In geometry the orientation, angular position, or attitude of an object such as a line, plane or rigid body is part of the description of how it is placed in the space it occupies. Namely, it is the imaginary rotation, needed to move the object from a reference placement to its current placement. A rotation may not be enough to reach the current placement, it may be necessary to add an imaginary translation, called the object's location. The location and orientation together describe how the object is placed in space; the above-mentioned imaginary rotation and translation may be thought to occur in any order, as the orientation of an object does not change when it translates, its location does not change when it rotates. Euler's rotation theorem shows that in three dimensions any orientation can be reached with a single rotation around a fixed axis; this gives one common way of representing the orientation using an axis–angle representation. Other used methods include rotation quaternions, Euler angles, or rotation matrices.
More specialist uses include Miller indices in crystallography and dip in geology and grade on maps and signs. The orientation is given relative to a frame of reference specified by a Cartesian coordinate system. In general the position and orientation in space of a rigid body are defined as the position and orientation, relative to the main reference frame, of another reference frame, fixed relative to the body, hence translates and rotates with it. At least three independent values are needed to describe the orientation of this local frame. Three other values are All the points of the body change their position during a rotation except for those lying on the rotation axis. If the rigid body has rotational symmetry not all orientations are distinguishable, except by observing how the orientation evolves in time from a known starting orientation. For example, the orientation in space of a line, line segment, or vector can be specified with only two values, for example two direction cosines. Another example is the position of a point on the earth described using the orientation of a line joining it with the earth's center, measured using the two angles of longitude and latitude.
The orientation of a plane can be described with two values as well, for instance by specifying the orientation of a line normal to that plane, or by using the strike and dip angles. Further details about the mathematical methods to represent the orientation of rigid bodies and planes in three dimensions are given in the following sections. In two dimensions the orientation of any object is given by a single value: the angle through which it has rotated. There is only one fixed point about which the rotation takes place. Several methods to describe orientations of a rigid body in three dimensions have been developed, they are summarized in the following sections. The first attempt to represent an orientation was owed to Leonhard Euler, he imagined three reference frames that could rotate one around the other, realized that by starting with a fixed reference frame and performing three rotations, he could get any other reference frame in the space. The values of these three rotations are called Euler angles.
These are three angles known as yaw and roll, Navigation angles and Cardan angles. Mathematically they constitute a set of six possibilities inside the twelve possible sets of Euler angles, the ordering being the one best used for describing the orientation of a vehicle such as an airplane. In aerospace engineering they are referred to as Euler angles. Euler realized that the composition of two rotations is equivalent to a single rotation about a different fixed axis. Therefore, the composition of the former three angles has to be equal to only one rotation, whose axis was complicated to calculate until matrices were developed. Based on this fact he introduced a vectorial way to describe any rotation, with a vector on the rotation axis and module equal to the value of the angle. Therefore, any orientation can be represented by a rotation vector that leads to it from the reference frame; when used to represent an orientation, the rotation vector is called orientation vector, or attitude vector.
A similar method, called axis–angle representation, describes a rotation or orientation using a unit vector aligned with the rotation axis, a separate value to indicate the angle. With the introduction of matrices, the Euler theorems were rewritten; the rotations were described by orthogonal matrices referred to as rotation matrices or direction cosine matrices. When used to represent an orientation, a rotation matrix is called orientation matrix, or attitude matrix; the above-mentioned Euler vector is the eigenvector of a rotation matrix. The product of two rotation matrices is the composition of rotations. Therefore, as before, the orientation can be given as the rotation from the initial frame to achieve the frame that we want to describe; the configuration space of a non-symmetrical object in n-dimensional space is SO × Rn. Orientation may be visualized by attaching a basis of tangent vectors to an object; the direction in which each vector points determines its orientation. Another way to describe rotations is using rotation quaternions called versors.
They are equivalent to rotation matrices and rotation vectors. With respect to rotation vectors, they can be more converted to and from matri
Earth is the third planet from the Sun and the only astronomical object known to harbor life. According to radiometric dating and other sources of evidence, Earth formed over 4.5 billion years ago. Earth's gravity interacts with other objects in space the Sun and the Moon, Earth's only natural satellite. Earth revolves around the Sun in a period known as an Earth year. During this time, Earth rotates about its axis about 366.26 times. Earth's axis of rotation is tilted with respect to its orbital plane; the gravitational interaction between Earth and the Moon causes ocean tides, stabilizes Earth's orientation on its axis, slows its rotation. Earth is the largest of the four terrestrial planets. Earth's lithosphere is divided into several rigid tectonic plates that migrate across the surface over periods of many millions of years. About 71% of Earth's surface is covered with water by oceans; the remaining 29% is land consisting of continents and islands that together have many lakes and other sources of water that contribute to the hydrosphere.
