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70-689: The Greenwich meridian is a prime meridian , a geographical reference line that passes through the Royal Observatory , Greenwich , in London , England. From 1884 to 1974, the Greenwich meridian was the international standard prime meridian, used worldwide for timekeeping and navigation. The modern standard, the IERS Reference Meridian , is based on the Greenwich meridian, but differs slightly from it. This prime meridian (at

140-552: A 360°-system ) form a great circle . This great circle divides a spheroid, like Earth, into two hemispheres : the Eastern Hemisphere and the Western Hemisphere (for an east-west notational system). For Earth's prime meridian, various conventions have been used or advocated in different regions throughout history. Earth's current international standard prime meridian is the IERS Reference Meridian . It

210-471: A and b ) are both perpendicular to a third line ( c ), all of the angles formed along the third line are right angles. Therefore, in Euclidean geometry , any two lines that are both perpendicular to a third line are parallel to each other, because of the parallel postulate . Conversely, if one line is perpendicular to a second line, it is also perpendicular to any line parallel to that second line. In

280-404: A circle is perpendicular to the tangent line to that circle at the point where the diameter intersects the circle. A line segment through a circle's center bisecting a chord is perpendicular to the chord. If the intersection of any two perpendicular chords divides one chord into lengths a and b and divides the other chord into lengths c and d , then a + b + c + d equals

350-447: A hyperbola is perpendicular to the conjugate axis and to each directrix. The product of the perpendicular distances from a point P on a hyperbola or on its conjugate hyperbola to the asymptotes is a constant independent of the location of P. A rectangular hyperbola has asymptotes that are perpendicular to each other. It has an eccentricity equal to 2 . {\displaystyle {\sqrt {2}}.} The legs of

420-404: A quadrilateral is a perpendicular to a side through the midpoint of the opposite side. An orthodiagonal quadrilateral is a quadrilateral whose diagonals are perpendicular. These include the square , the rhombus , and the kite . By Brahmagupta's theorem , in an orthodiagonal quadrilateral that is also cyclic , a line through the midpoint of one side and through the intersection point of

490-406: A right triangle are perpendicular to each other. The altitudes of a triangle are perpendicular to their respective bases . The perpendicular bisectors of the sides also play a prominent role in triangle geometry. The Euler line of an isosceles triangle is perpendicular to the triangle's base. The Droz-Farny line theorem concerns a property of two perpendicular lines intersecting at

560-493: A spherical Earth , the book described the thousands years old customs of the prime meridian , or zero longitude, as passing through Avanti , the ancient name for the historic city of Ujjain , and Rohitaka , the ancient name for Rohtak ( 28°54′N 76°38′E  /  28.900°N 76.633°E  / 28.900; 76.633  ( Rohitaka (Rohtak) ) ), a city near the Kurukshetra . Ptolemy's Geographia

630-523: A territorial claim of Norway , on its way from the North Pole to the South Pole . It crosses the maritime exclusive economic zones of: Prime meridian A prime meridian is an arbitrarily chosen meridian (a line of longitude ) in a geographic coordinate system at which longitude is defined to be 0°. Together, a prime meridian and its anti-meridian (the 180th meridian in

700-560: A line of longitude. In 1541, Mercator produced his famous 41 cm terrestrial globe and drew his prime meridian precisely through Fuerteventura (14°1'W) in the Canaries. His later maps used the Azores, following the magnetic hypothesis. But by the time that Ortelius produced the first modern atlas in 1570, other islands such as Cape Verde were coming into use. In his atlas longitudes were counted from 0° to 360°, not 180°W to 180°E as

770-427: A line segment that is perpendicular to a tangent line to the curve at the nearest point on the curve. The distance from a point to a plane is measured as the length from the point along a segment that is perpendicular to the plane, meaning that it is perpendicular to all lines in the plane that pass through the nearest point in the plane to the given point. Other instances include: Perpendicular regression fits

