Line (geometry)

Lines in a Cartesian plane or, more generally, in affine coordinates, can be described algebraically by linear equations.

In two dimensions, the equation for non-vertical lines is often given in the slope-intercept form:

${\displaystyle y=mx+b}$

where:

m is the slope or gradient of the line.

b is the y-intercept of the line.

x is the independent variable of the function y = f(x).

The slope of the line through points ${\displaystyle A(x_{a},y_{a})}$ and ${\displaystyle B(x_{b},y_{b})}$, when ${\displaystyle x_{a}\neq x_{b}}$, is given by ${\displaystyle m=(y_{b}-y_{a})/(x_{b}-x_{a})}$ and the equation of this line can be written ${\displaystyle y=m(x-x_{a})+y_{a}}$.

In ${\displaystyle \mathbb {R^{2}} }$, every line ${\displaystyle L}$ (including vertical lines) is described by a linear equation of the form

${\displaystyle L=\{(x,y)\mid ax+by=c\}}$

with fixed real coefficients a, b and c such that a and b are not both zero. Using this form, vertical lines correspond to the equations with b = 0.

There are many variant ways to write the equation of a line which can all be converted from one to another by algebraic manipulation. These forms (see Linear equation for other 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 ${\displaystyle P_{0}(x_{0},y_{0})}$ and ${\displaystyle P_{1}(x_{1},y_{1})}$ may be written as

${\displaystyle (y-y_{0})(x_{1}-x_{0})=(y_{1}-y_{0})(x-x_{0})}$.

If x₀ ≠ x₁, this equation may be rewritten as

${\displaystyle y=(x-x_{0})\,{\frac {y_{1}-y_{0}}{x_{1}-x_{0}}}+y_{0}}$

or

${\displaystyle y=x\,{\frac {y_{1}-y_{0}}{x_{1}-x_{0}}}+{\frac {x_{1}y_{0}-x_{0}y_{1}}{x_{1}-x_{0}}}\,.}$

In three dimensions, lines can not be described by a single linear equation, so they are frequently described by parametric equations:

${\displaystyle x=x_{0}+at}$

${\displaystyle y=y_{0}+bt}$

${\displaystyle z=z_{0}+ct}$

where:

x, y, and z are all functions of the independent variable t which ranges over the real numbers.

(x₀, y₀, z₀) is any point on the line.

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

They may also be described as the simultaneous solutions of two linear equations

${\displaystyle a_{1}x+b_{1}y+c_{1}z-d_{1}=0}$

${\displaystyle a_{2}x+b_{2}y+c_{2}z-d_{2}=0}$

such that ${\displaystyle (a_{1},b_{1},c_{1})}$ and ${\displaystyle (a_{2},b_{2},c_{2})}$ are not proportional (the relations ${\displaystyle a_{1}=ta_{2},b_{1}=tb_{2},c_{1}=tc_{2}}$ imply ${\displaystyle t=0}$). 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.

The normal form (also called the Hesse normal form,[11] 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:

${\displaystyle y\sin \theta +x\cos \theta -p=0,}$

where θ is the angle of inclination of the normal segment (the oriented angle from the unit vector of the x axis to this segment), and p is the (positive) length of the normal segment. The normal form can be derived from the general form ${\displaystyle ax+by=c}$ by dividing all of the coefficients by

${\displaystyle {\frac {c}{|c|}}{\sqrt {a^{2}+b^{2}}}.}$

Unlike the slope-intercept and intercept forms, this form can represent any line but also requires only two finite parameters, θ and p, to be specified. If p > 0, then θ is uniquely defined modulo 2π. On the other hand, if the line is through the origin (c = 0, p = 0), one drops the c/|c| term to compute sin θ and cos θ, and θ is only defined modulo π.

In polar coordinates on the Euclidean plane the slope-intercept form of the equation of a line is expressed as:

${\displaystyle r={\frac {mr\cos \theta +b}{\sin \theta }},}$

where m is the slope of the line and b is the y-intercept. When θ = 0 the graph will be undefined. The equation can be rewritten to eliminate discontinuities in this manner:

${\displaystyle r\sin \theta =mr\cos \theta +b.}$

In polar coordinates on the Euclidean plane, the intercept form of the equation of a line that is non-horizontal, non-vertical, and does not pass through pole may be expressed as,

