The line or rope that is used to control the main sail is called the Halyard, and this name is derived from the phrase "Haul the Yard".
Sheets.Correction:Each sail has its own name based on its position on the boat and upon its usage. A "sheet" is a woven line used to control a sail, typically by controlling the clew, and is not a sail at all.
The "rope used to raise and lower sail" is no longer a rope when it has an assigned task aboard ship. It is more correctly referred to as a line, and its' assigned title is "halyard".It is also utilized to hold the sail in place when it is not raising or lowering the sail. The name derives from "haul yard", referring to the movement (hauling) of "yards of cloth".
in a sailing dinghy, the name given to ropes used for hoisting the sail is halyards.
sail
The name "line" is actually used in math.The name "line" is actually used in math.The name "line" is actually used in math.The name "line" is actually used in math.
The Nina, the Pinta and the Santa Maria
In the case of most offshore yachts, yes, they can be "reefed" (furled and tied down) to reduce the amount of surface area exposed to the wind. However this does depend on the type of furling gear used, some (mostly jibs) can only be used fully unfurled (In general a solid forestay is required for a roller-reefing jib.)
There is no specific name - other than what you used: "a pair of parallel line segments".
directrix
Rhythms are generally used to determine what the name of a line of poetry will be. Repeating aural patters are also often used.
The name of the line for the compass alignment is the index line. The line and the numbers around the outside edge can be used to plot a course to a distant object.