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A 1 abA 2 A 1 abA 2 – Two stanzas, where the first lines of both stanzas are exactly the same, and the last lines of both stanzas are the same. The second lines of the two stanzas are different, but rhyme at the end with the first and last lines. (In other words, all the "A" and "a" lines rhyme with each other, but not with the "b" lines.)
In poetry, a couplet (/ ˈ k ʌ p l ə t / CUP-lət) or distich (/ ˈ d ɪ s t ɪ k / DISS-tick) is a pair of successive lines that rhyme and have the same metre. A couplet may be formal (closed) or run-on (open). In a formal (closed) couplet, each of the two lines is end-stopped, implying that there is a grammatical pause at the end of a line ...
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry.As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement.
Given a line a and two distinct intersecting lines m and n, each different from a, there exists a line g which intersects a and m, but not n. The splitting of the parallel postulate into the conjunction of these incidence-geometric axioms is possible only in the presence of absolute geometry .
Two lines are parallel if and only if the two angles of any pair of consecutive interior angles of any transversal are supplementary (sum to 180°). Proposition 1.28 of Euclid's Elements , a theorem of absolute geometry (hence valid in both hyperbolic and Euclidean Geometry ), proves that if the angles of a pair of consecutive interior angles ...
These short and sweet messages make a big statement.
Assume that we want to find intersection of two infinite lines in 2-dimensional space, defined as a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0. We can represent these two lines in line coordinates as U 1 = (a 1, b 1, c 1) and U 2 = (a 2, b 2, c 2). The intersection P′ of two lines is then simply given by [4]
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