Ingeometry, astraight line, usually abbreviatedline, is an infinitely long object with no width, depth, orcurvature. It is a special case of acurve and an idealization of such physical objects as astraightedge, a taut string, or aray of light. Lines arespaces ofdimension one, which may beembedded in spaces ofdimension two, three, or higher. The wordline may also refer, in everyday life, to aline segment, which is a part of a line delimited by twopoints (itsendpoints).
Euclid'sElements defines a straight line as a "breadthless length" that "lies evenly with respect to the points on itself", and introduced severalpostulates as basic unprovable properties on which the rest of geometry was established.Euclidean line andEuclidean geometry are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such asnon-Euclidean,projective, andaffine geometry.
In theGreek deductive geometry ofEuclid'sElements, a generalline (now called acurve) is defined as a "breadthless length", and astraight line (now called aline segment) was defined as a line "which lies evenly with the points on itself".[1]: 291 These definitions appeal to readers' physical experience, relying on terms that are not themselves defined, and the definitions are never explicitly referenced in the remainder of the text. In modern geometry, a line is usually either taken as aprimitive notion with properties given byaxioms,[1]: 95 or else defined as aset of points obeying a linear relationship, for instance whenreal numbers are taken to be primitive and geometry is establishedanalytically in terms of numericalcoordinates.
In an axiomatic formulation of Euclidean geometry, such as that ofHilbert (modern mathematicians added to Euclid's original axioms to fill perceived logical gaps),[1]: 108 a line is stated to have certain properties that relate it to other lines andpoints. For example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect at most at one point.[1]: 300 In twodimensions (i.e., the Euclideanplane), two lines that do not intersect are calledparallel. In higher dimensions, two lines that do not intersect are parallel if they are contained in aplane, orskew if they are not.
On aEuclidean plane, a line can be represented as a boundary between two regions.[2]: 104 Any collection of finitely many lines partitions the plane intoconvex polygons (possibly unbounded); this partition is known as anarrangement of lines.
Inthree-dimensional space, afirst degree equation in the variablesx,y, andz defines a plane, so two such equations, provided the planes they give rise to are not parallel, define a line which is the intersection of the planes. More generally, inn-dimensional spacen−1 first-degree equations in thencoordinate variables define a line under suitable conditions.
In more generalEuclidean space,Rn (and analogously in every otheraffine space), the lineL passing through two different pointsa andb is the subsetThedirection of the line is from a reference pointa (t = 0) to another pointb (t = 1), or in other words, in the direction of the vectorb − a. Different choices ofa andb can yield the same line.
Three or more points are said to becollinear if they lie on the same line. If three points are not collinear, there is exactly oneplane that contains them.
Inaffine coordinates, inn-dimensional space the pointsX = (x1,x2, ...,xn),Y = (y1,y2, ...,yn), andZ = (z1,z2, ...,zn) are collinear if thematrixhas arank less than 3. In particular, for three points in the plane (n = 2), the above matrix is square and the points are collinear if and only if itsdeterminant is zero.
Equivalently for three points in a plane, the points are collinear if and only if the slope between one pair of points equals the slope between any other pair of points (in which case the slope between the remaining pair of points will equal the other slopes). By extension,k points in a plane are collinear if and only if any (k–1) pairs of points have the same pairwise slopes.
The pointsa,b andc are collinear if and only ifd(x,a) =d(c,a) andd(x,b) =d(c,b) impliesx =c.
However, there are other notions of distance (such as theManhattan distance) for which this property is not true.
In the geometries where the concept of a line is aprimitive notion, as may be the case in somesynthetic geometries, other methods of determining collinearity are needed.
Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot.
In Euclidean geometry, all lines arecongruent, meaning that every line can be obtained by moving a specific line. However, lines may play special roles with respect to othergeometric objects and can be classified according to that relationship.
In the context of determiningparallelism in Euclidean geometry, atransversal is a line that intersects two other lines that may or not be parallel to each other.
For ahexagon with vertices lying on a conic we have thePascal line and, in the special case where the conic is a pair of lines, we have thePappus line.
Parallel lines are lines in the same plane that never cross.Intersecting lines share a single point in common. Coincidental lines coincide with each other—every point that is on either one of them is also on the other.
