1.
Centre (geometry)
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In geometry, a centre of an object is a point in some sense in the middle of the object. According to the definition of centre taken into consideration, an object might have no centre. If geometry is regarded as the study of groups then a centre is a fixed point of all the isometries which move the object onto itself. The centre of a circle is the point equidistant from the points on the edge, similarly the centre of a sphere is the point equidistant from the points on the surface, and the centre of a line segment is the midpoint of the two ends. For objects with several symmetries, the centre of symmetry is the point left unchanged by the symmetric actions, so the centre of a square, rectangle, rhombus or parallelogram is where the diagonals intersect, this being the fixed point of rotational symmetries. Similarly the centre of an ellipse or a hyperbola is where the axes intersect. For an equilateral triangle, these are the point, which lies at the intersection of the three axes of symmetry of the triangle, one third of the distance from its base to its apex. e. F=thf for some real power h, thus the position of a centre is independent of scale. F is symmetric in its last two arguments, i. e. f= f, thus position of a centre in a triangle is the mirror-image of its position in the original triangle. This strict definition excludes pairs of points such as the Brocard points. The Encyclopedia of Triangle Centers lists over 9,000 different triangle centres, a tangential polygon has each of its sides tangent to a particular circle, called the incircle or inscribed circle. The centre of the incircle, called the incentre, can be considered a centre of the polygon, a cyclic polygon has each of its vertices on a particular circle, called the circumcircle or circumscribed circle. The centre of the circumcircle, called the circumcentre, can be considered a centre of the polygon, if a polygon is both tangential and cyclic, it is called bicentric. The incentre and circumcentre of a polygon are not in general the same point. The centre of a general polygon can be defined in different ways. The vertex centroid comes from considering the polygon as being empty, the side centroid comes from considering the sides to have constant mass per unit length. The usual centre, called just the centroid comes from considering the surface of the polygon as having constant density and these three points are in general not all the same point. Centrepoint Centre of mass Chebyshev centre Fixed points of isometry groups in Euclidean space

2.
Aerodynamic center
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The aerodynamic center is the point at which the pitching moment coefficient for the airfoil does not vary with lift coefficient, so this choice makes analysis simpler. D C m d C L =0 where C L is the lift coefficient. Forces can be summed up to act through a single point, the center of pressure location however, changes significantly with a change in angle of attack and is thus impractical for analysis. Thus the 25% chord position, or assumed aerodynamic center, is taken about which the forces, at the 25% chord position the moment generated was found and proven to be nearly constant with varying angle of attack. The concept of the center is important in aerodynamics. It is fundamental in the science of stability of aircraft in flight, please note that in highly theoretical/analytical analysis the aerodynamic center does vary slightly and changes location. In most literature however the center is taken at the 25% chord position. This conversely means that if one keeps the assumed AC fixed at 25% chord, a large portion of cambered airfoils have non constant moments about the 25% chord position because the AC does vary slightly. For most analysis the non constant moment about the 25% chord position is not significant enough to warrant consideration, for symmetric airfoils in subsonic flight the aerodynamic center is located approximately 25% of the chord from the leading edge of the airfoil. This point is described as the quarter-chord point and this result also holds true for thin-airfoils. For non-symmetric airfoils the quarter-chord is only an approximation for the aerodynamic center, a similar concept is that of center of pressure. The location of the center of pressure varies with changes of lift coefficient and this makes the center of pressure unsuitable for use in analysis of longitudinal static stability. Read about movement of centre of pressure, Y A C = Y r e f + c d C l d C z + c d C n d C x

