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Since these are equivalent properties, any one of them could be taken as the definition of parallel lines in Euclidean space, but the first and third properties involve measurement, and so, are "more complicated" than the second. Thus, the second property is the one usually chosen as the defining property of parallel lines in Euclidean geometry ...
This postulate does not specifically talk about parallel lines; [1] it is only a postulate related to parallelism. Euclid gave the definition of parallel lines in Book I, Definition 23 [2] just before the five postulates. [3] Euclidean geometry is the study of geometry that satisfies all of Euclid's axioms, including the parallel postulate.
Clifford's original definition was of curved parallel lines, but the concept generalizes to Clifford parallel objects of more than one dimension. [2] In 4-dimensional Euclidean space Clifford parallel objects of 1, 2, 3 or 4 dimensions are related by isoclinic rotations.
The number of vertices is smaller when some lines are parallel, or when some vertices are crossed by more than two lines. [4] An arrangement can be rotated, if necessary, to avoid axis-parallel lines. After this step, each ray that forms an edge of the arrangement extends either upward or downward from its endpoint; it cannot be horizontal.
As affine geometry deals with parallel lines, one of the properties of parallels noted by Pappus of Alexandria has been taken as a premise: [9] [10] Suppose A, B, C are on one line and A', B', C' on another. If the lines AB' and A'B are parallel and the lines BC' and B'C are parallel, then the lines CA' and C'A are parallel.
If σ is a collineation of a projective plane, a point P with P = P σ is called a fixed point of σ, and a line m with m = m σ is called a fixed line of σ. The points on a fixed line need not be fixed points, their images under σ are just constrained to lie on this line.
The parallel muscle architecture is found in muscles where the fibers are parallel to the force-generating axis. [1] These muscles are often used for fast or extensive movements and can be measured by the anatomical cross-sectional area (ACSA). [3] Parallel muscles can be further defined into three main categories: strap, fusiform, or fan-shaped.
A parallel of a curve is the envelope of a family of congruent circles centered on the curve. It generalises the concept of parallel (straight) lines. It can also be defined as a curve whose points are at a constant normal distance from a given curve. [1]
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