Insynthetic geometry, the concept of a line is often considered as aprimitive notion,[1]: 95 meaning it is not being defined by using other concepts, but it is defined by the properties, calledaxioms, that it must satisfy.[9]
However, theaxiomatic definition of a line does not explain the relevance of the concept and is often too abstract for beginners. So, the definition is often replaced or completed by amental image orintuitive description that allows understanding of what a line is. Such descriptions are sometimes referred to as definitions, but are not true definitions since they cannot be used inmathematical proofs. The "definition" of a line inEuclid's Elements falls into this category;[1]: 95 and is never used in proofs of theorems.
Line graphs of linear equations on the Cartesian plane
Lines in a Cartesian plane or, more generally, inaffine coordinates, are characterized by linear equations. More precisely, every line (including vertical lines) is the set of all points whosecoordinates (x,y) satisfy a linear equation; that is,wherea,b andc are fixedreal numbers (calledcoefficients) such thata andb are not both zero. Using this form, vertical lines correspond to equations withb = 0.
One can further suppose eitherc = 1 orc = 0, by dividing everything byc if it is not zero.
There are many variant ways to write the equation of a line which can all be converted from one to another by algebraic manipulation. The above form is sometimes called thestandard form. If the constant term is put on the left, the equation becomesand this is sometimes called thegeneral form of the equation. However, this terminology is not universally accepted, and many authors do not distinguish these two forms.
These forms are generally named by the type of information (data) about the line that is needed to write down the form. Some of the important data of a line is its slope,x-intercept, known points on the line and y-intercept.
The equation of the line passing through two different points and may be written asIfx0 ≠x1, this equation may be rewritten asorIntwo dimensions, the equation for non-vertical lines is often given in theslope–intercept form:
The slope of the line through points and, when, is given by and the equation of this line can be written.
As a note, lines in three dimensions may also be described as the simultaneous solutions of twolinear equationssuch that and are not proportional (the relations imply). This follows since in three dimensions a single linear equation typically describes aplane and a line is what is common to two distinct intersecting planes.
Parametric equations are also used to specify lines, particularly in those inthree dimensions or more because in more than two dimensions linescannot be described by a single linear equation.
In three dimensions lines are frequently described by parametric equations:where:
x,y, andz are all functions of the independent variablet which ranges over the real numbers.
(x0,y0,z0) is any point on the line.
a,b, andc are related to the slope of the line, such that the directionvector (a,b,c) is parallel to the line.
Parametric equations for lines in higher dimensions are similar in that they are based on the specification of one point on the line and a direction vector.
Distance from the origin O to the line E calculated with the Hesse normal form. Normal vector in red, line in green, point O shown in blue.
Thenormal form (also called theHesse normal form,[10] after the German mathematicianLudwig Otto Hesse), is based on thenormal segment for a given line, which is defined to be the line segment drawn from theorigin perpendicular to the line. This segment joins the origin with the closest point on the line to the origin. The normal form of the equation of a straight line on the plane is given by:where is the angle of inclination of the normal segment (the oriented angle from the unit vector of thex-axis to this segment), andp is the (positive) length of the normal segment. The normal form can be derived from the standard form by dividing all of the coefficients byand also multiplying through by if
Unlike the slope-intercept and intercept forms, this form can represent any line but also requires only two finite parameters, andp, to be specified. Ifp > 0, then is uniquely defined modulo2π. On the other hand, if the line is through the origin (c =p = 0), one drops thec/|c| term to compute and, and it follows that is only defined moduloπ.
In polar coordinates, the equation of a line not passing through theorigin—the point with coordinates(0, 0)—can be writtenwithr > 0 and Here,p is the (positive) length of theline segment perpendicular to the line and delimited by the origin and the line, and is the (oriented) angle from thex-axis to this segment.
It may be useful to express the equation in terms of the angle between thex-axis and the line. In this case, the equation becomeswithr > 0 and
These equations can be derived from thenormal form of the line equation by setting and and then applying theangle difference identity for sine or cosine.
These equations can also be provengeometrically by applyingright triangle definitions of sine and cosine to theright triangle that has a point of the line and the origin as vertices, and the line and its perpendicular through the origin as sides.
The previous forms do not apply for a line passing through the origin, but a simpler formula can be written: the polar coordinates of the points of a line passing through the origin and making an angle of with thex-axis, are the pairs such that
In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, inanalytic geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a givenlinear equation, but in a more abstract setting, such asincidence geometry, a line may be an independent object, distinct from the set of points which lie on it.
