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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 ...
Almgren–Pitts min-max theory; Approximation theory; Arakelov theory; Asymptotic theory; Automata theory; Bass–Serre theory; Bifurcation theory; Braid theory; Brill–Noether theory; Catastrophe theory; Category theory; Chaos theory; Character theory; Choquet theory; Class field theory; Cobordism theory; Coding theory; Cohomology theory ...
If m is an ideal of the ring of integers of a number field K and S is a subset of the real places, then the ray class group of m and S is the quotient group / where I m is the group of fractional ideals co-prime to m, and the "ray" P m is the group of principal ideals generated by elements a with a ≡ 1 mod m that are positive at the places of S.
In topology, a curve is defined by a function from an interval of the real numbers to another space. [49] In differential geometry, the same definition is used, but the defining function is required to be differentiable. [53] Algebraic geometry studies algebraic curves, which are defined as algebraic varieties of dimension one. [54]
Geometrical optics, or ray optics, is a model of optics that describes light propagation in terms of rays. The ray in geometrical optics is an abstraction useful for approximating the paths along which light propagates under certain circumstances. The simplifying assumptions of geometrical optics include that light rays:
Denote by h′ a ray of the straight line a′ emanating from a point O′ of this line. Then in the plane α ′ there is one and only one ray k ′ such that the angle ∠ ( h , k ) , or ∠ ( k , h ) , is congruent to the angle ∠ ( h ′, k ′) and at the same time all interior points of the angle ∠ ( h ′, k ′) lie upon the given ...
Set theory of the real line is an area of mathematics concerned with the application of set theory to aspects of the real numbers. For example, one knows that all countable sets of reals are null , i.e. have Lebesgue measure 0; one might therefore ask the least possible size of a set which is not Lebesgue null.
Starting with the real numbers, the corresponding projective "line" is geometrically a circle, and then the extra point / gives the shape that is the source of the term "wheel". Or starting with the complex numbers instead, the corresponding projective "line" is a sphere (the Riemann sphere ), and then the extra point gives a 3-dimensional ...