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840-769: A line to data points by minimizing the sum of squared perpendicular distances from the data points to the line. Other geometric curve fitting methods using perpendicular distance to measure the quality of a fit exist, as in total least squares . The concept of perpendicular distance may be generalized to In the two-dimensional plane, right angles can be formed by two intersected lines if the product of their slopes equals −1. Thus for two linear functions y 1 ( x ) = m 1 x + b 1 {\displaystyle y_{1}(x)=m_{1}x+b_{1}} and y 2 ( x ) = m 2 x + b 2 {\displaystyle y_{2}(x)=m_{2}x+b_{2}} ,

910-421: A powerful green laser shining north across the London night sky. The Global Positioning System (GPS) receivers show that the marking strip for the prime meridian at Greenwich is not exactly at zero degrees, zero minutes, and zero seconds but at approximately 5.3 seconds of arc to the west of the meridian (meaning that the meridian appears to be 102 metres east). In the past, this offset has been attributed to

980-412: A sphere. The downward extended plumb lines don't even all intersect the rotation axis of Earth; this much smaller effect is due to the uneven distribution of Earth's mass. To make computations feasible, scientists defined ellipsoids of revolution, more closely emulating the shape of Earth, modified for a particular zone; a published ellipsoid would be a good base line for measurements. The difference between

1050-400: A triangle's orthocenter . Harcourt's theorem concerns the relationship of line segments through a vertex and perpendicular to any line tangent to the triangle's incircle . In a square or other rectangle , all pairs of adjacent sides are perpendicular. A right trapezoid is a trapezoid that has two pairs of adjacent sides that are perpendicular. Each of the four maltitudes of

1120-611: Is deflected slightly from the normal, or line perpendicular, to the reference ellipsoid used to define geodetic latitude and longitude in the International Terrestrial Reference Frame (which is nearly the same as the WGS84 system used by the GPS ). While Airy's local vertical, set by the apparent centre of gravity of Earth still points to (aligns with) the modern celestial meridian (the intersection of

1190-477: Is derived, but differs slightly, from the Greenwich Meridian , the previous standard. A prime meridian for a planetary body not tidally locked (or at least not in synchronous rotation) is entirely arbitrary, unlike an equator , which is determined by the axis of rotation. However, for celestial objects that are tidally locked (more specifically, synchronous), their prime meridians are determined by

1260-409: Is extended in both directions to form an infinite line, these two resulting lines are perpendicular in the sense above. In symbols, A B ¯ ⊥ C D ¯ {\displaystyle {\overline {AB}}\perp {\overline {CD}}} means line segment AB is perpendicular to line segment CD. A line is said to be perpendicular to a plane if it

1330-420: Is not necessarily at the bottom. More precisely, let A be a point and m a line. If B is the point of intersection of m and the unique line through A that is perpendicular to m , then B is called the foot of this perpendicular through A . To make the perpendicular to the line AB through the point P using compass-and-straightedge construction , proceed as follows (see figure left): To prove that

1400-442: Is perpendicular to every line in the plane that it intersects. This definition depends on the definition of perpendicularity between lines. Two planes in space are said to be perpendicular if the dihedral angle at which they meet is a right angle. The word foot is frequently used in connection with perpendiculars. This usage is exemplified in the top diagram, above, and its caption. The diagram can be in any orientation. The foot

1470-626: Is roughly 43 metres (47 yards) to the west of the Airy Transit Circle, a distance equivalent to roughly 2 seconds of longitude. It was Airy's transit circle that was adopted in principle (with French delegates, who pressed for adoption of the Paris meridian abstaining) as the Prime Meridian of the world at the 1884 International Meridian Conference. All of these Greenwich meridians were located via an astronomic observation from

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1540-419: Is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its normal vector . A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1)

1610-630: Is the same as that of its orbit. East longitudes are used if the rotation is retrograde . The notion of longitude for Greeks was developed by the Greek Eratosthenes (c.   276 – 195   BCE) in Alexandria , and Hipparchus (c.   190 – 120   BCE) in Rhodes , and applied to a large number of cities by the geographer Strabo (64/63   BCE – c.   24   CE). But it

1680-448: Is usual today. This practice was followed by navigators well into the 18th century. In 1634, Cardinal Richelieu used the westernmost island of the Canaries, El Hierro , 19° 55' west of Paris, as the choice of meridian. The geographer Delisle decided to round this off to 20°, so that it simply became the meridian of Paris disguised. In the early 18th century the battle was on to improve the determination of longitude at sea, leading to