3.
Central angle
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Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one. The central angle is known as the arcs angular distance. The size of a central angle Θ is 0° < Θ < 360° оr 0 < Θ < 2π. When defining or drawing a central angle, in addition to specifying the points A and B, equivalently, one must specify whether the movement from point A to point B is clockwise or counterclockwise. If the intersection points A and B of the legs of the angle with the form a diameter. Let L be the arc of the circle between points A and B, and let R be the radius of the circle. If the central angle Θ is subtended by L, then 0 ∘ < Θ <180 ∘, Θ = ∘ = L R. If the central angle Θ is not subtended by the minor arc L, if a tangent at A and a tangent at B intersect at the exterior point P, then denoting the center as O, the angles ∠BOA and ∠BPA are supplementary. A regular polygon with n sides has a circle upon which all its vertices lie. The central angle of the polygon is formed at the center by the radii to two adjacent vertices. The measure of this angle is 2 π / n, Inscribed angle Great-circle navigation Central angle. Interactive Inscribed and Central Angles in a Circle

4.
Center of balance (horse)
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In horsemanship, the center of balance of a horse is a position on the horses back which correlates closely to the center of gravity of the horse itself. The term may refer to the horses center of gravity. For the best performance by the horse, as well as for better balance of the rider, the location of the horses center of balance depends on a combination of speed and degree of collection. For a standing or quietly walking horse, it is slightly behind the heart girth, if a horse is moving at a trot or canter, the center of balance shifts slightly forward, and it moves even more forward when the horse is galloping or jumping. If a horse is highly collected, the center of balance will be back, regardless of gait. For example, a close contact style of English saddle, designed for show jumping, places the riders seat farther forward than does a dressage style English saddle

5.
Cardinal point (optics)
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In Gaussian optics, the cardinal points consist of three pairs of points located on the optical axis of a rotationally symmetric, focal, optical system. These are the points, the principal points, and the nodal points. The only ideal system that has achieved in practice is the plane mirror. Cardinal points provide a way to simplify a system with many components. The cardinal points lie on the axis of the optical system. Each point is defined by the effect the optical system has on rays that pass through that point, the paraxial approximation assumes that rays travel at shallow angles with respect to the optical axis, so that sin θ ≈ θ and cos θ ≈1. Aperture effects are ignored, rays that do not pass through the stop of the system are not considered in the discussion below. The front focal point of a system, by definition, has the property that any ray that passes through it will emerge from the system parallel to the optical axis. The rear focal point of the system has the reverse property, the front and rear focal planes are defined as the planes, perpendicular to the optic axis, which pass through the front and rear focal points. An object infinitely far from the optical system forms an image at the focal plane. For objects a finite distance away, the image is formed at a different location, but rays that leave the object parallel to one another cross at the rear focal plane. A diaphragm or stop at the focal plane can be used to filter rays by angle, since. No matter where on the object the ray comes from, the ray will pass through the aperture as long as the angle at which it is emitted from the object is small enough. Note that the aperture must be centered on the axis for this to work as indicated. Using a sufficiently small aperture in the plane will make the lens telecentric. Similarly, the range of angles on the output side of the lens can be filtered by putting an aperture at the front focal plane of the lens. This is important for DSLR cameras having CCD sensors, the pixels in these sensors are more sensitive to rays that hit them straight on than to those that strike at an angle. A lens that does not control the angle of incidence at the detector will produce pixel vignetting in the images and this means that the lens can be treated as if all of the refraction happened at the principal planes