${\displaystyle r={\frac {1}{{\frac {\cos \theta }{x_{o}}}+{\frac {\sin \theta }{y_{o}}}}}}$

where ${\displaystyle x_{o}}$ and ${\displaystyle y_{o}}$ represent the x and y intercepts respectively. The above equation is not applicable for vertical and horizontal lines because in these cases one of the intercepts does not exist. Moreover, it is not applicable on lines passing through the pole since in this case, both x and y intercepts are zero (which is not allowed here since ${\displaystyle x_{o}}$ and ${\displaystyle y_{o}}$ are denominators). A vertical line that doesn't pass through the pole is given by the equation

${\displaystyle r\cos \theta =x_{o}.}$

Similarly, a horizontal line that doesn't pass through the pole is given by the equation

${\displaystyle r\sin \theta =y_{o}.}$

The equation of a line which passes through the pole is simply given as:

${\displaystyle \tan \theta =m}$

where m is the slope of the line.

The vector equation of the line through points A and B is given by ${\displaystyle \mathbf {r} =\mathbf {OA} +\lambda \,\mathbf {AB} }$ (where λ is a scalar).

If a is vector OA and b is vector OB, then the equation of the line can be written: ${\displaystyle \mathbf {r} =\mathbf {a} +\lambda (\mathbf {b} -\mathbf {a} )}$.

A ray starting at point A is described by limiting λ. One ray is obtained if λ ≥ 0, and the opposite ray comes from λ ≤ 0.

In three-dimensional space, a first degree equation in the variables x, y, and z 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 n-dimensional space n-1 first-degree equations in the n coordinate variables define a line under suitable conditions.

In more general Euclidean space, Rⁿ (and analogously in every other affine space), the line L passing through two different points a and b (considered as vectors) is the subset

${\displaystyle L=\{(1-t)\,a+t\,b\mid t\in \mathbb {R} \}}$

The direction of the line is from a (t = 0) to b (t = 1), or in other words, in the direction of the vector b − a. Different choices of a and b can yield the same line.

Three points are said to be collinear if they lie on the same line. Three points usually determine a plane, but in the case of three collinear points this does not happen.

In affine coordinates, in n-dimensional space the points X=(x₁, x₂, ..., x_n), Y=(y₁, y₂, ..., y_n), and Z=(z₁, z₂, ..., z_n) are collinear if the matrix

${\displaystyle {\begin{bmatrix}1&x_{1}&x_{2}&\dots &x_{n}\\1&y_{1}&y_{2}&\dots &y_{n}\\1&z_{1}&z_{2}&\dots &z_{n}\end{bmatrix}}}$

has a rank 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 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, k points in a plane are collinear if and only if any (k–1) pairs of points have the same pairwise slopes.

In Euclidean geometry, the Euclidean distance d(a,b) between two points a and b may be used to express the collinearity between three points by:[12][13]

The points a, b and c are collinear if and only if d(x,a) = d(c,a) and d(x,b) = d(c,b) implies x=c.

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.

In a sense,[14] all lines in Euclidean geometry are equal, in that, without coordinates, one can not tell them apart from one another. However, lines may play special roles with respect to other objects in the geometry and be divided into types 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.

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:

• i-secant lines, meeting the curve in i 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.

Given a line and any point A on it, we may consider A as decomposing this line into two parts. Each such part is called a ray and the point A is called its initial point. It is also known as half-line, a one-dimensional half-space. The point A is considered to be a member of the ray.[16] Intuitively, a ray consists of those points on a line passing through A and proceeding indefinitely, starting at A, 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 points A and B, they determine a unique ray with initial point A. As two points define a unique line, this ray consists of all the points between A and B (including A and B) and all the points C on the line through A and B such that B is between A and C.[17] This is, at times, also expressed as the set of all points C such that A is not between B and C.[18] A point D, on the line determined by A and B but not in the ray with initial point A determined by B, will determine another ray with initial point A. With respect to the AB ray, the AD ray is called the opposite ray.

Ray

Thus, we would say that two different points, A and B, define a line and a decomposition of this line into the disjoint union of an open segment (A, B) and two rays, BC and AD (the point D is not drawn in the diagram, but is to the left of A on the line AB). These are not opposite rays since they have different initial points.

In Euclidean geometry two rays with a common endpoint form an angle.

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, typically Euclidean geometry or affine geometry over an ordered field. On the other hand, rays do not exist in projective geometry nor in a geometry over a non-ordered field, like the complex numbers or any finite field.

A line 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 are coplanar and either do not intersect or are collinear.

The "shortness" and "straightness" of a line, interpreted as the property that the distance along the line between any two of its points is minimized (see triangle inequality), can be generalized and leads to the concept of geodesics in metric spaces.