Ecliptic

The ecliptic is the mean plane of the apparent path in the Earth's sky that the Sun follows over the course of one year. This plane of reference is coplanar with Earth's orbit around the Sun; the ecliptic is not noticeable from Earth's surface because the planet's rotation carries the observer through the daily cycles of sunrise and sunset, which obscure the Sun's apparent motion against the background of stars during the year. The motions as described above are simplifications. Due to the movement of Earth around the Earth–Moon center of mass, the apparent path of the Sun wobbles with a period of about one month. Due to further perturbations by the other planets of the Solar System, the Earth–Moon barycenter wobbles around a mean position in a complex fashion; the ecliptic is the apparent path of the Sun throughout the course of a year. Because Earth takes one year to orbit the Sun, the apparent position of the Sun takes one year to make a complete circuit of the ecliptic. With more than 365 days in one year, the Sun moves a little less than 1° eastward every day.

This small difference in the Sun's position against the stars causes any particular spot on Earth's surface to catch up with the Sun about four minutes each day than it would if Earth would not orbit. Again, this is a simplification, based on a hypothetical Earth that orbits at uniform speed around the Sun; the actual speed with which Earth orbits the Sun varies during the year, so the speed with which the Sun seems to move along the ecliptic varies. For example, the Sun is north of the celestial equator for about 185 days of each year, south of it for about 180 days; the variation of orbital speed accounts for part of the equation of time. Because Earth's rotational axis is not perpendicular to its orbital plane, Earth's equatorial plane is not coplanar with the ecliptic plane, but is inclined to it by an angle of about 23.4°, known as the obliquity of the ecliptic. If the equator is projected outward to the celestial sphere, forming the celestial equator, it crosses the ecliptic at two points known as the equinoxes.

The Sun, in its apparent motion along the ecliptic, crosses the celestial equator at these points, one from south to north, the other from north to south. The crossing from south to north is known as the vernal equinox known as the first point of Aries and the ascending node of the ecliptic on the celestial equator; the crossing from north to south is descending node. The orientation of Earth's axis and equator are not fixed in space, but rotate about the poles of the ecliptic with a period of about 26,000 years, a process known as lunisolar precession, as it is due to the gravitational effect of the Moon and Sun on Earth's equatorial bulge; the ecliptic itself is not fixed. The gravitational perturbations of the other bodies of the Solar System cause a much smaller motion of the plane of Earth's orbit, hence of the ecliptic, known as planetary precession; the combined action of these two motions is called general precession, changes the position of the equinoxes by about 50 arc seconds per year.

Once again, this is a simplification. Periodic motions of the Moon and apparent periodic motions of the Sun cause short-term small-amplitude periodic oscillations of Earth's axis, hence the celestial equator, known as nutation; this adds a periodic component to the position of the equinoxes. Obliquity of the ecliptic is the term used by astronomers for the inclination of Earth's equator with respect to the ecliptic, or of Earth's rotation axis to a perpendicular to the ecliptic, it is about 23.4° and is decreasing 0.013 degrees per hundred years due to planetary perturbations. The angular value of the obliquity is found by observation of the motions of Earth and other planets over many years. Astronomers produce new fundamental ephemerides as the accuracy of observation improves and as the understanding of the dynamics increases, from these ephemerides various astronomical values, including the obliquity, are derived; until 1983 the obliquity for any date was calculated from work of Newcomb, who analyzed positions of the planets until about 1895: ε = 23° 27′ 08″.26 − 46″.845 T − 0″.0059 T2 + 0″.00181 T3 where ε is the obliquity and T is tropical centuries from B1900.0 to the date in question.

From 1984, the Jet Propulsion Laboratory's DE series of computer-generated ephemerides took over as the fundamental ephemeris of the Astronomical Almanac. Obliquity based on DE200, which analyzed observations from 1911 to 1979, was calculated: ε = 23° 26′ 21″.45 − 46″.815 T − 0″.0006 T2 + 0″.00181 T3 where hereafter T is Julian centuries from J2000.0. JPL's fundamental ephemerides have been continually updated; the Astronomical Almanac for 2010 specifies:ε = 23° 26′ 21″.406 − 46″.836769 T − 0″.0001831 T2 + 0″.00200340 T3 − 0″.576×10−6 T4 − 4″.34×10−8 T5 These expressions for the obliquity are intended for high precision over a short time span ± several centuries. J. Laskar computed an expression to order T10 good to 0″.04/1000 years over 10,000 years. All of these expressions are for the mean obliquity, that is, without the nutation of the equator included; the true or instantaneous obliquity includes the nutation. Most of the major bodies of the Solar System o

Orbit

In physics, an orbit is the gravitationally curved trajectory of an object, such as the trajectory of a planet around a star or a natural satellite around a planet. Orbit refers to a repeating trajectory, although it may refer to a non-repeating trajectory. To a close approximation and satellites follow elliptic orbits, with the central mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion. For most situations, orbital motion is adequately approximated by Newtonian mechanics, which explains gravity as a force obeying an inverse-square law. However, Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of spacetime, with orbits following geodesics, provides a more accurate calculation and understanding of the exact mechanics of orbital motion; the apparent motions of the planets were described by European and Arabic philosophers using the idea of celestial spheres. This model posited the existence of perfect moving spheres or rings to which the stars and planets were attached.