6.
Centroid
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In mathematics and physics, the centroid or geometric center of a plane figure is the arithmetic mean position of all the points in the shape. The definition extends to any object in space, its centroid is the mean position of all the points in all of the coordinate directions. Informally, it is the point at which a cutout of the shape could be balanced on the tip of a pin. While in geometry the term barycenter is a synonym for centroid, in astrophysics and astronomy, in physics, the center of mass is the arithmetic mean of all points weighted by the local density or specific weight. If a physical object has uniform density, then its center of mass is the same as the centroid of its shape, in geography, the centroid of a radial projection of a region of the Earths surface to sea level is known as the regions geographical center. The geometric centroid of an object always lies in the object. A non-convex object might have a centroid that is outside the figure itself, the centroid of a ring or a bowl, for example, lies in the objects central void. If the centroid is defined, it is a point of all isometries in its symmetry group. In particular, the centroid of an object lies in the intersection of all its hyperplanes of symmetry. The centroid of many figures can be determined by this principle alone, in particular, the centroid of a parallelogram is the meeting point of its two diagonals. This is not true for other quadrilaterals, for the same reason, the centroid of an object with translational symmetry is undefined, because a translation has no fixed point. The centroid of a triangle is the intersection of the three medians of the triangle and it lies on the triangles Euler line, which also goes through various other key points including the orthocenter and the circumcenter. Any of the three medians through the centroid divides the area in half. Let P be any point in the plane of a triangle with vertices A, B, and C and centroid G. The sum of the squares of the sides equals three times the sum of the squared distances of the centroid from the vertices, A B2 + B C2 + C A2 =3. A triangles centroid is the point that maximizes the product of the distances of a point from the triangles sidelines. For other properties of a centroid, see below. The body is held by the pin inserted at a point near the perimeter, in such a way that it can freely rotate around the pin

7.
Concentric objects
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In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another, in the Euclidean plane, two circles that are concentric necessarily have different radii from each other. However, circles in three-dimensional space may be concentric, and have the radius as each other. For example, two different meridians of a terrestrial globe are concentric with each other and with the globe of the earth, more generally, every two great circles on a sphere are concentric with each other and with the sphere. The circumcircle and the incircle of a regular n-gon, and the regular n-gon itself, are concentric, for the circumradius-to-inradius ratio for various n, see Bicentric polygon#Regular polygons. The region of the plane between two circles is an annulus, and analogously the region of space between two concentric spheres is a spherical shell. For a given point c in the plane, the set of all circles having c as their forms a pencil of circles. Each two circles in the pencil are concentric, and have different radii, every point in the plane, except for the shared center, belongs to exactly one of the circles in the pencil. Every two disjoint circles, and every hyperbolic pencil of circles, may be transformed into a set of circles by a Möbius transformation. The ripples formed by dropping an object into still water naturally form an expanding system of concentric circles. Evenly spaced circles on the used in target archery or similar sports provide another familiar example of concentric circles. Coaxial cable is a type of cable in which the combined neutral. Johannes Keplers Mysterium Cosmographicum envisioned a system formed by concentric regular polyhedra. Concentric circles are found in diopter sights, a type of mechanic sights commonly found on target rifles. They usually feature a disk with a small-diametre hole near the shooters eye. When these sights are aligned, the point of impact will be in the middle of the front sight circle. Centered cube number Homoeoid Focaloid Circular symmetry Magic circle Geometry, Concentric circles demonstration With interactive animation

8.
Center of curvature
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In geometry, the center of curvature of a curve is found at a point that is at a distance from the curve equal to the radius of curvature lying on the normal vector. It is the point at infinity if the curvature is zero, the osculating circle to the curve is centered at the centre of curvature. Cauchy defined the center of curvature C as the point of two infinitely close normal lines to the curve. The locus of centers of curvature for each point on the curve comprise the evolute of the curve, curvature Differential geometry of curves Hilbert, David, Cohn-Vossen, Stephan, Geometry and the Imagination, New York, Chelsea, ISBN 978-0-8284-0087-9

9.
Epicenter
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The term was coined by the Irish seismologist Robert Mallet. The word, however, is misused to mean center. In seismology, the epicenter is the point on the Earths surface directly above the point where the fault begins to rupture, however, in larger events, the length of the fault rupture is much longer, and damage can be spread across the rupture zone. During an earthquake seismic waves propagate spherically out from the hypocenter, outside of the seismic shadow zone both types of wave can be detected but, due to their different velocities and paths through the Earth, they arrive at different times. This distance is called the distance, commonly measured in °. Once epicentral distances have been calculated from at least three seismographic measuring stations, it is a matter to find out where the epicenter was located using trilateration. Epicentral distance is used in calculating seismic magnitudes developed by Richter. Epicenter is frequently misused when not employed in the context of seismology and it is often utilized as an alternative to centre. For example, Travel is restricted in the Chinese province thought to be the epicentre of the SARS outbreak, garners Modern American Usage gives several examples of such misuse. However, Garner notes that these misusages may be metaphorical uses of the term to describe focal points of unstable and potentially destructive environments

