# Definition:Parallel (Geometry)

*This page is about Parallel in the context of Geometry. For other uses, see Parallel.*

## Definition

### Lines

In the words of Euclid:

**Parallel**straight lines are straight lines which, being in the same plane and being produced indefinitely in either direction, do not meet one another in either direction.

(*The Elements*: Book $\text{I}$: Definition $23$)

The contemporary interpretation of the concept of parallelism declares that a straight line is parallel to itself.

### Planes

Two planes are **parallel** if and only if, when produced indefinitely, do not intersect at any point.

In the words of Euclid:

**Parallel planes**are those which do not meet.

(*The Elements*: Book $\text{XI}$: Definition $8$)

The contemporary interpretation of the concept of parallelism declares that a plane is parallel to itself.

### Line Parallel to Plane

Let $L$ be a straight line.

Let $P$ be a plane.

Then $L$ and $P$ are **parallel** if and only if, when produced indefinitely, they do not intersect at any point.

### Surfaces

Let $S_1$ and $S_2$ be surfaces in ordinary space.

Let $S_1$ and $S_2$ have the property that:

- for every point $P$ on $S_1$, a normal vector passing through $P$ is also a normal vector to $S_2$

and:

- for every point $Q$ on $S_2$, a normal vector passing through $Q$ is also a normal vector to $S_1$.

Then $S_1$ and $S_2$ are **parallel**.

## Also see

- Results about
**parallel**can be found**here**.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**parallel** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**parallel** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**parallel**