Definition:Line
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Definition
As Euclid defined it:
- A line is breadthless length.
(The Elements: Book 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
As Euclid defined it:
- A straight line is a line which lies evenly with the points on itself.
(The Elements: Book I: Definition $4$)
Curve
A curve (or curved line) is a line which is not straight.
More generally, it is a line whose straightness (or not) is either not relevant or unimportant. Hence the word curve is used in certain branches of geometry as a synonym for line.
Arc
An arc is a specific part of a curve.
Line Segment
An alternative definition for straight line which is frequently encountered is:
- A straight line is the shortest distance between two points.
This is all very well but it assumes that the line is therefore terminated at two particular points, that is, at either end.
However, this is a perfectly valid definition of the concept of a straight line segment.
As Euclid defined it:
- The extremities of a line are points.
(The Elements: Book I: Definition $3$)
His first postulate states:
- A straight line segment can be drawn joining any two points.
A line segment, therefore, is any line (straight or not) which terminates at two points.
Also, we have:
- The extremities of a surface are lines.
(The Elements: Book I: Definition $6$)
Endpoint
Each of the points at either end of a line segment is called an endpoint (or end point).
Similarly, the point at which an infinite half-line terminates is called the endpoint of that line.
Infinite Line
An infinite line is a line which has no endpoints.
Infinite Half-Line
An infinite half-line is a line which starts at (or terminates at, depending on how you want to look at it) a point at one end, but goes on to infinity at the other.
Another term for this is a ray.
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.