1750-683: The International Meridian Conference in Washington, D.C. , 22 countries voted to adopt the Greenwich meridian as the prime meridian of the world. The French argued for a neutral line, mentioning the Azores and the Bering Strait , but eventually abstained and continued to use the Paris meridian until 1911. The current international standard Prime Meridian is the IERS Reference Meridian . The International Hydrographic Organization adopted an early version of

1820-650: The Royal Observatory, Greenwich . "Maskelyne's tables not only made the lunar method practicable, they also made the Greenwich meridian the universal reference point. Even the French translations of the Nautical Almanac retained Maskelyne's calculations from Greenwich – in spite of the fact that every other table in the Connaissance des Temps considered the Paris meridian as the prime." In 1884, at

1890-406: The perpendicular distance between two objects is the distance from one to the other, measured along a line that is perpendicular to one or both. The distance from a point to a line is the distance to the nearest point on that line. That is the point at which a segment from it to the given point is perpendicular to the line. Likewise, the distance from a point to a curve is measured by

1960-409: The point of intersection called a foot . The condition of perpendicularity may be represented graphically using the perpendicular symbol , ⟂. Perpendicular intersections can happen between two lines (or two line segments), between a line and a plane, and between two planes. Perpendicularity is one particular instance of the more general mathematical concept of orthogonality ; perpendicularity

2030-859: The 18th century most countries in Europe adapted their own prime meridian, usually through their capital, hence in France the Paris meridian was prime, in Prussia it was the Berlin meridian, in Denmark the Copenhagen meridian, and in United Kingdom the Greenwich meridian . Between 1765 and 1811, Nevil Maskelyne published 49 issues of the Nautical Almanac based on the meridian of

2100-662: The BBC website, that the gap between astronomical and geodetic coordinates means that any measurements of transit time across the IRTF zero meridian will occur precisely 0.352 seconds (or 0.353 sidereal seconds) before the transit across the "intended meridian" are based on a failure of understanding. The explanation by Malys et al . on the other hand is more studied and correct. The Greenwich meridian passes through eight countries in Europe and Africa from north to south: It also passes through Antarctica , only touching Queen Maud Land ,

2170-520: The Earth, prime meridians must be arbitrarily defined. Often a landmark such as a crater is used; other times a prime meridian is defined by reference to another celestial object, or by magnetic fields . The prime meridians of the following planetographic systems have been defined: Perpendicular In geometry , two geometric objects are perpendicular if their intersection forms right angles ( angles that are 90 degrees or π/2 radians wide) at

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2240-553: The IERS Reference Meridian, the plane of which passes through the centre of mass of the Earth. This differs from the plane established by the Airy transit, which is affected by vertical deflection (the local vertical is affected by influences such as nearby mountains). The change from relying on the local vertical to using a meridian based on the centre of the Earth caused the modern prime meridian to be 5.3″ east of

2310-677: The IRM in 1983 for all nautical charts. It was adopted for air navigation by the International Civil Aviation Organization on 3 March 1989. Since 1984, the international standard for the Earth's prime meridian is the IERS Reference Meridian. Between 1884 and 1984, the meridian of Greenwich was the world standard. These meridians are very close to each other. In October 1884 the Greenwich Meridian

2380-553: The PQ is perpendicular to AB, use the SSS congruence theorem for QPA' and QPB' to conclude that angles OPA' and OPB' are equal. Then use the SAS congruence theorem for triangles OPA' and OPB' to conclude that angles POA and POB are equal. To make the perpendicular to the line g at or through the point P using Thales's theorem , see the animation at right. The Pythagorean theorem can be used as

2450-512: The astronomic Greenwich prime meridian through the Airy Transit Circle. At the latitude of Greenwich, this amounts to 102 metres (112 yards). This was officially accepted by the Bureau International de l'Heure (BIH) in 1984 via its BTS84 (BIH Terrestrial System) that later became WGS84 (World Geodetic System 1984) and the various International Terrestrial Reference Frames (ITRFs). Due to the movement of Earth's tectonic plates ,

