Dreaptă (matematică)

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Dreapta, în matematică, este linie ce poate fi definită ca având doar o dimensiune, lungimea. Orice dreaptă este de lungime infinită, conţine o infinitate de puncte, este de grosime zero şi este o curbă perfect "dreaptă". În geometria euclidiană, pentru două puncte fixate există o dreaptă şi numai una ce trece prin amândouă. Folosind metrica standard, linia dreaptă reprezintă drumul cel mai scurt dintre două puncte.

În cazul bidimensional, două drepte diferite pot fi confundate (au toate punctele comune), paralele (sunt disjuncte, nu au nici un punct comun) sau concurente (se intersectează, întotdeauna într-un punct şi numai unul). În cazurile tri- sau multi-dimensionale, dreptele pot fi oarecare între ele, însemnând nu numai că nu se intersectează, dar şi că nu definesc un plan. Două plane distincte se pot intersecta având minimum o dreaptă comună. Trei sau mai multe puncte care aparţin uneia şi aceleiaşi drepte se numesc coliniare.

Cuprins 1 Exemple 2 Definiţii formale 3 Proprietăţi 4 Semi-dreaptă 5 Vedeţi şi 6 Legături externe

[modifică] Exemple

Lines in a Cartesian plane can be described algebraically by linear equations and linear functions. In two dimensions, the characteristic equation is often given by the slope-intercept form:

where:

m is the slope of the line.

b is the y-intercept of the line.

x is the independent variable of the function y.

In three dimensions, a line is often described by parametric equations:

where:

x, y, and z are all functions of the independent variable t.

x₀, y₀, and z₀ are the initial values of each respective variable.

a, b, and c are related to the slope of the line, such that the vector (a, b, c) is a parallel to the line.

[modifică] Definiţii formale

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. 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.

[modifică] Proprietăţi

In a two-dimensional space, such as the plane, two different lines must either be parallel lines or must intersect at one point. In higher-dimensional spaces however, two lines may do neither, and two such lines are called skew lines.

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 (see Linear equation for other forms). Important properties of these lines are their slope, x-intercept and y-intercept. The eccentricity of a straight line is infinity.

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.

[modifică] Semi-dreaptă

In Euclidean geometry, a ray, or half-line, given two distinct points A (the origin) and B on the ray, is the set of points C on the line containing points A and B such that A is not strictly between C and B. In geometry, a ray starts at one point, then goes on forever in one direction.

[modifică] Vedeţi şi

• Line segment
• Affine function
• Diffraction
• Glossary of Riemannian and metric geometry#R for its meaning in Riemannian geometry.
• Incidence (geometry)
• Minimal line representation

[modifică] Legături externe

• Detailed explanation of the line at MathWorld Encyclopedia
• Equations of the Straight Line at cut-the-knot

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