Plane (geometry)

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a line and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with a room's walls extended infinitely far, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry; when working in two-dimensional Euclidean space, the definite article is used, so, the plane refers to the whole space. Many fundamental tasks in mathematics, trigonometry, graph theory, graphing are performed in a two-dimensional space, or, in other words, in the plane. Euclid set forth the first great landmark of mathematical thought, an axiomatic treatment of geometry, he selected a small core of undefined terms and postulates which he used to prove various geometrical statements. Although the plane in its modern sense is not directly given a definition anywhere in the Elements, it may be thought of as part of the common notions.

Euclid never used numbers to measure angle, or area. In this way the Euclidean plane is not quite the same as the Cartesian plane. A plane is a ruled surface; this section is concerned with planes embedded in three dimensions: in R3. In a Euclidean space of any number of dimensions, a plane is uniquely determined by any of the following: Three non-collinear points. A line and a point not on that line. Two distinct but intersecting lines. Two parallel lines; the following statements hold in three-dimensional Euclidean space but not in higher dimensions, though they have higher-dimensional analogues: Two distinct planes are either parallel or they intersect in a line. A line intersects it at a single point, or is contained in the plane. Two distinct lines perpendicular to the same plane must be parallel to each other. Two distinct planes perpendicular to the same line must be parallel to each other. In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it to indicate its "inclination".

Let r0 be the position vector of some point P0 =, let n = be a nonzero vector. The plane determined by the point P0 and the vector n consists of those points P, with position vector r, such that the vector drawn from P0 to P is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be described as the set of all points r such that n ⋅ = 0. Expanded this becomes a + b + c = 0, the point-normal form of the equation of a plane; this is just a linear equation a x + b y + c z + d = 0, where d = −. Conversely, it is shown that if a, b, c and d are constants and a, b, c are not all zero the graph of the equation a x + b y + c z + d = 0, is a plane having the vector n = as a normal; this familiar equation for a plane is called the general form of the equation of the plane. Thus for example a regression equation of the form y = d + ax + cz establishes a best-fit plane in three-dimensional space when there are two explanatory variables.

Alternatively, a plane may be described parametrically as the set of all points of the form r = r 0 + s v + t w, where s and t range over all real numbers, v and w are given linearly independent vectors defining the plane, r0 is the vector representing the position of an arbitrary point on the plane. The vectors v and w can be visualized as vectors starting at r0 and pointing in different directions along the plane; the vectors v and w can not be parallel. Let p1=, p2=, p3= be non-collinear points; the plane passing through p1, p2, p3 can be described as the set of all points that satisfy the following determinant equations: | x − x 1 y − y 1 z − z 1 x 2 − x 1 y 2 − y

European maritime exploration of Australia

The maritime European exploration of Australia consisted of several waves of white European seafarers that sailed the edges of the Australian continent. Dutch navigators were the first Europeans known to have explored and mapped the Australian coastline; the first documented encounter was that of Dutch navigator Willem Janszoon, in 1606. Dutch seafarers visited the west and north coasts of the continent, as did French explorers; the most famous expedition was that of Royal Navy Lieutenant James Cook 164 years after Janszoon's sighting. After an assignment to make observations of the 1769 Venus Transit, Cook followed Admiralty instructions to explore the south Pacific for the reported Terra Australis and on 19 April 1770 sighted the south-eastern coast of Australia and became the first recorded European to explore the eastern coastline. Explorers by land and sea continued to survey the continent for some years after settlement; some writers have advanced the theory that the Portuguese were the first Europeans to sight Australia in the 1520s.

A number of relics and remains have been interpreted as evidence that the Portuguese reached Australia. The primary evidence advanced to support this theory is the representation of the continent of Jave la Grande, which appears on a series of French world maps, the Dieppe maps, that may, in part, be based on Portuguese charts. However, most historians do not accept this theory, the interpretation of the Dieppe maps is contentious. In the early 20th century, Lawrence Hargrave argued that Spain had established a colony in Botany Bay in the 16th century. Five coins from the Kilwa Sultanate were found on Marchinbar Island, in the Wessel Islands in 1945 by RAAF radar operator Morry Isenberg. In 2018 another coin thought to be from Kilwa, was found on a beach on Elcho Island, another of the Wessel Islands, by archaeologist and member of the Past Masters, Mike Hermes. Hermes speculated that the coins may suggest trade between indigenous Australians and Kilwa, or may have arrived via Makassan contact with Australia.

