A natural satellite or moon is, in the most common usage, an astronomical body that orbits a planet or minor planet. In the Solar System there are six planetary satellite systems containing 185 known natural satellites. Four IAU-listed dwarf planets are known to have natural satellites: Pluto, Haumea and Eris; as of September 2018, there are 334 other minor planets known to have moons. The Earth–Moon system is unique in that the ratio of the mass of the Moon to the mass of Earth is much greater than that of any other natural-satellite–planet ratio in the Solar System. At 3,474 km across, the Moon is 0.27 times the diameter of Earth. The first known natural satellite was the Moon, but it was considered a "planet" until Copernicus' introduction of De revolutionibus orbium coelestium in 1543; until the discovery of the Galilean satellites in 1610, there was no opportunity for referring to such objects as a class. Galileo chose to refer to his discoveries as Planetæ, but discoverers chose other terms to distinguish them from the objects they orbited.
The first to use of the term satellite to describe orbiting bodies was the German astronomer Johannes Kepler in his pamphlet Narratio de Observatis a se quatuor Iouis satellitibus erronibus in 1610. He derived the term from the Latin word satelles, meaning "guard", "attendant", or "companion", because the satellites accompanied their primary planet in their journey through the heavens; the term satellite thus became the normal one for referring to an object orbiting a planet, as it avoided the ambiguity of "moon". In 1957, the launching of the artificial object Sputnik created a need for new terminology. Sputnik was created by Soviet Union, it was the first satellite ever; the terms man-made satellite and artificial moon were quickly abandoned in favor of the simpler satellite, as a consequence, the term has become linked with artificial objects flown in space – including, sometimes those not in orbit around a planet. Because of this shift in meaning, the term moon, which had continued to be used in a generic sense in works of popular science and in fiction, has regained respectability and is now used interchangeably with natural satellite in scientific articles.
When it is necessary to avoid both the ambiguity of confusion with Earth's natural satellite the Moon and the natural satellites of the other planets on the one hand, artificial satellites on the other, the term natural satellite is used. To further avoid ambiguity, the convention is to capitalize the word Moon when referring to Earth's natural satellite, but not when referring to other natural satellites. Many authors define "satellite" or "natural satellite" as orbiting some planet or minor planet, synonymous with "moon" – by such a definition all natural satellites are moons, but Earth and other planets are not satellites. A few recent authors define "moon" as "a satellite of a planet or minor planet", "planet" as "a satellite of a star" – such authors consider Earth as a "natural satellite of the sun". There is no established lower limit on what is considered a "moon"; every natural celestial body with an identified orbit around a planet of the Solar System, some as small as a kilometer across, has been considered a moon, though objects a tenth that size within Saturn's rings, which have not been directly observed, have been called moonlets.
Small asteroid moons, such as Dactyl, have been called moonlets. The upper limit is vague. Two orbiting bodies are sometimes described as a double planet rather than satellite. Asteroids such as 90 Antiope are considered double asteroids, but they have not forced a clear definition of what constitutes a moon; some authors consider the Pluto–Charon system to be a double planet. The most common dividing line on what is considered a moon rests upon whether the barycentre is below the surface of the larger body, though this is somewhat arbitrary, because it depends on distance as well as relative mass; the natural satellites orbiting close to the planet on prograde, uninclined circular orbits are thought to have been formed out of the same collapsing region of the protoplanetary disk that created its primary. In contrast, irregular satellites are thought to be captured asteroids further fragmented by collisions. Most of the major natural satellites of the Solar System have regular orbits, while most of the small natural satellites have irregular orbits.
The Moon and Charon are exceptions among large bodies in that they are thought to have originated by the collision of two large proto-planetary objects. The material that would have been placed in orbit around the central body is predicted to have reaccreted to form one or more orbiting natural satellites; as opposed to planetary-sized bodies, asteroid moons are thought to form by this process. Triton is another exception; the capture of an asteroid from a heliocentric orbit is not always permanent. According to simulations, temporary satellites should be a common phenomenon; the only observed example is 2006 RH120, a temporary satellite of Earth for nine months in 2006 and 2007. Most regular moons (natural satellites following close and prograde orbits with small orb
A circle is a simple closed shape. It is the set of all points in a plane; the distance between any of the points and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted. A circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior. A circle may be defined as a special kind of ellipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter squared, using calculus of variations. A circle is a plane figure bounded by one line, such that all right lines drawn from a certain point within it to the bounding line, are equal; the bounding line is called the point, its centre. Annulus: a ring-shaped object, the region bounded by two concentric circles.