10.
Focus (geometry)
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In geometry, focuses or foci, singular focus, are special points with reference to which any of a variety of curves is constructed. For example, one or two foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola. In addition, two foci are used to define the Cassini oval and the Cartesian oval, and more than two foci are used in defining an n-ellipse. An ellipse can be defined as the locus of points for each of which the sum of the distances to two given foci is a constant, a circle is the special case of an ellipse in which the two foci coincide with each other. Thus, a circle can be simply defined as the locus of points each of which is a fixed distance from a single given focus. A circle can also be defined as the circle of Apollonius, a parabola is a limiting case of an ellipse in which one of the foci is a point at infinity. A hyperbola can be defined as the locus of points for each of which the value of the difference between the distances to two given foci is a constant. It is also possible to describe all conic sections in terms of a focus and a single directrix. A conic is defined as the locus of points for each of which the distance to the focus divided by the distance to the directrix is a positive constant. If e is zero and one the conic is an ellipse, if e=1 the conic is a parabola. If the distance to the focus is fixed and the directrix is a line at infinity, so the eccentricity is zero and it is also possible to describe all the conic sections as loci of points that are equidistant from a single focus and a single, circular directrix. The ellipse thus generated has its focus at the center of the directrix circle. For the parabola, the center of the moves to the point at infinity. The directrix circle becomes a curve with zero curvature, indistinguishable from a straight line. To generate a hyperbola, the radius of the circle is chosen to be less than the distance between the center of this circle and the focus, thus, the focus is outside the directrix circle. The two branches of a hyperbola are thus the two halves of a curve closed over infinity, in projective geometry, all conics are equivalent in the sense that every theorem that can be stated for one can be stated for the others. Plutos ellipse is entirely inside Charons ellipse, as shown in animation of the system. The barycenter is about three-quarters of the distance from Earths center to its surface, moreover, the Pluto-Charon system moves in an ellipse around its barycenter with the Sun, as does the Earth-Moon system

11.
Galactic Center
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The Galactic Center is the rotational center of the Milky Way. The estimates for its range from 7.6 to 8.7 kiloparsecs from Earth in the direction of the constellations Sagittarius, Ophiuchus. There is strong evidence consistent with the existence of a black hole at the Galactic Center of the Milky Way. Because of interstellar dust along the line of sight, the Galactic Center cannot be studied at visible, the available information about the Galactic Center comes from observations at gamma ray, hard X-ray, infrared, sub-millimetre and radio wavelengths. In the early 1940s Walter Baade at Mount Wilson Observatory took advantage of wartime conditions in nearby Los Angeles to conduct a search for the center with the 100 inch Hooker Telescope. This gap has been known as Baades Window ever since, by 1954 they had built an 80 feet fixed dish antenna and used it to make a detailed study of an extended, extremely powerful belt of radio emission that was detected in Sagittarius. In 1958 the International Astronomical Union decided to adopt the position of Sagittarius A as the true zero co-ordinate point for the system of latitude and longitude. In the equatorial system the location is, RA 17h 45m 40. 04s. The exact distance between the Solar System and the Galactic Center is not certain, although estimates since 2000 have remained within the range 7. 2–8.8 kpc. The nature of the Milky Ways bar, which extends across the Galactic Center, is also debated, with estimates for its half-length. Certain authors advocate that the Milky Way features two bars, one nestled within the other. The bar is delineated by red-clump stars, however, RR Lyr variables do not trace a prominent Galactic bar. The bar may be surrounded by a called the 5-kpc ring that contains a large fraction of the molecular hydrogen present in the Milky Way. Viewed from the Andromeda Galaxy, it would be the brightest feature of the Milky Way, accretion of gas onto the black hole, probably involving a disk around it, would release energy to power the radio source, itself much larger than the black hole. The latter is too small to see with present instruments, a study in 2008 which linked radio telescopes in Hawaii, Arizona and California measured the diameter of Sagittarius A* to be 44 million kilometers. For comparison, the radius of Earths orbit around the Sun is about 150 million kilometers, thus the diameter of the radio source is slightly less than the distance from Mercury to the Sun.3 million solar masses. On 5 January 2015, NASA reported observing an X-ray flare 400 times brighter than usual, the central cubic parsec around Sagittarius A* contains around 10 million stars. Although most of them are old red giant stars, the Galactic Center is also rich in massive stars, more than 100 OB and Wolf–Rayet stars have been identified there so far