2520-441: The basis of methods of constructing right angles. For example, by counting links, three pieces of chain can be made with lengths in the ratio 3:4:5. These can be laid out to form a triangle, which will have a right angle opposite its longest side. This method is useful for laying out gardens and fields, where the dimensions are large, and great accuracy is not needed. The chains can be used repeatedly whenever required. If two lines (

2590-482: The cardinal points; North, East, South, West (NESW) The line N-S is perpendicular to the line W-E and the angles N-E, E-S, S-W and W-N are all 90° to one another. Perpendicularity easily extends to segments and rays . For example, a line segment A B ¯ {\displaystyle {\overline {AB}}} is perpendicular to a line segment C D ¯ {\displaystyle {\overline {CD}}} if, when each

2660-423: The development of the marine chronometer by John Harrison . But it was the development of accurate star charts, principally by the first British Astronomer Royal , John Flamsteed between 1680 and 1719 and disseminated by his successor Edmund Halley , that enabled navigators to use the lunar method of determining longitude more accurately using the octant developed by Thomas Godfrey and John Hadley . In

2730-427: The direction of a plumb line or vertical, and a line perpendicular to the surface of the ellipsoid of revolution – a normal to said ellipsoid – at a particular observatory, is the deflection of the vertical. When the Airy transit circle was built, a mercury basin was used to align the telescope to the perpendicular. Thus the circle was aligned with the local vertical or plumb line, which

2800-400: The east of the Greenwich meridian. At around the time of the 1884 conference, scientists were making measurements to determine the deflection of the vertical on a large scale. One might expect that plumb lines set up in various locations, if extended downward, would all pass through a single point, the centre of Earth, but this is not the case, primarily due to Earth being an ellipsoid , not

2870-402: The ellipse. The major axis of an ellipse is perpendicular to the directrix and to each latus rectum . In a parabola , the axis of symmetry is perpendicular to each of the latus rectum, the directrix, and the tangent line at the point where the axis intersects the parabola. From a point on the tangent line to a parabola's vertex, the other tangent line to the parabola is perpendicular to

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2940-623: The establishment of reference meridians for space-based location systems such as WGS-84 (which the GPS relies on) or to the fact that errors gradually crept into the International Time Bureau timekeeping process. The actual reason for the discrepancy is that the difference between precise GNSS coordinates and astronomically determined coordinates everywhere remains a localized gravity effect due to vertical deflection ; thus, no systematic rotation of global longitudes occurred between

3010-502: The existing one. This was to allow uninterrupted observation during each new construction. The final meridian was established as an imaginary line from the North Pole to the South Pole passing through the Airy transit circle. This became the United Kingdom's meridian in 1851. For all practical purposes of the period, the changes as the meridian was moved went unnoticed. Transit instruments are installed to be perpendicular to

3080-475: The face always inward of the orbit (a planet facing its star, or a moon facing its planet), just as equators are determined by rotation. Longitudes for the Earth and Moon are measured from their prime meridian (at 0°) to 180° east and west. For all other Solar System bodies, longitude is measured from 0° (their prime meridian) to 360°. West longitudes are used if the rotation of the body is prograde (or 'direct', like Earth), meaning that its direction of rotation

3150-433: The fact that the inner product vanishes for perpendicular vectors: Both proofs are valid for horizontal and vertical lines to the extent that we can let one slope be ε {\displaystyle \varepsilon } , and take the limit that ε → 0. {\displaystyle \varepsilon \rightarrow 0.} If one slope goes to zero, the other goes to infinity. Each diameter of

3220-405: The figure at the right, all of the orange-shaded angles are congruent to each other and all of the green-shaded angles are congruent to each other, because vertical angles are congruent and alternate interior angles formed by a transversal cutting parallel lines are congruent. Therefore, if lines a and b are parallel, any of the following conclusions leads to all of the others: In geometry ,

3290-542: The former astronomical system and the current geodetic system. Before the establishment of a common meridian, most maritime countries established their own prime meridian, usually passing through the country in question. In 1721, Great Britain established its own meridian passing through an early transit circle at the newly established Royal Observatory at Greenwich. The meridian was moved around 10 metres or so east on three occasions as transit circles with newer and better instruments were built, on each occasion next door to

