Points, Lines, Segments, and Rays

What is a point? What a wonderful question.

Ok, let's first see an example:

Now that we've seen what a point is not, let's see what a point is. As we just saw, a point is not a thing, like a dot. Rather, a point is a location, and the dot you see above represents a location.

We generally denote points with a single capital letter, like so

...and of course you can always have multiple points, as long as they don't have the same name

And you reference them like "Point $H$ is at $(0,0)$."

We say that points are zero-dimensional (as opposed to lines for example, which are "one dimensional"). Though we haven't defined what it means for something to be zero dimensional, this notion may be intuitively helpful in the future.

Hope you found this interesting!

Lines, Line Segments and Rays

Correct, the line extends indefinitely in both directions.

Correct, line segments are finite. (between the two points)

Correct, ray extends indefinitely in only one direction.

Hope this makes you differentiate between lines, line segments and rays.

Definitions (or lack thereof)

A point is a dot in space with no dimension. Maybe something seems amiss here. Shouldn't we be able to do better than the above statement? Yes, it gets the idea across, but isn't math supposed to be more rigorous than this? Well...

Think of the simplest notion in all of geometry: a shape? No, that can be defined in terms of lines. A line? Lines can be defined in terms of points. But a point? A point is a "primitive" notion, meaning we cannot define it because we have nothing to use in the definition. You cannot build rigor without some sort of context.

That being said, the intuitive idea of a point, ie. a dot in space with no dimension, should suffice.

A Zero-Dimensional Object

Points are sometimes referred to as zero-dimensional, since you don't need any n dimensions to describe the notion of a point. Also, if you were to land on a point, you wouldn't be able to move at all . You'd just be, well, a point floating around in space.

Lines

A line is an geometric object defined by two distinct points ("distinct" here means that the points aren't in the same location). Along with these points, a line is made up of the infinitely many other points it passes through.

Line Segments and Rays

Line segments and rays are parts of a line set off by points. Two distinct points on a line define a line segment (the line segment has a length, the distance between the two points). Similarly, one distinct point on a line defines two rays (intuitively, the ray starting from the point going left along the line, and the ray starting from the point going right along the line).

Dimension

Lines, line segments, and rays are all one-dimensional objects: in order to have the concept of a line make sense in an argument, the context in which you are arguing must have at least one dimension.

Examples

Number lines are a type of line. The edges on polygons (two-dimensional shapes), polyhedra (three-dimensional shapes), and polytopes (4+ dimensional shapes) are examples of line segments. Taking the number line example from before, an example of a ray would be the part of the number line lying to the left of zero (the negative numbers), or the part of the number line lying to the right of zero (the positive numbers).

Collinear Points

Three distinct points are said to be collinear if all three of them contained within the same line. Note that any two points in space are always collinear.

Introduction

You've probably seen lines, line segments, and rays (at least if you know me) before. In this exploration, we'll see

1. How do we define a line?

2. What are the differences between lines, line segments, and rays?

1. How do we define a line?

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Let's begin! First, let's see if you can come up with the definition of a line on your own.

Your definition might have looked something like

A straight thing that extends infinitely long and is infinitely thin

However, what do each of these words, "straight", "infinitely long", and "infinitely thin" mean? You probably have a pretty good idea of what these terms mean, but let's think about how we can improve the current definition.

Good, so far the only geometric object we have seen to far is a point, so it seems natural to define a line in terms of points. Now we can refine our definition a bit:

A line is a collection of points that is straight, extends infinitely long, and is infinitely thin

The idea that a line is a collection of points helps give us a better idea of what it actually is.

We still haven't defined any of straight, infinitely long, or infinitely thin. One way you might go about defining straight is in terms of slope, but we will leave the discussion at this for now.

1. What are the differences between lines, line segments, and rays?

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It's important to make this distinction clear. Here are the formal definitions of line, segment, and ray.

A line is a set of points that is straight, infinitely thin, and that extends infinitely far in one direction

A line segment (or just segment) is a part of a line bounded between two endpoints

A ray is a part of a line bounded on one side by a point and extends infinitely far in the other direction.

...but this is best shown by examples.

Well done!

A line goes on forever. A ray goes on forever on one end and stops somewhere on the other end. A line segment stops somewhere on both ends.

"A point is a dot in space with no dimension" In other words, if we are far away from a point, in order to give it any appearance at all, it may need to be as big as a sun (from millions of miles away), and the closer we get to that point, the more it shrinks, and shrinks, and becomes smaller yet, until we boink our nose into it, at which point it is so small we cannot perceive it at all, because it has no dimension whatsoever. You gotta love math, or else, uhm, what's the point!

(Video) Points, Lines, and Planes

by MathAntics

This video explains the difference between a line, a segment and a ray. It also shows some ways to annotate them. Correct notation is important in geometry, it establishes a simple way of communicating ideas and definitions.

A point is a dot that occupies the space without any mass.

A point is where lines can form an angle, or mark a position.