12.
Center of gravity of an aircraft
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The center of gravity of an aircraft is the point over which the aircraft would balance. Its position is calculated after supporting the aircraft on at least two sets of weighing scales or load cells and noting the weight shown on set of scales or load cells. The center of gravity affects the stability of the aircraft, to ensure the aircraft is safe to fly, the center of gravity must fall within specified limits established by the aircraft manufacturer. Ballast Ballast is removable or permanently installed weight in a used to bring the center of gravity into the allowable range. Center-of-Gravity Limits Center of gravity limits are specified longitudinal and/or lateral limits within which the center of gravity must be located during flight. The CG limits are indicated in the flight manual. The area between the limits is called the CG range of the aircraft, different maximum weights may be defined for different situations, for example, large aircraft may have maximum landing weights that are lower than maximum take-off weights. The center of gravity may change over the duration of the flight as the weight changes due to fuel burn or by passengers moving forward or aft in the cabin. Reference Datum The reference datum is a plane that allows accurate. The location of the datum is established by the manufacturer and is defined in the aircraft flight manual. There is no fixed rule for its location, and it may be located forward of the nose of the aircraft, for helicopters, it may be located at the rotor mast, the nose of the helicopter, or even at a point in space ahead of the helicopter. This is to all the computed values positive. The lateral reference datum is usually located at the center of the helicopter, arm The arm is the horizontal distance from the reference datum to the center of gravity of an item. The algebraic sign is plus if measured aft of the datum or to the side of the center line when considering a lateral calculation. The algebraic sign is minus if measured forward of the datum or the side of the center line when considering a lateral calculation. Moment The moment is the moment of force, or torque, moment is also referred to as the tendency of an object to rotate or pivot about a point. The further an object is from this point, the greater the force it exerts, moment is calculated by multiplying the weight of an object by its arm. Mean Aerodynamic Chord A specific chord line of a tapered wing, at the mean aerodynamic chord, the center of pressure has the same aerodynamic force, position, and area as it does on the rest of the wing

13.
Homothetic center
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In geometry, a homothetic center is a point from which at least two geometrically similar figures can be seen as a dilation/contraction of one another. If the center is external, the two figures are similar to one another, their angles have the same rotational sense. If the center is internal, the two figures are scaled mirror images of one another, their angles have the opposite sense, the homothetic center and the two figures need not lie in the same plane, they can be related by a projection from the homothetic center. Homothetic centers may be external or internal, if the center is internal, the two geometric figures are scaled mirror images of one another, in technical language, they have opposite chirality. A clockwise angle in one figure would correspond to an angle in the other. Conversely, if the center is external, the two figures are similar to one another, their angles have the same sense. Circles are geometrically similar to one another and mirror symmetric, hence, a pair of circles has both types of homothetic centers, internal and external, unless the centers are equal or the radii are equal, these exceptional cases are treated after general position. These two homothetic centers lie on the joining the centers of the two given circles, which is called the line of centers. Circles with radius zero can also be included, and negative radius can also be used, for a given pair of circles, the internal and external homothetic centers may be found in various ways. More generally, taking both radii with the sign yields the inner center, while taking the radii with opposite signs yields the outer center. Note that the equation for the center is valid for any values. In synthetic geometry, two diameters are drawn, one for each circle, these make the same angle α with the line of centers. The lines A1A2 and B1B2 drawn through corresponding endpoints of those radii, which are points, intersect each other. Conversely, the lines A1B2 and B1A2 drawn through one endpoint and the endpoint of its counterpart intersects each other. If the circles fall on opposite sides of the line, it passes through the homothetic center. Conversely, if the fall on the same side of the line. An external center can be defined in the plane to be the point at infinity corresponding to the slope of this line. This is also the limit of the center if the centers of the circles are fixed