The majority of Earth's polar regions are covered in ice, including the Antarctic ice sheet and the sea ice of the Arctic ice pack. Earth's interior remains active with a solid iron inner core, a liquid outer core that generates the Earth's magnetic field, a convecting mantle that drives plate tectonics. Within the first billion years of Earth's history, life appeared in the oceans and began to affect the Earth's atmosphere and surface, leading to the proliferation of aerobic and anaerobic organisms; some geological evidence indicates. Since the combination of Earth's distance from the Sun, physical properties, geological history have allowed life to evolve and thrive. In the history of the Earth, biodiversity has gone through long periods of expansion punctuated by mass extinction events. Over 99% of all species that lived on Earth are extinct. Estimates of the number of species on Earth today vary widely. Over 7.6 billion humans live on Earth and depend on its biosphere and natural resources for their survival.
Humans have developed diverse cultures. The modern English word Earth developed from a wide variety of Middle English forms, which derived from an Old English noun most spelled eorðe, it has cognates in every Germanic language, their proto-Germanic root has been reconstructed as *erþō. In its earliest appearances, eorðe was being used to translate the many senses of Latin terra and Greek γῆ: the ground, its soil, dry land, the human world, the surface of the world, the globe itself; as with Terra and Gaia, Earth was a personified goddess in Germanic paganism: the Angles were listed by Tacitus as among the devotees of Nerthus, Norse mythology included Jörð, a giantess given as the mother of Thor. Earth was written in lowercase, from early Middle English, its definite sense as "the globe" was expressed as the earth. By Early Modern English, many nouns were capitalized, the earth became the Earth when referenced along with other heavenly bodies. More the name is sometimes given as Earth, by analogy with the names of the other planets.
House styles now vary: Oxford spelling recognizes the lowercase form as the most common, with the capitalized form an acceptable variant. Another convention capitalizes "Earth" when appearing as a name but writes it in lowercase when preceded by the, it always appears in lowercase in colloquial expressions such as "what on earth are you doing?" The oldest material found in the Solar System is dated to 4.5672±0.0006 billion years ago. By 4.54±0.04 Bya the primordial Earth had formed. The bodies in the Solar System evolved with the Sun. In theory, a solar nebula partitions a volume out of a molecular cloud by gravitational collapse, which begins to spin and flatten into a circumstellar disk, the planets grow out of that disk with the Sun. A nebula contains gas, ice grains, dust. According to nebular theory, planetesimals formed by accretion, with the primordial Earth taking 10–20 million years to form. A subject of research is the formation of some 4.53 Bya. A leading hypothesis is that it was formed by accretion from material loosed from Earth after a Mars-sized object, named Theia, hit Earth.
In this view, the mass of Theia was 10 percent of Earth, it hit Earth with a glancing blow and some of its mass merged with Earth. Between 4.1 and 3.8 Bya, numerous asteroid impacts during the Late Heavy Bombardment caused significant changes to the greater surface environment of the Moon and, by inference, to that of Earth. Earth's atmosphere and oceans were formed by volcanic outgassing. Water vapor from these sources condensed into the oceans, augmented by water and ice from asteroids and comets. In this model, atmospheric "greenhouse gases" kept the oceans from freezing when the newly forming Sun had only 70% of its current luminosity. By 3.5 Bya, Earth's magnetic field was established, which helped prevent the atmosphere from being stripped away by the solar wind. A crust formed; the two models that explain land mass propose either a steady growth to the present-day forms or, more a rapid growth early in Earth history followed by a long-term steady continental area. Continents formed by plate tectonics
Projective geometry is a topic in mathematics. It is the study of geometric properties that are invariant with respect to projective transformations; this means that, compared to elementary geometry, projective geometry has a different setting, projective space, a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, that geometric transformations are permitted that transform the extra points to Euclidean points, vice versa. Properties meaningful for projective geometry are respected by this new idea of transformation, more radical in its effects than can be expressed by a transformation matrix and translations; the first issue for geometers is. It is not possible to refer to angles in projective geometry as it is in Euclidean geometry, because angle is an example of a concept not invariant with respect to projective transformations, as is seen in perspective drawing. One source for projective geometry was indeed the theory of perspective.
Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, once the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of projective geometry in two dimensions. While the ideas were available earlier, projective geometry was a development of the 19th century; this included the theory of complex projective space, the coordinates used being complex numbers. Several major types of more abstract mathematics were based on projective geometry, it was a subject with a large number of practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry; the topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry and projective differential geometry.
Projective geometry is an elementary non-metrical form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of configurations of lines; that there is indeed some geometric interest in this sparse setting was first established by Desargues and others in their exploration of the principles of perspective art. In higher dimensional spaces there are considered hyperplanes, other linear subspaces, which exhibit the principle of duality; the simplest illustration of duality is in the projective plane, where the statements "two distinct points determine a unique line" and "two distinct lines determine a unique point" show the same structure as propositions. Projective geometry can be seen as a geometry of constructions with a straight-edge alone. Since projective geometry excludes compass constructions, there are no circles, no angles, no measurements, no parallels, no concept of intermediacy, it was realised. For example, the different conic sections are all equivalent in projective geometry, some theorems about circles can be considered as special cases of these general theorems.
During the early 19th century the work of Jean-Victor Poncelet, Lazare Carnot and others established projective geometry as an independent field of mathematics. Its rigorous foundations were addressed by Karl von Staudt and perfected by Italians Giuseppe Peano, Mario Pieri, Alessandro Padoa and Gino Fano during the late 19th century. Projective geometry, like affine and Euclidean geometry, can be developed from the Erlangen program of Felix Klein. After much work on the large number of theorems in the subject, the basics of projective geometry became understood; the incidence structure and the cross-ratio are fundamental invariants under projective transformations. Projective geometry can be modeled by the affine plane plus a line "at infinity" and treating that line as "ordinary". An algebraic model for doing projective geometry in the style of analytic geometry is given by homogeneous coordinates. On the other hand, axiomatic studies revealed the existence of non-Desarguesian planes, examples to show that the axioms of incidence can be modelled by structures not accessible to reasoning through homogeneous coordinate systems.
In a foundational sense, projective geometry and ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine and Euclidean geometry. Projective geometry is not ``; the first geometrical properties of a projective nature were discovered during the 3rd century by Pappus of Alexandria. Filippo Brunelleschi started investigating the geometry of perspective during 1425. Johannes Kepler and Gérard Desargues independently developed the concept of the "point at infinity". Desarg
In astronomy, the geocentric model is a superseded description of the Universe with Earth at the center. Under the geocentric model, the Sun, Moon and planets all orbited Earth; the geocentric model served as the predominant description of the cosmos in many ancient civilizations, such as those of Aristotle and Ptolemy. Two observations supported the idea. First, from anywhere on Earth, the Sun appears to revolve around Earth once per day. While the Moon and the planets have their own motions, they appear to revolve around Earth about once per day; the stars appeared to be fixed on a celestial sphere rotating once each day about an axis through the geographic poles of Earth. Second, Earth seems to be unmoving from the perspective of an earthbound observer. Ancient Greek, ancient Roman, medieval philosophers combined the geocentric model with a spherical Earth, in contrast to the older flat Earth model implied in some mythology; the ancient Jewish Babylonian uranography pictured a flat Earth with a dome-shaped, rigid canopy called the firmament placed over it.
However, the ancient Greeks believed that the motions of the planets were circular and not elliptical, a view, not challenged in Western culture until the 17th century, when Johannes Kepler postulated that orbits were heliocentric and elliptical. In 1687, Newton showed; the astronomical predictions of Ptolemy's geocentric model were used to prepare astrological and astronomical charts for over 1500 years. The geocentric model held sway into the early modern age, but from the late 16th century onward, it was superseded by the heliocentric model of Copernicus and Kepler. There was much resistance to the transition between these two theories; some Christian theologians were reluctant to reject a theory. Others felt a unknown theory could not subvert an accepted consensus for geocentrism; the geocentric model entered Greek philosophy at an early point. In the 6th century BC, Anaximander proposed a cosmology with Earth shaped like a section of a pillar, held aloft at the center of everything; the Sun and planets were holes in invisible wheels surrounding Earth.
