Parallelism (geometry)

In Euclidean geometry

Elements of Euclide

The concept of parallelism already exists in the elements of Euclide. It is important to stress that, for Euclide, are connected rather with one.

Definition 35: the parallels are lines which, being located in the same plan, and being prolonged ad infinitum on both sides, meet neither on a side nor other
Proposition 27: If a line, falling on two lines, fact of the equal alternate angles between them, these two lines will be parallel
Proposition 31: By a given point, it passes at least a parallel on a given line

Postulate 5 If a line, falling on two lines, forms the interior angles on the same side smaller than two rights, these lines, prolonged ad infinitum, will meet side where the angles are smaller than two rights makes it possible to prove

the unicity of the parallel on a given line passing by a given point.
proposal 29: If two lines are parallel, very right-hand side cutting one and the other, form with this one of the equal alternate angles.

Propostition 30: Two distinct lines parallel on the same line are parallel between them

Proposition 32: the sum of the angles of a triangle is equal to 180°

Proposition 33 : Two lines which unite same sides of parallel straight lines and of the same length are parallel and of the same length (the figure traced in proposal 33 is a Parallélogramme))

Proposal 34: the opposite sides and the angles opposed in a parallelogram are equal, and the diagonal cuts the parallelogram in two equal triangles.

Relation of equivalence

In short, while agreeing to regard lines confused as parallels, one can see that the relation of parallelism is then

reflexive: a line is parallel with itself
symmetrical: If a line (d) is parallel on a line (d') then the line (d') is parallel to the right-hand side (d)
transitive: : If a line (d) is parallel to (of) and if (of) is parallel to (d") then (d) is parallel to (d")

What makes it possible to say that the relation of parallelism is a Relation of equivalence whose classes of equivalence are the directions lines.

In Géométrie refines

In geometry closely connected plane

A line is defined by a point and a directing vector. Two lines are known as parallels if and only if their directing vectors are colinéaires. It appears whereas two confused lines are parallel according to this definition whereas they were not it according to the definition of Euclide. Two parallel distinct lines are then called strictly parallel.

In a space refines dimension 3

In a space refines, two plans are defined by a directing point and two vectors not colinéaires.

Two plans are parallel if and only if the four directing vectors are Coplanaire S. In a space of dimension three, two plans are or parallel (without points commun run or confused) or secant according to a line.

A line is parallel to a plan if and only if the three directing vectors (both of the plan and that of the right-hand side) are coplanar. In a space of dimension 3, being given a line and a plan, or the line are parallel to the plan, or the line and the plan are secant according to a point.

In a Espace dimension N refines

A space closely connected of dimension p is defined using a point and of a vectorial subspace of dimension p called direction of space closely connected. Two subspace closely connected of dimension p are parallel if and only if they have the same vectorial subspace like direction. Two subspace closely connected parallels are disjoined or confused.

The relation of parallelism remains a relation of equivalence on the whole of the subspaces closely connected of dimension p

Simple: Parallel (geometry)

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