Definition:Line

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Definition

In the words of Euclid:

A line is breadthless length.

(The Elements: Book $\text{I}$: Definition $2$)


This can be interpreted to mean that a line is a construct that has no thickness.

This mathematical abstraction can not of course be actualised in reality because however thin you make your line, it will have some finite width.

It can be considered as a continuous succession of points.


The word line is frequently used to mean infinite straight line. The context ought to make it clear if this is the case.


Straight Line

In the words of Euclid:

A straight line is a line which lies evenly with the points on itself.

(The Elements: Book $\text{I}$: Definition $4$)


Curve

A curve is a line which is not straight.

More generally, it is a line whose straightness (or not) is either not relevant, not specified or unimportant.


Hence the word curve is used in certain branches of geometry to mean the same thing as line.


Arc

An arc is a specific part of a curve.


Line Segment

A line segment is any line (straight or not) which terminates at two points.


Straight Line Segment

A straight line segment is a line segment which is straight.


In the words of Euclid:

A straight line segment can be drawn joining any two points.

(The Elements: Postulates: Euclid's Second Postulate)


Endpoint

Each of the points at either end of a line segment is called an endpoint of that line segment.

Similarly, the point at which an infinite half-line terminates is called the endpoint of that line.


In the words of Euclid:

The extremities of a line are points.

(The Elements: Book $\text{I}$: Definition $3$)


Midpoint

Let $L = AB$ be a line segment whose endpoints are $A$ and $B$.

Let $M$ be a point on $L$ such that the line segment $AM$ is equal to the line segment $MB$.


Then $M$ is the midpoint of $L$.


Infinite Line

An infinite line is a line which has no endpoints.


Infinite Half-Line

An infinite half-line is a line which terminates at an endpoint at one end, but has no such endpoint at the other.


Infinite Straight Line

An infinite straight line is a straight line which has no endpoints, or equally, a straight line which is infinite.


Equality of Line Segments

Two line segments are equal iff they have the same length.


Also see

In the words of Euclid:

The extremities of a surface are lines.

(The Elements: Book $\text{I}$: Definition $6$)


  • Results about lines can be found here.


Sources