Line (geometry)

• 1. The Breadthless Length

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  A taut string, a ray of light, the edge of a straightedge. Each is a physical hint at something that cannot physically exist. In geometry, a line is an infinitely long object with no width, no depth, and no curvature. It stretches forever in both directions and occupies a space of exactly one dimension. Euclid's Elements reached for this same idea with a strange phrase. It called a straight line a breadthless length, one that lies evenly with respect to the points on itself. Why would a definition built on undefined words survive for thousands of years? Why does something so simple need rays of light and stretched strings just to be described? And what happens when this perfectly straight thing is bent onto a sphere, or asked to pass through the origin of a number line? The answers reveal a figure that is far less settled than it first appears.

• 2. When A Definition Fails

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  Euclid's phrase, lies evenly with the points on itself, leans entirely on the reader's physical experience. It uses terms that are never themselves defined. Stranger still, the definition is never referenced again anywhere in the rest of the Elements, and it is never used to prove a single theorem. Modern mathematicians treated this as a logical gap to be filled. In an axiomatic formulation such as Hilbert's, mathematicians added new axioms to Euclid's originals. A line is no longer described by an image. It is described by what it must do: for any two distinct points, there is a unique line containing them, and any two distinct lines intersect at most at one point. Some descriptions still survive as mental images, called definitions out of habit, even though they cannot be used in proofs. The terms Euclidean line and Euclidean geometry were themselves coined for a defensive reason. They distinguish the classical idea from the generalizations introduced since the end of the 19th century, including non-Euclidean, projective, and affine geometry.

• 3. Lines Among Their Neighbors

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  All lines in Euclidean geometry are congruent, meaning any line can be obtained by moving another. Yet a line earns a name from the company it keeps. Set a line against a conic, that is a circle, ellipse, parabola, or hyperbola, and its role decides its title. A tangent line touches the conic at a single point. A secant line cuts through the interior, meeting the conic at two points. An exterior line never meets the conic at all on the Euclidean plane. A directrix uses its distance from a point to help establish whether that point lies on the conic. Triangles supply their own roster, including the Euler line, the Simson lines, and the central lines. A convex quadrilateral with at most two parallel sides has a Newton line, joining the midpoints of its two diagonals. A hexagon with vertices on a conic carries the Pascal line, and when that conic collapses into a pair of lines, the Pappus line appears. For broader algebraic curves there are asymptotes, which a curve approaches arbitrarily closely without ever touching.

• 4. Crossing, Missing, And Skew

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  Two distinct lines can do only so much to each other, and geometry has a word for each outcome. Intersecting lines share a single point in common. Coincidental lines coincide completely, so every point on one is also on the other. Perpendicular lines intersect at right angles. On a Euclidean plane, two lines that never cross are parallel. Climb into higher dimensions and a new possibility opens up. Two lines that do not intersect are parallel only if they sit together in a plane. If they do not share a plane, they are skew, passing each other in three-dimensional space without ever meeting. A single line can also act on the plane itself, serving as a boundary between two regions. Gather finitely many lines and they partition the plane into convex polygons, some possibly unbounded, in a figure known as an arrangement of lines.

• 5. Three Points On One Line

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  Three or more points are called collinear when they lie on the same line. The moment three points refuse to be collinear, exactly one plane contains them. Mathematicians turned this intuition into machinery. In affine coordinates, three points are collinear when a certain matrix has rank less than three. For three points in the plane, that matrix is square, and the points line up if and only if its determinant is zero. There is a more visual test as well. Three points in a plane are collinear if and only if the slope between one pair equals the slope between any other pair. The idea extends cleanly: k points in a plane are collinear if and only if any k minus 1 pairs share the same pairwise slopes. Distance offers yet another route, using the Euclidean distance between points to express collinearity. That route has limits worth noting. For other notions of distance, such as the Manhattan distance, the property simply fails.

• 6. Writing The Line Down

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  Every line in a Cartesian plane is the set of all points whose coordinates satisfy a linear equation, with coefficients a, b, and c where a and b are not both zero. Vertical lines, often the troublemakers, are handled here by setting b to zero. This standard form has a sibling called the general form, though many authors refuse to distinguish them. For non-vertical lines, the slope-intercept form names two pieces of data directly: m, the slope or gradient, and b, the y-intercept. Parametric equations take over where single equations fail. In three dimensions or higher, a line cannot be described by one linear equation, so it is built from a point on the line and a direction vector parallel to it. The Hesse normal form, named after the German mathematician Ludwig Otto Hesse, takes a different anchor entirely. It uses the normal segment, drawn from the origin perpendicular to the line, joining the origin to the closest point on the line. Unlike the slope-intercept and intercept forms, this version can represent any line while needing only two finite parameters.

• 7. Lines That Bend

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  In the spherical representation of elliptic geometry, a line is not straight at all. It is a great circle of a sphere, with diametrically opposite points identified as one. A second model of the same geometry abandons the sphere and represents lines as Euclidean planes passing through the origin. The two pictures look nothing alike, yet both satisfy the rule that two points determine a unique line. This is the freedom of leaving a line undefined as a primitive object. The properties of lines come only from the axioms that refer to them. In differential geometry a line becomes a geodesic, the shortest path between points, and the notions of shortness and straightness generalize into geodesics in metric spaces. The flexibility reaches past mathematics. It permits physicists to treat the path of a light ray as a line.

• 8. Pieces Of A Line

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  Take a line and mark any point A on it, and A splits the line into two parts. Each part is a ray, with A as its initial point, a one-dimensional half-space that runs indefinitely in one direction. Two rays sharing a common endpoint form an angle. Rays carry a hidden requirement: they depend on the notion of betweenness for points on a line. So rays exist only where betweenness exists, typically Euclidean geometry or affine geometry over an ordered field. They vanish in projective geometry, and over the complex numbers or any finite field. A line segment is the bounded cousin, a part of a line held between two distinct end points and containing every point in between. The number line gives the line a final job. Each point corresponds to a real number, with positive numbers on the right and negative on the left. Draw a perpendicular line of imaginary numbers through zero, and the two lines together form the complex plane, a geometric image of the complex numbers.