 25 inches off the floor, 37 inches from the left wall, and 49 inches from the front wall. A point has no size and no thickness. The best real-world example is an electron, which seems to be a point charge with no size, although an electron never stands still long enough for us to measure its position precisely.

A line segment is the shortest path between two points. It starts at one point and heads straight to the other. On the surface of the earth, lines are curved, because you aren't allowed to tunnel through the earth. So the shortest distance from Canada To China is a curved arc that cuts across the north pole, and that's how airplanes fly. But in euclidean geometry, line segments are always straight.

A line is an infinite extension of a line segment. If the segment joins a and b, the point c is also in the line ab if b is contained in the segment ac, or a is contained in the segment bc. (Of course c could already be in the segment ab.)

A ray is an extension of a segment in one direction. The ray ab starts at a, moves along the segment ab, and extends beyond b, like a ray of light streaming out forever. Formally, c is in the ray ab if b is contained in the segment ac.

A plane is defined by a line l and a point p not on the line. All lines passing through p and intersecting l form the plane. In other words, a plane is like a piece of paper, but it has no thickness, and it goes on forever. The above is not a perfect definition, as it misses the line through p parallel to l. It is also not terribly rigorous. But Euclid's entire world was the plane, at least at the outset, so there was no pressing need to define it. Besides, "line segment" isn't defined rigorously either, until you have a distance metric, and a real vector space, and continuous paths, and so forth. Note that 3 noncolinear (not in a line) points determine a plane. The first two points determine a line and the third point, which is not part of that line, determines the plane.