![]() A ray is a line that has a starting point and ends at an opposite point. The Definition of Opposite Rays in Geometry In addition to defining angles, rays define lines that cross intersections. They are named based on their directions and endpoints. In addition to angles, rays also form sides of a circle. A line that extends to the same direction is called an oblique angle. A right-angled triangle is an example of an angle. More generally, in n-dimensional space n−1 first-degree equations in the n coordinate variables define a line under suitable conditions.The oblique rays define the angle of a triangle. 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. : 104 Any collection of finitely many lines partitions the plane into convex polygons (possibly unbounded) this partition is known as an arrangement of lines. On a Euclidean plane, a line can be represented as a boundary between two regions. In higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not. : 300 In two dimensions (i.e., the Euclidean plane), two lines that do not intersect are called parallel. 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 an axiomatic formulation of Euclidean geometry, such as that of Hilbert (modern mathematicians added to Euclid's original axioms to fill perceived logical gaps), : 108 a line is stated to have certain properties that relate it to other lines and points. In modern geometry, a line is usually either taken as a primitive notion with properties given by axioms, : 95 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. : 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 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". ![]() 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. 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).Įuclid'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. Lines are spaces of dimension one, which may be embedded in spaces of dimension two, three, or higher. In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical objects as a straightedge, a taut string, or a ray of light.
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