It assumed the heavens were fixed apart from the motion of the spheres, was developed without any understanding of gravity. After the planets' motions were more measured, theoretical mechanisms such as deferent and epicycles were added. Although the model was capable of reasonably predicting the planets' positions in the sky and more epicycles were required as the measurements became more accurate, hence the model became unwieldy. Geocentric it was modified by Copernicus to place the Sun at the centre to help simplify the model; the model was further challenged during the 16th century, as comets were observed traversing the spheres. The basis for the modern understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion. First, he found that the orbits of the planets in our Solar System are elliptical, not circular, as had been believed, that the Sun is not located at the center of the orbits, but rather at one focus. Second, he found that the orbital speed of each planet is not constant, as had been thought, but rather that the speed depends on the planet's distance from the Sun.

Third, Kepler found a universal relationship between the orbital properties of all the planets orbiting the Sun. For the planets, the cubes of their distances from the Sun are proportional to the squares of their orbital periods. Jupiter and Venus, for example, are about 5.2 and 0.723 AU distant from the Sun, their orbital periods about 11.86 and 0.615 years. The proportionality is seen by the fact that the ratio for Jupiter, 5.23/11.862, is equal to that for Venus, 0.7233/0.6152, in accord with the relationship. Idealised orbits meeting these rules are known as Kepler orbits. Isaac Newton demonstrated that Kepler's laws were derivable from his theory of gravitation and that, in general, the orbits of bodies subject to gravity were conic sections. Newton showed that, for a pair of bodies, the orbits' sizes are in inverse proportion to their masses, that those bodies orbit their common center of mass. Where one body is much more massive than the other, it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body.

Advances in Newtonian mechanics were used to explore variations from the simple assumptions behind Kepler orbits, such as the perturbations due to other bodies, or the impact of spheroidal rather than spherical bodies. Lagrange developed a new approach to Newtonian mechanics emphasizing energy more than force, made progress on the three body problem, discovering the Lagrangian points. In a dramatic vindication of classical mechanics, in 1846 Urbain Le Verrier was able to predict the position of Neptune based on unexplained perturbations in the orbit of Uranus. Albert Einstein in his 1916 paper The Foundation of the General Theory of Relativity explained that gravity was due to curvature of space-time and removed Newton's assumption that changes propagate instantaneously; this led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy in understanding orbits. In relativity theory, orbits follow geodesic trajectories which are approximated well by the Newtonian predictions but the differences are measurable.

All the experimental evidence that can distinguish between the theories agrees with relativity theory to within experimental measurement accuracy. The original vindication of general relativity is that it was able to account for the remaining unexplained amount in precession of Mercury's perihelion first noted by Le Verrier. However, Newton's solution is still used for most short term purposes since it is easier to use and sufficiently accurate. Within a planetary system, dwarf planets and other minor planets and space debris orbit the system's barycenter in elliptical orbits. A comet in a parabolic or hyperbolic orbit about a barycenter is not gravitationally bound to the star and therefore is not considered part of the star's planetary system. Bodies which are gravitationally bound to one of the planets in a planetary system, either natural or artificial satellites, follow orbits about a barycenter near or within that planet. Owing to mutual gravitational perturbations, the eccentricities of the planetary orbits vary over time.

Mercury, the smallest planet in the Solar System, has the most eccentric orbit

ECEF

ECEF known as ECR, is a geographic and Cartesian coordinate system and is sometimes known as a "conventional terrestrial" system. It represents positions as X, Y, Z coordinates; the point is defined as the center of mass of Earth, hence the term geocentric coordinates. The distance from a given point of interest to the center of Earth is called the geocentric radius or geocentric distance, its axes are aligned with the international reference pole and international reference meridian that are fixed with respect to Earth's surface, hence the descriptor earth-fixed. This term can cause confusion, since Earth does not rotate about the z-axis, is therefore alternatively called ECR; the z-axis extends through true north, which does not coincide with the instantaneous Earth rotational axis. The slight "wobbling" of the rotational axis is known as polar motion; the x-axis intersects the sphere of the earth at 0 ° longitude. This means that ECEF rotates with the earth, therefore coordinates of a point fixed on the surface of the earth do not change.

Conversion from a WGS84 datum to ECEF can be used as an intermediate step in converting velocities to the north east down coordinate system. Conversions between ECEF and geodetic coordinates are discussed at geographic coordinate conversion. Geocentric coordinates can be used for locating astronomical objects in the Solar System in three dimensions along the Cartesian X, Y, Z axes, they are differentiated from topocentric coordinates, which use the observer's location as the reference point for bearings in altitude and azimuth. However, neither system takes Earth's constant motion into account, which requires the addition of a time component to fix objects. For nearby stars, astronomers use heliocentric coordinates, with the center of the Sun as the origin; the plane of reference can be aligned with the Earth's celestial equator, the ecliptic, or the Milky Way's galactic equator. These 3D celestial coordinate systems add actual distance as the Z axis to the equatorial and galactic coordinate systems used in spherical astronomy.

