In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature. It is a special case of a curve and an idealisation 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. The word line may also refer, in everyday life, to a line segment, which is a part of a line delimited by two points (its endpoints).
Euclid's Elements defines a straight line as a "breadthless length" that "lies evenly with respect to the points on itself", and introduced several postulates as basic unprovable properties on which the rest of geometry was established. Euclidean line and Euclidean geometry are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such as non-Euclidean, projective, and affine geometry.
Properties In the Greek deductive geometry of Euclid's Elements, a general line (now called a curve) is defined as a "breadthless length", and a straight line (now called a line segment) was defined as a line "which lies evenly with the points on itself". 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 a primitive notion with properties given by axioms, or else defined as a set of points obeying a linear relationship, for instance when real numbers are taken to be primitive and geometry is established analytically in terms of numerical coordinates.
In an axiomatic formulation of Euclidean geometry, such as that of Hilbert (modern mathematicians added to Euclid's original axioms to fill perceived logical gaps), a line is stated to have certain properties that relate it to other lines and points. 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. In two dimensions (i.e., the Euclidean plane), two lines that do not intersect are called parallel. In higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not.
On a Euclidean plane, a line can be represented as a boundary between two regions. Any collection of finitely many lines partitions the plane into convex polygons (possibly unbounded); this partition is known as an arrangement of lines.
In higher dimensions In three-dimensional space, a first degree equation in the variables , , and 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, in -dimensional space first-degree equations in the coordinate variables define a line under suitable conditions.
In more general Euclidean space, (and analogously in every other affine space), the line passing through two different points and is the subset
The orientation or direction of the oriented line (or directed line) above is from a reference point () to a target point (), or in other words, in the direction of the relative vector . A directed line is also called an axis, especially if it plays a distinctive role (e.g., an axis of rotation). Swapping points and yields the opposite directed line.
Collinear points
Three or more points are said to be collinear if they lie on the same line. If three points are not collinear, there is exactly one plane that contains them.
In affine coordinates, in -dimensional space the points
are collinear if the matrix
has a rank less than 3. In particular, for three points in the plane (), the above matrix is square and the points are collinear if and only if its determinant 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, points in a plane are collinear if and only if any () pairs of points have the same pairwise slopes.
In Euclidean geometry, the Euclidean distance between two points and may be used to express the collinearity between three points by:
However, there are other notions of distance (such as the Manhattan distance) for which this property is not true.
In the geometries where the concept of a line is a primitive notion, as may be the case in some synthetic geometries, other methods of determining collinearity are needed.
Relationship with other figures
In Euclidean geometry, all lines are congruent, meaning that every line can be obtained by moving a specific line. However, lines may play special roles with respect to other geometric objects and can be classified according to that relationship.
For instance, with respect to a conic (a circle, ellipse, parabola, or hyperbola), lines can be: tangent lines, which touch the conic at a single point; secant lines, which intersect the conic at two points and pass through its interior; exterior lines, which do not meet the conic at any point of the Euclidean plane; or a directrix, whose distance from a point helps to establish whether the point is on the conic. a coordinate line, a linear coordinate dimension
In the context of determining parallelism in Euclidean geometry, a transversal is a line that intersects two other lines that may or not be parallel to each other.
For more general algebraic curves, lines could also be: -secant lines, meeting the curve in points counted without multiplicity, or asymptotes, which a curve approaches arbitrarily closely without touching it. With respect to triangles we have: the Euler line, the Simson lines, and central lines.
For a convex quadrilateral with at most two parallel sides, the Newton line is the line that connects the midpoints of the two diagonals. ()
For a hexagon with vertices lying on a conic we have the Pascal line and, in the special case where the conic is a pair of lines, we have the Pappus 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.
Perpendicular lines are lines that intersect at right angles.
In three-dimensional space, skew lines are lines that are not in the same plane and thus do not intersect each other.
In axiomatic systems In synthetic geometry, the concept of a line is often considered as a primitive notion, meaning it is not being defined by using other concepts, but it is defined by the properties, called axioms, that it must satisfy.
However, the axiomatic 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 a mental image or intuitive 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 in mathematical proofs. The "definition" of a line in Euclid's Elements falls into this category; and is never used in proofs of theorems.
Definition
Linear equation
Lines in a Cartesian plane or, more generally, in affine coordinates, are characterized by linear equations. More precisely, every line (including vertical lines) is the set of all points whose coordinates satisfy a linear equation; that is,
where , and are fixed real numbers (called coefficients) such that and are not both zero. Using this form, vertical lines correspond to equations with .
One can further suppose either or , by dividing everything by 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 the standard form. If the constant term is put on the left, the equation becomes
and this is sometimes called the general 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, -intercept, known points on the line and -intercept.
The equation of the line passing through two different points and may be written as
If , this equation may be rewritten as
In two dimensions, the equation for non-vertical lines is often given in the slope–intercept form:
where is the slope or gradient of the line. is the y-intercept of the line. is the independent variable of the function .
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 two linear equations
such that and are not proportional (the relations , , imply ). This follows since in three dimensions a single linear equation typically describes a plane and a line is what is common to two distinct intersecting planes.
Parametric equation
Parametric equations are also used to specify lines, particularly in those in three dimensions or more, because in more than two dimensions lines cannot be described by a single linear equation.
In three dimensions, lines are frequently described by parametric equations:
where: , , and are all functions of the independent variable which ranges over the real numbers. is any point on the line. , , and are related to the slope of the line, such that the direction vector 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.
Hesse normal form
The normal form (also called the Hesse normal form, after the German mathematician Ludwig Otto Hesse), is based on the normal segment for a given line, which is defined to be the line segment drawn from the origin 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 the -axis to this segment), and is the (positive) length of the normal segment. The normal form can be derived from the standard form by dividing all of the coefficients by
and also multiplying through by –1 if .
Unlike the slope-intercept and intercept forms, this form can represent any line but also requires only two finite parameters, and , to be specified. If , then is uniquely defined modulo . On the other hand, if the line is through the origin (), one drops the term to compute and , and it follows that is only defined modulo .
Other representations
Vectors The vector equation of the line through points and is given by (where is a scalar).
If is vector and is vector , then the equation of the line can be written:
A ray starting at point is described by limiting . One ray is obtained if , and the opposite ray comes from .
Polar coordinates
In a Cartesian plane, polar coordinates are related to Cartesian coordinates by the parametric equations:
In polar coordinates, the equation of a line not passing through the origin—the point with coordinates —can be written
with and Here, is the (positive) length of the line segment perpendicular to the line and delimited by the origin and the line, and is the (oriented) angle from the -axis to this segment.
It may be useful to express the equation in terms of the angle between the -axis and the line. In this case, the equation becomes
with and .