About the same time, Pythagoras thought that the Earth was a sphere, but not at the center. These views were combined, so most educated Greeks from the 4th century BC on thought that the Earth was a sphere at the center of the universe. In the 4th century BC, two influential Greek philosophers and his student Aristotle, wrote works based on the geocentric model. According to Plato, the Earth was a sphere; the stars and planets were carried around the Earth on spheres or circles, arranged in the order: Moon, Venus, Mars, Saturn, fixed stars, with the fixed stars located on the celestial sphere. In his "Myth of Er", a section of the Republic, Plato describes the cosmos as the Spindle of Necessity, attended by the Sirens and turned by the three Fates. Eudoxus of Cnidus, who worked with Plato, developed a less mythical, more mathematical explanation of the planets' motion based on Plato's dictum stating that all phenomena in the heavens can be explained with uniform circular motion. Aristotle elaborated on Eudoxus' system.
In the developed Aristotelian system, the spherical Earth is at the center of the universe, all other heavenly bodies are attached to 47–55 transparent, rotating spheres surrounding the Earth, all concentric with it. These spheres, known as crystalline spheres, all moved at different uniform speeds to create the revolution of bodies around the Earth, they were composed of an incorruptible substance called aether. Aristotle believed that the Moon was in the innermost sphere and therefore touches the realm of Earth, causing the dark spots and the ability to go through lunar phases, he further described his system by explaining the natural tendencies of the terrestrial elements: Earth, fire, air, as well as celestial aether. His system held that Earth was the heaviest element, with the strongest movement towards the center, thus water formed a layer surrounding the sphere of Earth; the tendency of air and fire, on the other hand, was to move upwards, away from the center, with fire being lighter than air.
Beyond the layer of fire, were the solid spheres of aether in which the celestial bodies were embedded. They, were entirely composed of aether. Adherence to the geocentric model stemmed from several important observations. First of all, if the Earth did move one ought to be able to observe the shifting of the fixed stars due to stellar parallax. In short, if the Earth was moving, the shapes of the constellations should change over the course of a year. If they did not appear to move, the stars are either much farther away than the Sun and the planets than conceived, making their motion undetectable, or in reality they are not moving at all; because the stars were much further away than Greek astronomers postulated, stellar parallax was not detected until the 19th century. Therefore, the Greeks chose the simpler of the two explanations. Another observ
In astronomy and celestial navigation, the hour angle is one of the coordinates used in the equatorial coordinate system to give the direction of a point on the celestial sphere. The hour angle of a point is the angle between two planes: one containing Earth's axis and the zenith, the other containing Earth's axis and the given point; the angle may be expressed as negative east of the meridian plane and positive west of the meridian plane, or as positive westward from 0° to 360°. The angle may be measured in time, with 24h = 360 ° exactly. In astronomy, hour angle is defined as the angular distance on the celestial sphere measured westward along the celestial equator from the meridian to the hour circle passing through a point, it may be given in time, or rotations depending on the application. In celestial navigation, the convention is to measure in degrees westward from the prime meridian, from the local meridian or from the first point of Aries; the hour angle is paired with the declination to specify the location of a point on the celestial sphere in the equatorial coordinate system.
The local hour angle of an object in the observer's sky is LHA object = LST − α object or LHA object = GST + λ observer − α object where LHAobject is the local hour angle of the object, LST is the local sidereal time, α object is the object's right ascension, GST is Greenwich sidereal time and λ observer is the observer's longitude. These angles can be measured in degrees -- one or the other, not both. Negative hour angles indicate the time until the next transit across the meridian. Observing the sun from earth, the solar hour angle is an expression of time, expressed in angular measurement degrees, from solar noon. At solar noon the hour angle is 0.000 degree, with the time before solar noon expressed as negative degrees, the local time after solar noon expressed as positive degrees. For example, at 10:30 AM local apparent time the hour angle is -22.5°. The cosine of the hour angle is used to calculate the solar zenith angle. At solar noon, h = 0.000 so cos=1, before and after solar noon the cos term = the same value for morning or afternoon, i.e. the sun is at the same altitude in the sky at 11:00AM and 1:00PM solar time, etc.