Arc: any connected part of a circle. Specifying two end points of an arc and a center allows for two arcs that together make up a full circle. Centre: the point equidistant from all points on the circle. Chord: a line segment whose endpoints lie on the circle, thus dividing a circle in two sements. Circumference: the length of one circuit along the circle, or the distance around the circle. Diameter: a line segment whose endpoints lie on the circle and that passes through the centre; this is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord for a given circle, its length is twice the length of a radius. Disc: the region of the plane bounded by a circle. Lens: the region common to two overlapping discs. Passant: a coplanar straight line that has no point in common with the circle. Radius: a line segment joining the centre of a circle with any single point on the circle itself. Sector: a region bounded by two radii of equal length with a common center and either of the two possible arcs, determined by this center and the endpoints of the radii.
Segment: a region bounded by a chord and one of the arcs connecting the chord's endpoints. The length of the chord imposes a lower boundary on the diameter of possible arcs. Sometimes the term segment is used only for regions not containing the center of the circle to which their arc belongs to. Secant: an extended chord, a coplanar straight line, intersecting a circle in two points. Semicircle: one of the two possible arcs determined by the endpoints of a diameter, taking its midpoint as center. In non-technical common usage it may mean the interior of the two dimensional region bounded by a diameter and one of its arcs, technically called a half-disc. A half-disc is a special case of a segment, namely the largest one. Tangent: a coplanar straight line that has one single point in common with a circle. All of the specified regions may be considered as open, that is, not containing their boundaries, or as closed, including their respective boundaries; the word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, meaning "hoop" or "ring".
The origins of the words circus and circuit are related. The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon, a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand; the circle is the basis for the wheel, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry and calculus. Early science geometry and astrology and astronomy, was connected to the divine for most medieval scholars, many believed that there was something intrinsically "divine" or "perfect" that could be found in circles; some highlights in the history of the circle are: 1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as an approximate value of π. 300 BCE – Book 3 of Euclid's Elements deals with the properties of circles. In Plato's Seventh Letter there is a detailed explanation of the circle.
Plato explains the perfect circle, how it is different from any drawing, definition or explanation. 1880 CE – Lindemann proves that π is transcendental settling the millennia-old problem of squaring the circle. The ratio of a circle's circumference to its diameter is π, an irrational constant equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by: C = 2 π r = π d; as proved by Archimedes, in his Measurement of a Circle, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, which comes to π multiplied by the radius squared: A r e a = π r 2. Equivalently, denoting diameter by d, A r e
An astronomical object or celestial object is a occurring physical entity, association, or structures that exists in the observable universe. In astronomy, the terms object and body are used interchangeably. However, an astronomical body or celestial body is a single bound, contiguous entity, while an astronomical or celestial object is a complex, less cohesively bound structure, which may consist of multiple bodies or other objects with substructures. Examples of astronomical objects include planetary systems, star clusters and galaxies, while asteroids, moons and stars are astronomical bodies. A comet may be identified as both body and object: It is a body when referring to the frozen nucleus of ice and dust, an object when describing the entire comet with its diffuse coma and tail; the universe can be viewed as having a hierarchical structure. At the largest scales, the fundamental component of assembly is the galaxy. Galaxies are organized into groups and clusters within larger superclusters, that are strung along great filaments between nearly empty voids, forming a web that spans the observable universe.
The universe has a variety of morphologies, with irregular and disk-like shapes, depending on their formation and evolutionary histories, including interaction with other galaxies, which may lead to a merger. Disc galaxies encompass lenticular and spiral galaxies with features, such as spiral arms and a distinct halo. At the core, most galaxies have a supermassive black hole, which may result in an active galactic nucleus. Galaxies can have satellites in the form of dwarf galaxies and globular clusters; the constituents of a galaxy are formed out of gaseous matter that assembles through gravitational self-attraction in a hierarchical manner. At this level, the resulting fundamental components are the stars, which are assembled in clusters from the various condensing nebulae; the great variety of stellar forms are determined entirely by the mass and evolutionary state of these stars. Stars may be found in multi-star systems. A planetary system and various minor objects such as asteroids and debris, can form in a hierarchical process of accretion from the protoplanetary disks that surrounds newly formed stars.
