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In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical objects as a straightedge, a taut string, or a ray of light. Lines are spaces of dimension one, which may be embedded in spaces of dimension two, three, or higher.
In studying geometry one concentrates on the position of points and on the length, orientation and curvature of lines. Geometrical–optical illusions then relate in the first instance to object characteristics as defined by geometry.
In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the general case situation, as opposed to some more special or coincidental cases that are possible, which is referred to as special position. Its precise meaning differs in different settings.
In geometry, a generalized circle, sometimes called a cline or circline, [1] is a straight line or a circle, the curves of constant curvature in the Euclidean plane. The natural setting for generalized circles is the extended plane, a plane along with one point at infinity through which every straight line is considered to pass. Given any three ...
With these modern definitions, every geometric shape is defined as a set of points; this is not the case in synthetic geometry, where a line is another fundamental object that is not viewed as the set of the points through which it passes. However, there are modern geometries in which points are not primitive objects, or even without points.
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The notions themselves may not necessarily need to be stated; Susan Haack (1978) writes, "A set of axioms is sometimes said to give an implicit definition of its primitive terms." [9] Euclidean geometry: Under Hilbert's axiom system the primitive notions are point, line, plane, congruence, betweeness, and incidence.
As any line in this extension of σ corresponds to a plane through O, and since any pair of such planes intersects in a line through O, one can conclude that any pair of lines in the extension intersect: the point of intersection lies where the plane intersection meets σ or the line at infinity. Thus the axiom of projective geometry, requiring ...