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Encyclopedia > Straight line

A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i.e. a curve that is long and straight. Given two points, in Euclidean geometry, one can always find exactly one line that passes through the two points; the line provides the shortest connection between the points. Three or more points that lie on the same line are called collinear. Two different lines can intersect in at most one point; two different planes can intersect in at most one line. This intuitive concept of a line can be formalized in various ways.

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 numbers for this purpose, or even the long line of topology.

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 geodesics on differentiable manifolds. Results from FactBites:

 How to get Moire from your computer. (3757 words) The equations in lines 44 and 46 are the generating functions, solved to determine the distance from the center; that distance is then compared against the list of bounding radii in the data statement, while a count is kept to determine the light/dark decision. The lines were close enough together that the ]['s associated with near intersections seemed to clutter up the picture, so I modified the print routine for the version shown in the fourth to eliminate them, and the result shows the moire lines much more clearly. In the third pattern (shown on page 174), radial straight lines of alternating color, the hoped for illusion was not striking, but the detail near the center of the pattern was very rewarding anyway.
 The UnMuseum - The Lines of Nazca (910 words) In and around the lines there are also trapezoidal zones, strange symbols, and pictures of birds and beasts all etched on a giant scale that can only be appreciated from the sky. The Nasca lines were created by clearing the darkened pampa stones to either side and exposing the lighter sand underneath. Two wooden stakes placed as a straight line would be used to guide the placement of a third stake along the line.
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