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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 ...
In geometry, the parallel postulate, also called Euclid 's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: If a line segment intersects two straight lines forming two interior angles on the same side that are less than two right ...
v. t. e. A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), along with two diverging ultra-parallel lines. In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai – Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
If there are closest parallels to a given line they are known as the limiting parallel, asymptotic parallel or horoparallel (horo from Greek: ὅριον — border). For rays, the relation of limiting parallel is an equivalence relation, which includes the equivalence relation of being coterminal. If, in a hyperbolic triangle, the pairs of ...
A vanishing point is a point on the image plane of a perspective rendering where the two-dimensional perspective projections of mutually parallel lines in three-dimensional space appear to converge. When the set of parallel lines is perpendicular to a picture plane, the construction is known as one-point perspective, and their vanishing point ...
Newton-Gauss line through the midpoints L, M, N of the diagonals. In geometry, the Newton–Gauss line (or Gauss–Newton line) is the line joining the midpoints of the three diagonals of a complete quadrilateral. The midpoints of the two diagonals of a convex quadrilateral with at most two parallel sides are distinct and thus determine a line ...
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.
By comparison, a quadrilateral with at least one pair of parallel sides is a trapezoid in American English or a trapezium in British English. The three-dimensional counterpart of a parallelogram is a parallelepiped. The word comes from the Greek παραλληλό-γραμμον, parallēló-grammon, which means a shape "of parallel lines".