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If geometry is developed axiomatically (as in Euclid's Elements and later in David Hilbert's Foundations of Geometry ), then lines are not defined at all, but characterized axiomatically by their properties. "Everything that satisfies the axioms for a line is a line." While Euclid did define a line as "length without breadth", he did not use this rather obscure definition in his later development.
In Euclidean space Rn (and analogously in all other vector spaces), we define a line L as a subset of the form
where a and b are given vectors in Rn with b non-zero. The vector b describes the direction of the line, and a is a point on the line. Different choices of a and b can yield the same line.
One can show that in R2, every line L is described by a linear equation of the form
with fixed real coefficients a, b and c such that a and b are not both zero. Important properties of these lines are their slope, x-intercept and y-intercept.
More abstractly, one usually thinks of the real line as the prototype of a line, and assumes that the points on a line stand in a one-to-one correspondence with the real numbers. However, one could also use the hyperreal numberIn mathematical logic, hyperreal numbers or nonstandard reals (usually denoted as R denote an ordered field which is a proper extension of the ordered field of real numbers and which have a transfer principle which allows true first order statements abouts for this purpose, or even the long lineIn topology, the long line is a topological space analogous to the real line, but much longer. Because it behaves locally just like the real line, but has different large-scale properties, it serves as one of the basic counterexamples of topology. Definit of topologyTopology is the study or science of places. It derives its name from the Greek words τοπος meaning place and λογος meaning study, talk. See also earth science, geography, human geography, g.
The "straightness" of a line, interpreted as the property that it minimizes distances between its points, can be generalized and leads to the concept of geodesicIn mathematics, a geodesic is a curve which is "straight" in some sense. It takes its name from the science of geodesy of measuring the size and shape of the earth, and was originally the shortest route between two points on the surface of the earth.s on differentiable manifoldIn mathematics, a manifold ''M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. Therefors.
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