3360-571: The graphs of the functions will be perpendicular if m 1 m 2 = − 1. {\displaystyle m_{1}m_{2}=-1.} The dot product of vectors can be also used to obtain the same result: First, shift coordinates so that the origin is situated where the lines cross. Then define two displacements along each line, r → j {\displaystyle {\vec {r}}_{j}} , for ( j = 1 , 2 ) . {\displaystyle (j=1,2).} Now, use

3430-565: The ideal International Terrestrial Reference System (ITRS) and its realization, the International Terrestrial Reference Frame (ITRF). A current convention on the Earth uses the line of longitude 180° opposite the IRM as the basis for the International Date Line . Download coordinates as: On Earth, starting at the North Pole and heading south to the South Pole , the IERS Reference Meridian (as of 2016) passes through 8 countries, 4 seas, 3 oceans and 1 channel: As on

3500-406: The line from that point through the parabola's focus . The orthoptic property of a parabola is that If two tangents to the parabola are perpendicular to each other, then they intersect on the directrix. Conversely, two tangents which intersect on the directrix are perpendicular. This implies that, seen from any point on its directrix, any parabola subtends a right angle. The transverse axis of

3570-456: The line of 0° longitude along the surface of the Earth has slowly moved toward the west from this shifted position by a few centimetres (inches); that is, towards the Airy Transit Circle (or the Airy Transit Circle has moved toward the east, depending on your point of view) since 1984 (or the 1960s). With the introduction of satellite technology, it became possible to create a more accurate and detailed global map. With these advances there also arose

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3640-620: The link between longitude and time. Based on observations to satellites and celestial compact radio sources (quasars) from various coordinated stations around the globe, Airy's transit circle drifts northeast about 2.5 centimetres (1 inch) per year relative to this Earth-centred 0° longitude. It is also the reference meridian of the Global Positioning System operated by the United States Department of Defense , and of WGS84 and its two formal versions,

3710-478: The local level (which is a plane perpendicular to a plumb line). In 1884, the International Meridian Conference took place in Washington, D.C. to establish an internationally-recognised single meridian. The meridian chosen was that which passed through the Airy transit circle at Greenwich, and it became the prime meridian of the world for a century. In 1984 it was superseded in that role by the IERS Reference Meridian which, at this latitude, runs about 102 metres to

3780-629: The location of the Airy Transit Circle ever since the first observation he took with it. Prior to that, it was defined by a succession of earlier transit instruments, the first of which was acquired by the second Astronomer Royal , Edmond Halley in 1721. It was set up in the extreme north-west corner of the Observatory between Flamsteed House and the Western Summer House. This spot, now subsumed into Flamsteed House,

3850-498: The meridian passing through Greenwich as the world standard prime meridian due to its popularity. However, France abstained from the vote, and French maps continued to use the Paris meridian for several decades. In the 18th century, London lexicographer Malachy Postlethwayt published his African maps showing the "Meridian of London" intersecting the Equator a few degrees west of the later meridian and Accra , Ghana . The plane of

3920-529: The necessity to define a reference meridian that, whilst being derived from the Airy Transit Circle, would also take into account the effects of plate movement and variations in the way that the Earth was spinning. As a result, the IERS Reference Meridian was established and is commonly used to denote the Earth's prime meridian (0° longitude) by the International Earth Rotation and Reference Systems Service , which defines and maintains

3990-494: The offset that have been proposed in the past are smaller than the current uncertainty in the deflection of the vertical, locally. The astronomical longitude of the Greenwich prime meridian was found to be 0.19″ ± 0.47″ E, i.e. the plane defined by the local vertical on the Greenwich prime meridian and the plane passing through Earth's rotation axis on the ITRF zero meridian are effectively parallel. Claims, such as that on

4060-483: The prime meridian is parallel to the local gravity vector at the Airy transit circle ( 51°28′40.1″N 0°0′5.3″W  /  51.477806°N 0.001472°W  / 51.477806; -0.001472  ( Airy Transit ) ) of the Greenwich observatory. The prime meridian was therefore long symbolised by a brass strip in the courtyard, now replaced by stainless steel, and since 16 December 1999, it has been marked by