Mike Owen, another member of the Past Masters group speculated that these coins may have arrived sometime after they had installed Muhammad Arcone on the Kilwa throne as a Portuguese vassal, from 1505 to 1506, or that the Portuguese had visited Wessel islands. The French navigator Binot Paulmier de Gonneville claimed to have landed at a land he described as "east of the Cape of Good Hope" in 1504, after being blown off course. For some time it had been thought he discovered Australia, but the place he landed has now been shown to be Brazil; the most significant exploration of Australia in the 17th century was by the Dutch. The Dutch East India Company was set up in 1602 and traded extensively with the islands which now form parts of Indonesia, hence were close to Australia already; the first documented and undisputed European sighting of and landing on Australia was in late February 1606, by the Dutch navigator Willem Janszoon aboard the Duyfken. Janszoon met with Aboriginal people. Janszoon followed the coast of New Guinea, missed Torres Strait, explored and charted part of the western side of Cape York, in the Gulf of Carpentaria, believing the land was still part of New Guinea.

On 26 February 1606, Janszoon and his party made landfall near the modern town of Weipa and the Pennefather River, but were promptly attacked by the Indigenous people. Janszoon proceeded down the coast for some 350 km, he stopped in some places, but was met by hostile natives and some of his men were killed. At the final place, he had friendly relations with the natives, but after he forced them to hunt for him and appropriated some of their women, violence broke out and there were many deaths on both sides; these events were recorded in Aboriginal oral history. Here Janszoon decided to turn back, the place being called Cape Keerweer, Dutch for "turnabout"; that same year, a Spanish expedition sailing in nearby waters and led by Pedro Fernández de Quiros landed in the New Hebrides and, believing such to be the fabled southern continent, named the land "Austrialia del Espiritu Santo", in honour of his queen Margaret of Austria, the wife of Philip III of Spain. That year, De Quiros' deputy Luís Vaez de Torres sailed to the north of Australia through Torres Strait, charting New Guinea's southern coast, sighting Cape York in October 1606.

In 1611 Hendrik Brouwer, working for VOC, discovered that sailing from Europe to Batavia was much quicker if the Roaring Forties were used. Up to that point, the Dutch had followed a route copied from Arab and Portuguese sailors who followed the coasts of Africa and Ceylon; the Brouwer Route involved sailing south from the Cape of Good Hope into the Roaring Forties sailing east before turning north to Java using the South Indian Ocean Current. The Brouwer Route became compulsory for Dutch vessels in 1617; the problem with the route, was that there was no easy way at the time to determine longitude, making Dutch landfalls on the west coast of Australia inevitable, as well as ships becoming wrecked on the shoals. Most of these landfalls were unplanned; the first such landfall was in 1616, when Dirk Hartog, employed by VOC, reached land at Shark Bay off the coast of Western Australia. Finding nothing of interest, Hartog continued sailing northwards along this undiscovered coastline of Western Australia, making nautical charts up to about 22° latitude south.

He left the coast and con

Acleris shepherdana

Acleris shepherdana, the meadow-sweet button, is a species of moth of the family Tortricidae. It is found in Europe, where it has been recorded from Great Britain, the Benelux, Denmark, Switzerland, the Czech Republic, Poland, Norway, Finland, the Baltic region and European Russia, it is found in the Russian Far East, Mongolia and Japan. The habitat consists of fens, river-banks and other damp areas; the wingspan is 13–16 mm. The wings are cinnamon-brown with a reticulate pattern and a blackish plumbeous outer margin of the costal blotch. Adults are on wing from mid-June to the end of September; the larvae feed on Spiraea ulmaria, Sanguisorba officinalis, Sanguisorba parviflora and Filipendula species. They live in spun shoots of their host plant. Larvae can be found from May to June. Pupation may take place in the larval feeding place, in a folded leaf-edge or on the ground