The various distinctive types of stars are shown by the Hertzsprung–Russell diagram —a plot of absolute stellar luminosity versus surface temperature. Each star follows an evolutionary track across this diagram. If this track takes the star through a region containing an intrinsic variable type its physical properties can cause it to become a variable star. An example of this is the instability strip, a region of the H-R diagram that includes Delta Scuti, RR Lyrae and Cepheid variables. Depending on the initial mass of the star and the presence or absence of a companion, a star may spend the last part of its life as a compact object; the table below lists the general categories of bodies and objects by their structure. List of light sources List of Solar System objects List of Solar System objects by size Lists of astronomical objects SkyChart, Sky & Telescope at the Library of Congress Web Archives Monthly skymaps for every location on Earth
The orbital eccentricity of an astronomical object is a parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values between 0 and 1 form an elliptic orbit, 1 is a parabolic escape orbit, greater than 1 is a hyperbola; the term derives its name from the parameters of conic sections, as every Kepler orbit is a conic section. It is used for the isolated two-body problem, but extensions exist for objects following a Klemperer rosette orbit through the galaxy. In a two-body problem with inverse-square-law force, every orbit is a Kepler orbit; the eccentricity of this Kepler orbit is a non-negative number. The eccentricity may take the following values: circular orbit: e = 0 elliptic orbit: 0 < e < 1 parabolic trajectory: e = 1 hyperbolic trajectory: e > 1 The eccentricity e is given by e = 1 + 2 E L 2 m red α 2 where E is the total orbital energy, L is the angular momentum, mred is the reduced mass, α the coefficient of the inverse-square law central force such as gravity or electrostatics in classical physics: F = α r 2 or in the case of a gravitational force: e = 1 + 2 ε h 2 μ 2 where ε is the specific orbital energy, μ the standard gravitational parameter based on the total mass, h the specific relative angular momentum.
For values of e from 0 to 1 the orbit's shape is an elongated ellipse. The limit case between an ellipse and a hyperbola, when e equals 1, is parabola. Radial trajectories are classified as elliptic, parabolic, or hyperbolic based on the energy of the orbit, not the eccentricity. Radial orbits hence eccentricity equal to one. Keeping the energy constant and reducing the angular momentum, elliptic and hyperbolic orbits each tend to the corresponding type of radial trajectory while e tends to 1. For a repulsive force only the hyperbolic trajectory, including the radial version, is applicable. For elliptical orbits, a simple proof shows that arcsin yields the projection angle of a perfect circle to an ellipse of eccentricity e. For example, to view the eccentricity of the planet Mercury, one must calculate the inverse sine to find the projection angle of 11.86 degrees. Next, tilt any circular object by that angle and the apparent ellipse projected to your eye will be of that same eccentricity; the word "eccentricity" comes from Medieval Latin eccentricus, derived from Greek ἔκκεντρος ekkentros "out of the center", from ἐκ- ek-, "out of" + κέντρον kentron "center".
"Eccentric" first appeared in English in 1551, with the definition "a circle in which the earth, sun. Etc. deviates from its center". By five years in 1556, an adjectival form of the word had developed; the eccentricity of an orbit can be calculated from the orbital state vectors as the magnitude of the eccentricity vector: e = | e | where: e is the eccentricity vector. For elliptical orbits it can be calculated from the periapsis and apoapsis since rp = a and ra = a, where a is the semimajor axis. E = r a − r p r a + r p = 1 − 2 r a r p + 1 where: ra is the radius at apoapsis. Rp is the radius at periapsis; the eccentricity of an elliptical orbit can be used to obtain the ratio of the periapsis to the apoapsis: r p r a = 1 − e 1 + e For Earth, orbital eccentricity ≈ 0.0167, apoapsis= aphelion and periapsis= perihelion relative to sun. For Earth's annual orbit path, ra/rp ratio = longest_radius / shortest_radius ≈ 1.034 relative to center point of path. The eccentricity of the Earth's orbit is about 0.0167.