When a geometry is described by a set ofaxioms, the notion of a line is usually left undefined (a so-calledprimitive object). The properties of lines are then determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry. Thus indifferential geometry, a line may be interpreted as ageodesic (shortest path between points), while in someprojective geometries, a line is a 2-dimensional vector space (all linear combinations of two independent vectors). This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line.
A great circle divides the sphere in two equal hemispheres, while also satisfying the "no curvature" property.
In many models ofprojective geometry, the representation of a line rarely conforms to the notion of the "straight curve" as it is visualised in Euclidean geometry. Inelliptic geometry we see a typical example of this.[1]: 108 In the spherical representation of elliptic geometry, lines are represented bygreat circles of a sphere with diametrically opposite points identified. In a different model of elliptic geometry, lines are represented by Euclideanplanes passing through the origin. Even though these representations are visually distinct, they satisfy all the properties (such as, two points determining a unique line) that make them suitable representations for lines in this geometry.
The "shortness" and "straightness" of a line, interpreted as the property that thedistance along the line between any two of its points is minimized (seetriangle inequality), can be generalized and leads to the concept ofgeodesics inmetric spaces.
Given a line and any pointA on it, we may considerA as decomposing this line into two parts.Each such part is called aray and the pointA is called itsinitial point. It is also known ashalf-line (sometimes, ahalf-axis if it plays a distinct role, e.g., as part of acoordinate axis). It is a one-dimensionalhalf-space. The point A is considered to be a member of the ray.[a] Intuitively, a ray consists of those points on a line passing throughA and proceeding indefinitely, starting atA, in one direction only along the line. However, in order to use this concept of a ray in proofs a more precise definition is required.
Given distinct pointsA andB, they determine a unique ray with initial pointA. As two points define a unique line, this ray consists of all the points betweenA andB (includingA andB) and all the pointsC on the line throughA andB such thatB is betweenA andC.[12] This is, at times, also expressed as the set of all pointsC on the line determined byA andB such thatA is not betweenB andC.[13] A pointD, on the line determined byA andB but not in the ray with initial pointA determined byB, will determine another ray with initial pointA. With respect to theAB ray, theAD ray is called theopposite ray.
Thus, we would say that two different points,A andB, define a line and a decomposition of this line into thedisjoint union of an open segment(A, B) and two rays,BC andAD (the pointD is not drawn in the diagram, but is to the left ofA on the lineAB). These are not opposite rays since they have different initial points.
In Euclidean geometry two rays with a common endpoint form anangle.[14]
The definition of a ray depends upon the notion of betweenness for points on a line. It follows that rays exist only for geometries for which this notion exists, typicallyEuclidean geometry oraffine geometry over anordered field. On the other hand, rays do not exist inprojective geometry nor in a geometry over a non-ordered field, like thecomplex numbers or anyfinite field.
Aline segment is a part of a line that is bounded by two distinct end points and contains every point on the line between its end points. Depending on how the line segment is defined, either of the two end points may or may not be part of the line segment. Two or more line segments may have some of the same relationships as lines, such as being parallel, intersecting, or skew, but unlike lines they may be none of these, if they arecoplanar and either do not intersect or arecollinear.
A number line, with variable x on the left and y on the right. Therefore, x is smaller than y.
A point on number line corresponds to areal number and vice versa.[15] Usually,integers are evenly spaced on the line, with positive numbers are on the right, negative numbers on the left. As an extension to the concept, animaginary line representingimaginary numbers can be drawn perpendicular to the number line at zero.[16] The two lines forms thecomplex plane, a geometrical representation of the set ofcomplex numbers.
^On occasion we may consider a ray without its initial point. Such rays are calledopen rays, in contrast to the typical ray which would be said to beclosed.
^Alsina, Claudi; Nelsen, Roger B. (2010),Charming Proofs: A Journey Into Elegant Mathematics, MAA, pp. 108–109,ISBN9780883853481 (online copy, p. 108, atGoogle Books)
^Torrence, Bruce F.; Torrence, Eve A. (29 Jan 2009),The Student's Introduction to MATHEMATICA: A Handbook for Precalculus, Calculus, and Linear Algebra,Cambridge University Press, p. 314,ISBN9781139473736
^Wylie Jr., C.R. (1964),Foundations of Geometry, New York: McGraw-Hill, p. 59, definition 3,ISBN0-07-072191-2{{citation}}:ISBN / Date incompatibility (help)
^Pedoe, Dan (1988),Geometry: A Comprehensive Course, Mineola, NY: Dover, p. 2,ISBN0-486-65812-0
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