The distances involved are so great compared to the relative velocities of the stars, that for most purposes, the time component can be neglected. Geodetic system Earth-centered inertial coordinate system International Terrestrial Reference System Orbital state vectors ECEF datum transformation Notes on converting ECEF coordinates to WGS-84 datum Datum Transformations of GPS Positions Application Note Clearer notes on converting ECEF coordinates to WGS-84 datum geodetic datum overview orientation of the coordinate system and additional information GeographicLib includes a utility CartConvert which converts between geodetic and geocentric or local Cartesian coordinates; this provides accurate results for all inputs including points close to the center of the earth. EPSG:4978

Anatomy

Anatomy is the branch of biology concerned with the study of the structure of organisms and their parts. Anatomy is a branch of natural science which deals with the structural organization of living things, it is an old science. Anatomy is inherently tied to developmental biology, comparative anatomy, evolutionary biology, phylogeny, as these are the processes by which anatomy is generated over immediate and long timescales. Anatomy and physiology, which study the structure and function of organisms and their parts, make a natural pair of related disciplines, they are studied together. Human anatomy is one of the essential basic sciences; the discipline of anatomy is divided into microscopic anatomy. Macroscopic anatomy, or gross anatomy, is the examination of an animal's body parts using unaided eyesight. Gross anatomy includes the branch of superficial anatomy. Microscopic anatomy involves the use of optical instruments in the study of the tissues of various structures, known as histology, in the study of cells.

The history of anatomy is characterized by a progressive understanding of the functions of the organs and structures of the human body. Methods have improved advancing from the examination of animals by dissection of carcasses and cadavers to 20th century medical imaging techniques including X-ray and magnetic resonance imaging. Derived from the Greek ἀνατομή anatomē "dissection", anatomy is the scientific study of the structure of organisms including their systems and tissues, it includes the appearance and position of the various parts, the materials from which they are composed, their locations and their relationships with other parts. Anatomy is quite distinct from physiology and biochemistry, which deal with the functions of those parts and the chemical processes involved. For example, an anatomist is concerned with the shape, position, blood supply and innervation of an organ such as the liver; the discipline of anatomy can be subdivided into a number of branches including gross or macroscopic anatomy and microscopic anatomy.

Gross anatomy is the study of structures large enough to be seen with the naked eye, includes superficial anatomy or surface anatomy, the study by sight of the external body features. Microscopic anatomy is the study of structures on a microscopic scale, along with histology, embryology. Anatomy can be studied using both invasive and non-invasive methods with the goal of obtaining information about the structure and organization of organs and systems. Methods used include dissection, in which a body is opened and its organs studied, endoscopy, in which a video camera-equipped instrument is inserted through a small incision in the body wall and used to explore the internal organs and other structures. Angiography using X-rays or magnetic resonance angiography are methods to visualize blood vessels; the term "anatomy" is taken to refer to human anatomy. However the same structures and tissues are found throughout the rest of the animal kingdom and the term includes the anatomy of other animals.

The term zootomy is sometimes used to refer to non-human animals. The structure and tissues of plants are of a dissimilar nature and they are studied in plant anatomy; the kingdom Animalia contains multicellular organisms that are motile. Most animals have bodies differentiated into separate tissues and these animals are known as eumetazoans, they have an internal digestive chamber, with two openings. Metazoans do not include the sponges. Unlike plant cells, animal cells have neither chloroplasts. Vacuoles, when present, are much smaller than those in the plant cell; the body tissues are composed of numerous types of cell, including those found in muscles and skin. Each has a cell membrane formed of phospholipids, cytoplasm and a nucleus. All of the different cells of an animal are derived from the embryonic germ layers; those simpler invertebrates which are formed from two germ layers of ectoderm and endoderm are called diploblastic and the more developed animals whose structures and organs are formed from three germ layers are called triploblastic.

All of a triploblastic animal's tissues and organs are derived from the three germ layers of the embryo, the ectoderm and endoderm. Animal tissues can be grouped into four basic types: connective, epithelial and nervous tissue. Connective tissues are fibrous and made up of cells scattered among inorganic material called the extracellular matrix. Connective tissue holds them in place; the main types are loose connective tissue, adipose tissue, fibrous connective tissue and bone. The extracellular matrix contains proteins, the chief and most abundant of, collagen. Collagen plays a major part in maintaining tissues; the matrix can be modified to form a skeleton to protect the body. An exoskeleton is a thickened, rigid cuticle, stiffened by mineralization, as in crustaceans or by the cross-linkin

Earth's orbit

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

Orbital eccentricity

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.

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Plane (geometry)

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

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

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

Alternatively, a plane may be described parametrically as the set of all points of the form r = r 0 + s v + t w, where s and t range over all real numbers, v and w are given linearly independent vectors defining the plane, r0 is the vector representing the position of an arbitrary point on the plane. The vectors v and w can be visualized as vectors starting at r0 and pointing in different directions along the plane. 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