The Sun is the star at the center of the Solar System. It is a nearly perfect sphere of hot plasma, with internal convective motion that generates a magnetic field via a dynamo process, it is by far the most important source of energy for life on Earth. Its diameter is about 1.39 million kilometers, or 109 times that of Earth, its mass is about 330,000 times that of Earth. It accounts for about 99.86% of the total mass of the Solar System. Three quarters of the Sun's mass consists of hydrogen; the Sun is a G-type main-sequence star based on its spectral class. As such, it is informally and not accurately referred to as a yellow dwarf, it formed 4.6 billion years ago from the gravitational collapse of matter within a region of a large molecular cloud. Most of this matter gathered in the center, whereas the rest flattened into an orbiting disk that became the Solar System; the central mass became so hot and dense that it initiated nuclear fusion in its core. It is thought that all stars form by this process.
The Sun is middle-aged. It fuses about 600 million tons of hydrogen into helium every second, converting 4 million tons of matter into energy every second as a result; this energy, which can take between 10,000 and 170,000 years to escape from its core, is the source of the Sun's light and heat. In about 5 billion years, when hydrogen fusion in its core has diminished to the point at which the Sun is no longer in hydrostatic equilibrium, its core will undergo a marked increase in density and temperature while its outer layers expand to become a red giant, it is calculated that the Sun will become sufficiently large to engulf the current orbits of Mercury and Venus, render Earth uninhabitable. After this, it will shed its outer layers and become a dense type of cooling star known as a white dwarf, no longer produce energy by fusion, but still glow and give off heat from its previous fusion; the enormous effect of the Sun on Earth has been recognized since prehistoric times, the Sun has been regarded by some cultures as a deity.
The synodic rotation of Earth and its orbit around the Sun are the basis of solar calendars, one of, the predominant calendar in use today. The English proper name Sun may be related to south. Cognates to English sun appear in other Germanic languages, including Old Frisian sunne, Old Saxon sunna, Middle Dutch sonne, modern Dutch zon, Old High German sunna, modern German Sonne, Old Norse sunna, Gothic sunnō. All Germanic terms for the Sun stem from Proto-Germanic *sunnōn; the Latin name for the Sun, Sol, is not used in everyday English. Sol is used by planetary astronomers to refer to the duration of a solar day on another planet, such as Mars; the related word solar is the usual adjectival term used for the Sun, in terms such as solar day, solar eclipse, Solar System. A mean Earth solar day is 24 hours, whereas a mean Martian'sol' is 24 hours, 39 minutes, 35.244 seconds. The English weekday name Sunday stems from Old English and is a result of a Germanic interpretation of Latin dies solis, itself a translation of the Greek ἡμέρα ἡλίου.
The Sun is a G-type main-sequence star. The Sun has an absolute magnitude of +4.83, estimated to be brighter than about 85% of the stars in the Milky Way, most of which are red dwarfs. The Sun is heavy-element-rich, star; the formation of the Sun may have been triggered by shockwaves from more nearby supernovae. This is suggested by a high abundance of heavy elements in the Solar System, such as gold and uranium, relative to the abundances of these elements in so-called Population II, heavy-element-poor, stars; the heavy elements could most plausibly have been produced by endothermic nuclear reactions during a supernova, or by transmutation through neutron absorption within a massive second-generation star. The Sun is by far the brightest object in the Earth's sky, with an apparent magnitude of −26.74. This is about 13 billion times brighter than the next brightest star, which has an apparent magnitude of −1.46. The mean distance of the Sun's center to Earth's center is 1 astronomical unit, though the distance varies as Earth moves from perihelion in January to aphelion in July.