The sidereal hour angle of a body on the celestial sphere is its angular distance west of the vernal equinox measured in degrees. An alternate definition is that SHA of a celestial body is the arc of the Equinoctial or the angle at the celestial pole contained between the celestial meridian of the First point of Aries and that through the body, measured westward from Aries; the SHA of a star changes and the SHA of a planet doesn't change quickly, so SHA is a convenient way to list their positions in an almanac. SHA is used in celestial navigation and navigational astronomy. Clock position
Earth orbits the Sun at an average distance of 149.60 million km, one complete orbit takes 365.256 days, during which time Earth has traveled 940 million km. Earth's orbit has an eccentricity of 0.0167. Since the Sun constitutes 99.76% of the mass of the Sun–Earth system, the center of the orbit is close to the center of the Sun. As seen from Earth, the planet's orbital prograde motion makes the Sun appear to move with respect to other stars at a rate of about 1° eastward per solar day. Earth's orbital speed averages about 30 km/s, fast enough to cover the planet's diameter in 7 minutes and the distance to the Moon in 4 hours. From a vantage point above the north pole of either the Sun or Earth, Earth would appear to revolve in a counterclockwise direction around the Sun. From the same vantage point, both the Earth and the Sun would appear to rotate in a counterclockwise direction about their respective axes. Heliocentrism is the scientific model that first placed the Sun at the center of the Solar System and put the planets, including Earth, in its orbit.
Heliocentrism is opposed to geocentrism, which placed the Earth at the center. Aristarchus of Samos proposed a heliocentric model in the 3rd century BC. In the 16th century, Nicolaus Copernicus' De revolutionibus presented a full discussion of a heliocentric model of the universe in much the same way as Ptolemy had presented his geocentric model in the 2nd century; this "Copernican revolution" resolved the issue of planetary retrograde motion by arguing that such motion was only perceived and apparent. "Although Copernicus's groundbreaking book...had been over a century earlier, Joan Blaeu was the first mapmaker to incorporate his revolutionary heliocentric theory into a map of the world." Because of Earth's axial tilt, the inclination of the Sun's trajectory in the sky varies over the course of the year. For an observer at a northern latitude, when the north pole is tilted toward the Sun the day lasts longer and the Sun appears higher in the sky; this results in warmer average temperatures. When the north pole is tilted away from the Sun, the reverse is true and the weather is cooler.
North of the Arctic Circle and south of the Antarctic Circle, an extreme case is reached in which there is no daylight at all for part of the year, continuous daylight during the opposite time of year. This is called midnight sun; this variation in the weather results in the seasons. By astronomical convention, the four seasons are determined by the equinoxes; the solstices and equinoxes divide the year up into four equal parts. In the northern hemisphere winter solstice occurs on or about December 21; the effect of the Earth's axial tilt in the southern hemisphere is the opposite of that in the northern hemisphere, thus the seasons of the solstices and equinoxes in the southern hemisphere are the reverse of those in the northern hemisphere. In modern times, Earth's perihelion occurs around January 3, the aphelion around July 4; the changing Earth–Sun distance results in an increase of about 6.9% in total solar energy reaching the Earth at perihelion relative to aphelion. Since the southern hemisphere is tilted toward the Sun at about the same time that the Earth reaches the closest approach to the Sun, the southern hemisphere receives more energy from the Sun than does the northern over the course of a year.
However, this effect is much less significant than the total energy change due to the axial tilt, most of the excess energy is absorbed by the higher proportion of water in the southern hemisphere. The Hill sphere of the Earth is about 1,500,000 kilometers in radius, or four times the average distance to the Moon; this is the maximal distance at which the Earth's gravitational influence is stronger than the more distant Sun and planets. Objects orbiting the Earth must be within this radius, otherwise they can become unbound by the gravitational perturbation of the Sun; the following diagram shows the relation between the line of solstice and the line of apsides of Earth's elliptical orbit. The orbital ellipse goes through each of the six Earth images, which are sequentially the perihelion on anywhere from January 2 to January 5, the point of March equinox on March 19, 20, or 21, the point of June solstice on June 20, 21, or 22, the aphelion on anywhere from July 3 to July 5, the September equinox on September 22, 23, or 24, the December solstice on December 21, 22, or 23.
The diagram shows an exaggerated shape of Earth's orbit. Because of the axial tilt of the Earth in its orbit, the maximal intensity of Sun rays hits the Earth 23.4 degrees north of equator at the June Solstice, 23.4 degrees south of equator at the December Solstice (at t
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