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