At this average distance, light travels from the Sun's horizon to Earth's horizon in about 8 minutes and 19 seconds, while light from the closest points of the Sun and Earth takes about two seconds less. The energy of this sunlight supports all life on Earth by photosynthesis, drives Earth's climate and weather; the Sun does not have a definite boundary, but its density decreases exponentially with increasing height above the photosphere. For the purpose of measurement, the Sun's radius is considered to be the distance from its center to the edge of the photosphere, the apparent visible surface of the Sun. By this measure, the Sun is a near-perfect sphere with an oblateness estimated at about 9 millionths, which means that its polar diameter differs from its equatorial diameter by only 10 kilometres; the tidal effect of the planets is weak and does not affect the shape of the Sun. The Sun rotates faster at its equator than at its poles; this differential rotation is caused by convective motion
In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphere, which may be centered on Earth or the observer. If centered on the observer, half of the sphere would resemble a hemispherical screen over the observing location; the celestial sphere is a practical tool for spherical astronomy, allowing astronomers to specify the apparent positions of objects in the sky if their distances are unknown or irrelevant. In the equatorial coordinate system, the celestial equator divides the celestial sphere into two halves: the northern and southern celestial hemispheres; because astronomical objects are at such remote distances, casual observation of the sky offers no information on their actual distances. All celestial objects seem far away, as if fixed onto the inside of a sphere with a large but unknown radius, which appears to rotate westward overhead.
For purposes of spherical astronomy, concerned only with the directions to celestial objects, it makes no difference if this is the case or if it is Earth, rotating while the celestial sphere is stationary. The celestial sphere can be considered to be infinite in radius; this means any point within it, including that occupied by the observer, can be considered the center. It means that all parallel lines, be they millimetres apart or across the Solar System from each other, will seem to intersect the sphere at a single point, analogous to the vanishing point of graphical perspective. All parallel planes will seem to intersect the sphere in a coincident great circle. Conversely, observers looking toward the same point on an infinite-radius celestial sphere will be looking along parallel lines, observers looking toward the same great circle, along parallel planes. On an infinite-radius celestial sphere, all observers see the same things in the same direction. For some objects, this is over-simplified.
Objects which are near to the observer will seem to change position against the distant celestial sphere if the observer moves far enough, from one side of planet Earth to the other. This effect, known as parallax, can be represented as a small offset from a mean position; the celestial sphere can be considered to be centered at the Earth's center, the Sun's center, or any other convenient location, offsets from positions referred to these centers can be calculated. In this way, astronomers can predict geocentric or heliocentric positions of objects on the celestial sphere, without the need to calculate the individual geometry of any particular observer, the utility of the celestial sphere is maintained. Individual observers can work out their own small offsets from the mean positions. In many cases in astronomy, the offsets are insignificant; the celestial sphere can thus be thought of as a kind of astronomical shorthand, is applied frequently by astronomers. For instance, the Astronomical Almanac for 2010 lists the apparent geocentric position of the Moon on January 1 at 00:00:00.00 Terrestrial Time, in equatorial coordinates, as right ascension 6h 57m 48.86s, declination +23° 30' 05.5".
Implied in this position is. For many rough uses, this position, as seen from the Earth's center, is adequate. For applications requiring precision, the Almanac gives formulae and methods for calculating the topocentric coordinates, that is, as seen from a particular place on the Earth's surface, based on the geocentric position; this abbreviates the amount of detail necessary in such almanacs, as each observer can handle their own specific circumstances. These concepts are important for understanding celestial coordinate systems, frameworks for measuring the positions of objects in the sky. Certain reference lines and planes on Earth, when projected onto the celestial sphere, form the bases of the reference systems; these include the Earth's equator and orbit. At their intersections with the celestial sphere, these form the celestial equator, the north and south celestial poles, the ecliptic, respectively; as the celestial sphere is considered arbitrary or infinite in radius, all observers see the celestial equator, celestial poles, ecliptic at the same place against the background stars.
From these bases, directions toward objects in the sky can be quantified by constructing celestial coordinate systems. Similar to geographic longitude and latitude, the equatorial coordinate system specifies positions relative to the celestial equator and celestial poles, using right ascension and declination; the ecliptic coordinate system specifies positions relative to the ecliptic, using ecliptic longitude and latitude. Besides the equatorial and ecliptic systems, some other celestial coordinate systems, like the galactic coordinate system, are more appropriate for particular purposes; the ancients assumed the literal truth of stars attached to a celestial sphere, revolving about the Earth in one day, a fixed Earth. The Eudoxan planetary model, on which the Aristotelian and Ptolemaic models were based, was the first geometric explanation for the "wandering" of the classical planets; the outer most of these "crystal spheres" was thought to carry the fixed stars. Eudoxus used 27 concentric spherical solids to answer Plato's challenge: "By the assumption of what uniform and orderly motions can the appa
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. Note that v and w can be perpendicular, but cannot 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