Parallel (geometry)
Parallel is a term in geometry and in everyday life that refers to a property of lines or planes. Parallel lines or planes are next to each other, but never touch each other. This means they never intersect at any point. If two lines [math]\displaystyle{ \ell_1 }[/math] and [math]\displaystyle{ \ell_2 }[/math] are parallel, then we describe this by writing [math]\displaystyle{ \ell_1 \parallel \ell_2 }[/math].[1][2] The slopes of parallel lines are always equal.
Even if these two line segments were extended to infinity, there would never be a point of intersection between the two of them. In fact, two parallel lines in the two-dimensional plane are always a fixed distance apart.[3]
Construction
Parallel (geometry) Media
Line art drawing of parallel lines and curves.
The problem: Draw a line through a parallel to l.
Intersecting, parallel and ultra parallel lines through a with respect to l in the hyperbolic plane. The parallel lines appear to intersect l just off the image. This is just an artifact of the visualisation. On a real hyperbolic plane the lines will get closer to each other and 'meet' in infinity.
On the sphere there is no such thing as a parallel line. Line a is a great circle, the equivalent of a straight line in spherical geometry. Line c is equidistant to line a but is not a great circle. It is a parallel of latitude. Line b is another geodesic which intersects a in two antipodal points. They share two common perpendiculars (one shown in blue).
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Related pages
References
- ↑ "List of Geometry and Trigonometry Symbols". Math Vault. 2020-04-17. Retrieved 2020-09-21.
- ↑ Weisstein, Eric W. "Parallel". mathworld.wolfram.com. Retrieved 2020-09-21.
- ↑ "Perpendicular and Parallel". www.mathsisfun.com. Retrieved 2020-09-21.