Line (geometry)
In ancient Greece, Euclid defined a straight line as a breadthless length that lies evenly with respect to the points on itself. This definition appeared in his work known as Elements over two thousand years ago. The text never explicitly referenced these definitions again during the remainder of the book. Modern mathematicians later introduced terms like Euclidean line and Euclidean geometry to distinguish original concepts from newer generalizations. These new categories emerged after the end of the 19th century when non-Euclidean geometries began to appear. Hilbert added axioms to fill logical gaps left by Euclid's original postulates. Any two distinct points determine a unique line containing them according to this system. Two distinct lines intersect at most at one point within this framework.
A linear equation characterizes every line in a Cartesian plane or affine coordinates. The set of all points whose coordinates satisfy this equation forms the line. Coefficients a, b, and c are fixed real numbers where a and b cannot both be zero. Vertical lines correspond to equations where b equals zero. The slope-intercept form uses m for the slope and b for the y-intercept. In three dimensions, a single linear equation typically describes a plane rather than a line. A line exists as what is common to two distinct intersecting planes. Parametric equations specify lines particularly in spaces with more than two dimensions. The direction vector relates to the slope of the line in three-dimensional space. Ludwig Otto Hesse developed the normal form based on a segment drawn from the origin perpendicular to the line.
Parallel lines exist in the same plane but never cross each other. Intersecting lines share a single point in common while coincidental lines overlap completely. Perpendicular lines meet at right angles within Euclidean geometry. Skew lines appear in three-dimensional space when they do not lie in the same plane. Tangent lines touch a conic such as a circle or ellipse at a single point. Secant lines intersect the conic at two points and pass through its interior. Exterior lines fail to meet the conic at any point of the Euclidean plane. Three or more points become collinear if they lie on the same line. If three points are not collinear, exactly one plane contains them. The Newton line connects midpoints of diagonals in a convex quadrilateral with at most two parallel sides.
In elliptic geometry, lines represent great circles of a sphere with diametrically opposite points identified. Projective geometries treat lines as 2-dimensional vector spaces containing all linear combinations of independent vectors. Differential geometry interprets a line as a geodesic representing the shortest path between points. Metric spaces generalize the concept of straightness by minimizing distance along the path. A great circle divides a sphere into two equal hemispheres while satisfying no curvature properties. These representations satisfy properties like two points determining a unique line despite visual differences. The shortness and straightness of a line lead to the concept of geodesics in metric spaces. Physicists think of light ray paths as lines within these abstract frameworks.
A ray decomposes a line into two parts when considering any point A upon it. This part is called a half-line or sometimes a half-axis if playing a distinct role. Two rays with a common endpoint form an angle in Euclidean geometry. Line segments bound a part of a line by two distinct endpoints. They contain every point on the line between those end points. Integers are evenly spaced on number lines with positive numbers on the right side. Negative numbers appear on the left side of this numerical representation. An imaginary line represents imaginary numbers perpendicular to the real number line at zero. These two lines together form the complex plane as a geometrical representation of complex numbers. Rays do not exist in projective geometry nor in geometries over non-ordered fields.
Common questions
What is the definition of a line in Euclid's Elements?
Euclid defined a straight line as a breadthless length that lies evenly with respect to the points on itself. This definition appeared in his work known as Elements over two thousand years ago.
When did non-Euclidean geometries emerge and change the concept of lines?
New categories emerged after the end of the 19th century when non-Euclidean geometries began to appear. Modern mathematicians later introduced terms like Euclidean line and Euclidean geometry to distinguish original concepts from newer generalizations.
How are parallel lines and intersecting lines defined in Euclidean geometry?
Parallel lines exist in the same plane but never cross each other while intersecting lines share a single point in common. Perpendicular lines meet at right angles within Euclidean geometry and skew lines appear in three-dimensional space when they do not lie in the same plane.
What mathematical equations characterize every line in a Cartesian plane?
A linear equation characterizes every line in a Cartesian plane or affine coordinates where coefficients a, b, and c are fixed real numbers. Vertical lines correspond to equations where b equals zero and the slope-intercept form uses m for the slope and b for the y-intercept.
How does elliptic geometry represent lines compared to standard definitions?
In elliptic geometry, lines represent great circles of a sphere with diametrically opposite points identified. A great circle divides a sphere into two equal hemispheres while satisfying no curvature properties.