In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted E 2 {\displaystyle {\textbf {E}}^{2}} or E 2 {\displaystyle \mathbb {E} ^{2}} . It is a geometric space in which two real numbers are required to determine the position of each point. It is an affine space, which includes in particular the concept of parallel lines. It has also metrical properties induced by a distance, which allows to define circles, and angle measurement.
A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane.The set R 2 {\displaystyle \mathbb {R} ^{2}} of the ordered pairs of real numbers (the real coordinate plane), equipped with the dot product, is often called the Euclidean plane, since every Euclidean plane is isomorphic to it.
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Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagorean theorem (Proposition 47), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area), among many other topics.
Later, the plane was described in a so-called Cartesian coordinate system, a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.
The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat, although Fermat also worked in three dimensions, and did not publish the discovery.[1] Both authors used a single (abscissa) axis in their treatments, with the lengths of ordinates measured along lines not-necessarily-perpendicular to that axis.[2] The concept of using a pair of fixed axes was introduced later, after Descartes' La Gomtrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work.[3]
In mathematics, analytic geometry (also called Cartesian geometry) describes every point in two-dimensional space by means of two coordinates. Two perpendicular coordinate axes are given which cross each other at the origin. They are usually labeled x and y. Relative to these axes, the position of any point in two-dimensional space is given by an ordered pair of real numbers, each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the other axis.
In Euclidean geometry, a plane is a flat two-dimensional surface that extends indefinitely. Euclidean planes often arise as subspaces of three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} .A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin.
The regular monogon (or henagon) {1} and regular digon {2} can be considered degenerate regular polygons and exist nondegenerately in non-Euclidean spaces like a 2-sphere, 2-torus, or right circular cylinder.
There exist infinitely many non-convex regular polytopes in two dimensions, whose Schlfli symbols consist of rational numbers {n/m}. They are called star polygons and share the same vertex arrangements of the convex regular polygons.
Another mathematical way of viewing two-dimensional space is found in linear algebra, where the idea of independence is crucial. The plane has two dimensions because the length of a rectangle is independent of its width. In the technical language of linear algebra, the plane is two-dimensional because every point in the plane can be described by a linear combination of two independent vectors.
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other.[9] Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.
Plane is a 2023 American action thriller film directed by Jean-Franois Richet from a screenplay by Charles Cumming and J. P. Davis.[4] The film stars Gerard Butler, Mike Colter, Yoson An, and Tony Goldwyn. The plot centers on a pilot (Butler) allying with a prisoner to save his passengers from a hostile territory in which they make an emergency landing.
Commercial pilot Brodie Torrance, a former RAF pilot from Scotland, flies Trailblazer Airlines Flight 119 with 14 passengers and three cabin crew members from Singapore to Honolulu via Tokyo. Among the passengers is fugitive homicide suspect Louis Gaspare, who is accompanied by an RCMP/GRC officer en route to Canada. Per a directive from one of his superiors, Brodie takes a shortcut across the South China Sea, but a lightning strike blows the plane's avionics and a flight attendant and the RCMP officer are killed during the turbulence. They make an emergency landing on a dirt strip of what turns out to be Jolo island in the Philippines.
In New York City, the board of Trailblazer calls David Scarsdale, their crisis manager. He dispatches a private military unit to rescue the passengers, as the authorities are unwilling to send troops into the rebel-controlled island. Brodie goes off into the jungle for help, accompanied by Louis. At an abandoned warehouse, Brodie wires the phone to call his superiors and daughter to tell their location. He succeeds but is attacked by a rebel whom he kills. Louis also kills other rebels in the building. They encounter a site used by the rebels to make ransom videos and race back to the plane, but are beaten to it by rebel leader Datu Junmar, who kills a couple who try to escape and takes the surviving passengers and crew hostage, intending to secure large ransoms from their families.
