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[a] 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.
Almgren–Pitts min-max theory; Approximation theory; Arakelov theory; Asymptotic theory; Automata theory; Bass–Serre theory; Bifurcation theory; Braid theory
Half-line (geometry) or ray, half of a line split at an initial point Directed half-line or ray, half of a directed or oriented line split at an initial point; Ray (graph theory), an infinite sequence of vertices such that each vertex appears at most once in the sequence and each two consecutive vertices in the sequence are the two endpoints of an edge in the graph
Mathematics and art are related in a variety of ways. For instance, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of projective geometry. [132] Artists have long used concepts of proportion in design.
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 ...
The theory of projective harmonic conjugates of points on a line can also be used to define this relationship. Using the same notation as above; Using the same notation as above; If a variable line through the point P is a secant of the conic C , the harmonic conjugates of P with respect to the two points of C on the secant all lie on the polar ...
Cantor–Bernstein–Schröder theorem (set theory, cardinal numbers) Cantor's theorem (set theory, Cantor's diagonal argument) Church–Rosser theorem (lambda calculus) Compactness theorem (mathematical logic) Conservativity theorem (mathematical logic) Craig's theorem (mathematical logic) Craig's interpolation theorem (mathematical logic)
The Fano plane cannot be represented in the Euclidean plane using only points and straight line segments (i.e., it is not realizable). This is a consequence of the Sylvester–Gallai theorem, according to which every realizable incidence geometry must include an ordinary line, a line containing only two points.