4130-433: The prime meridian plane with the celestial sphere), it does not pass through Earth's rotation axis. As a result of this, the ITRF zero meridian, defined by a plane passing through Earth's rotation axis, is 102.478 metres to the east of the prime meridian. A 2015 analysis by Malys et al. shows the offset between the former and the latter can be explained by this deflection of the vertical alone; other possible sources of

4200-404: The same point on a circle but going through opposite endpoints of a diameter are perpendicular. This is equivalent to saying that any diameter of a circle subtends a right angle at any point on the circle, except the two endpoints of the diameter. The major and minor axes of an ellipse are perpendicular to each other and to the tangent lines to the ellipse at the points where the axes intersect

4270-410: The square of the diameter. The sum of the squared lengths of any two perpendicular chords intersecting at a given point is the same as that of any other two perpendicular chords intersecting at the same point, and is given by 8 r – 4 p (where r is the circle's radius and p is the distance from the center point to the point of intersection). Thales' theorem states that two lines both through

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4340-432: The surface of the Earth, oriented via a plumb line along the direction of gravity at the surface. This astronomic Greenwich meridian was disseminated around the world, first via the lunar distance method , then by chronometers carried on ships, then via telegraph lines carried by submarine communications cables , then via radio time signals. One remote longitude ultimately based on the Greenwich meridian using these methods

4410-541: The time, one of many ) was first established by Sir George Airy in 1851, and by 1884, over two-thirds of all ships and tonnage used it as the reference meridian on their charts and maps. In October of that year, at the invitation of the President of the United States , 41 delegates from 25 nations met in Washington, D.C. , United States, for the International Meridian Conference . This conference selected

4480-509: The two lines meet; and (2) at the point of intersection the straight angle on one side of the first line is cut by the second line into two congruent angles . Perpendicularity can be shown to be symmetric , meaning if a first line is perpendicular to a second line, then the second line is also perpendicular to the first. For this reason, we may speak of two lines as being perpendicular (to each other) without specifying an order. A great example of perpendicularity can be seen in any compass, note

4550-500: The western tip of Africa (17.5° W) as negative numbers were not yet in use. His prime meridian corresponds to 18° 40' west of Winchester (about 20°W) today. At that time the chief method of determining longitude was by using the reported times of lunar eclipses in different countries. One of the earliest known descriptions of standard time in India appeared in the 4th century CE astronomical treatise Surya Siddhanta . Postulating

4620-602: Was Ptolemy (c.   90 – 168   CE) who first used a consistent meridian for a world map in his Geographia . Ptolemy used as his basis the " Fortunate Isles ", a group of islands in the Atlantic , which are usually associated with the Canary Islands (13° to 18°W), although his maps correspond more closely to the Cape Verde islands (22° to 25° W). The main point is to be comfortably west of

4690-487: Was eventually settled at 370 leagues (2,193 kilometers, 1,362 statute miles, or 1,184 nautical miles) west of Cape Verde . This is shown in the copies of Spain's Padron Real made by Diogo Ribeiro in 1527 and 1529. São Miguel Island (25.5°W) in the Azores was still used for the same reason as late as 1594 by Christopher Saxton , although by then it had been shown that the zero magnetic declination line did not follow

4760-483: Was first printed with maps at Bologna in 1477, and many early globes in the 16th century followed his lead. But there was still a hope that a "natural" basis for a prime meridian existed. Christopher Columbus reported (1493) that the compass pointed due north somewhere in mid-Atlantic, and this fact was used in the important Treaty of Tordesillas of 1494, which settled the territorial dispute between Spain and Portugal over newly discovered lands. The Tordesillas line

4830-413: Was selected by delegates (forty-one delegates representing twenty-five nations) to the International Meridian Conference held in Washington, D.C. , United States to be the common zero of longitude and standard of time reckoning throughout the world. The position of the historic prime meridian, based at the Royal Observatory, Greenwich , was established by Sir George Airy in 1851. It was defined by

4900-615: Was that of the North American Datum 1927 or NAD27, an ellipsoid whose surface best matches mean sea level under the United States . Beginning in 1973 the International Time Bureau and later the International Earth Rotation and Reference Systems Service changed from reliance on optical instruments like the Airy Transit Circle to techniques such as lunar laser ranging , satellite laser ranging , and very-long-baseline interferometry . The new techniques resulted in

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