After the group leaves, Brodie and Louis overpower remaining rebels and force them to reveal the location of their lair. Before leaving, Brodie leaves a note to inform rescue teams of the situation. At the rebel hideout, Brodie and Louis kill the guards and sneak the passengers and crew onto a bus. Brodie decides to stay behind to distract the rebels. As he is about to be executed, the rescue team arrives, inflicting heavy casualties on Junmar's men. The rescue team tells Brodie that the airline cannot send in a formal team to remove them from the island for another 24 hours, and they have insufficient contingency money to negotiate their way off the island. Brodie tells the rescue team that he has another plan.
Back at the plane, Brodie and his copilot Samuel Dele manage to fire up the plane and gather everyone inside for takeoff. Scarsdale's group sets up a Barrett M82 anti-materiel rifle for increased firepower and together with Louis fight off Junmar's men. Louis chooses to stay behind to distract Junmar's forces and allow Scarsdale's team to board the plane, preventing an attempt by Junmar and a terrorist to blow up the plane with an RPG, and he flees into the jungle with a bag of ransom money brought by the mercenaries. Angered and desperate, Junmar attempts to use another RPG in a last ditch effort to destroy the plane and wounds Brodie again, this time, in the shoulder. But Brodie (despite his injuries) is able to put the plane on full throttle, allowing it to take off safely and hit Junmar with the wheels, killing him.
The plane is too damaged and too low on fuel to make a long journey, but Brodie manages to land the plane at the neighboring friendly island of Siasi. As the passengers and crew are tended to by the island's rescue team, Brodie phones his daughter, telling her he is coming home.
On July 13, 2016, MadRiver Pictures acquired The Plane, an original pitch from novelist Charles Cumming, with Marc Butan and Di Bonaventura Pictures' Lorenzo di Bonaventura and Mark Vahradian announced as producers.[5] In October 2019, it was reported that Gerard Butler had joined the cast and would also produce alongside Alan Siegel.[6]
In November 2019, Lionsgate Films acquired distribution rights to the film,[7][8] but in November 2020, it abandoned the project after failing to obtain production insurance that would cover a COVID-19 outbreak, as the studio did not want to risk the film's original $50 million budget,[9] and Solstice Studios acquired the rights to the film.[10] However, in May 2021, Lionsgate re-acquired the rights to The Plane, in what Andreas Wiseman from Deadline Hollywood described as a "case of high-profile Hollywood volleyball."[11]
In August 2021, Kelly Gale,[12] Mike Colter,[13] Daniella Pineda,[14] Yoson An,[15] Remi Adeleke, Haleigh Hekking, Lilly Krug, Joey Slotnick, and Oliver Trevena joined the cast.[16] Production began that same month in Puerto Rico.[17] Michael Cho, Tim Lee, Gary Raskin, Alastair Burlingame, and Vicki Dee Rock were attached to the film as Executive Producers.[18] On a podcast, Colter said the film was going to focus more on characterization than action sequences.[19] On October 11, 2021, it was reported that filming was close to finishing and that Tony Goldwyn and Paul Ben-Victor would also star.[20]
Filipino actor and senator Robin Padilla condemned the film's portrayal of the Philippines, pointing out how the film depicted Jolo as being run by separatists and militia, and the Philippine Army as "cowards". In real life, the Jihadist group Abu Sayyaf established their base of operations in Jolo but were not able to push out the Philippine Government's authority in the island. Their presence has also declined significantly since their peak in the 2000s.[30][31] Padilla's criticism was supported by fellow senator Ronald dela Rosa and Senate President Migz Zubiri, who argued the film could damage the country's tourism.[32] Meanwhile, the Directors' Guild of the Philippines opposed the proposed ban, saying that it constitutes as censorship, and argued that the ban could set a precedent on the freedom of artistic expression of any depiction of the country. They also argued that the film was not a reliable commentary on the country's affairs, stating that Plane was just "mindless B-movie entertainment".[33][34] 152ee80cbc
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