14.
Hypocenter
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A hypocenter is the point of origin of an earthquake or a subsurface nuclear explosion. It is a synonym of the focus, confusingly, the term hypocenter is also used as a synonym for ground zero, the surface point directly beneath a nuclear airburst. An earthquakes hypocenter is the position where the energy stored in the rock is first released. This occurs directly beneath the epicenter, at a known as the focal or hypocentral depth. The focal depth can be calculated from measurements based on wave phenomena. Computing the hypocenters of foreshocks, main shock, and aftershocks of earthquakes allows the three-dimensional plotting of the fault along which movement is occurring, the wave reaches each station based upon how far away it was from the hypocenter. A number of things need to be taken into account, most importantly variations in the speed based upon the materials that it is passing through. With adjustments for velocity changes, the estimate of the hypocenter is made, then a series of linear equations is set up. The equations express the difference between the arrival times and those calculated from the initial estimated hypocenter. The system iterates until the location is pinpointed within the margin of error for the velocity computations, the dictionary definition of hypocenter at Wiktionary Media related to Hypocenters at Wikimedia Commons

15.
Locating the center of mass
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There are several methods for locating the center of mass of a body. This method is useful when one wishes to find the centroid of a complex shape with unknown dimensions. It relies on finding the center of mass of a body of homogeneous density having the same shape as the complex planar shape. This is a method of determining the center of mass of an L-shaped object, divide the shape into two rectangles, as shown in fig 2. Find the center of masses of two rectangles by drawing the diagonals. Draw a line joining the centers of mass, the center of mass of the shape must lie on this line AB. Divide the shape into two rectangles, as shown in fig 3. Find the centers of mass of two rectangles by drawing the diagonals. Draw a line joining the centers of mass, the center of mass of the L-shape must lie on this line CD. As the center of mass of the shape must lie along AB and also along CD, it is obvious that it is at the intersection of these two lines, at O. This method is useful when one wishes to find the location of the centroid or center of mass of an object that is divided into elementary shapes. Here the center of mass will only be found in the x direction, the same procedure may be followed to locate the center of mass in the y direction. It is easily divided into a square, triangle, and circle, note that the circle will have negative area. From the List of centroids, we note the coordinates of the individual centroid. From equation 1 above,3 × +5 ×102 +13.33 ×1022 −2.52 π +102 +1022 ≈8.5 units. The center of mass of this figure is at a distance of 8.5 units from the corner of the figure. A direct development of the known as an integraph, or integerometer. This method can be applied to a shape with an irregular and it was regularly used by ship builders to ensure the ship would not capsize

16.
Center of mass
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The distribution of mass is balanced around the center of mass and the average of the weighted position coordinates of the distributed mass defines its coordinates. Calculations in mechanics are simplified when formulated with respect to the center of mass. It is a point where entire mass of an object may be assumed to be concentrated to visualise its motion. In other words, the center of mass is the equivalent of a given object for application of Newtons laws of motion. In the case of a rigid body, the center of mass is fixed in relation to the body. The center of mass may be located outside the body, as is sometimes the case for hollow or open-shaped objects. In the case of a distribution of separate bodies, such as the planets of the Solar System, in orbital mechanics, the equations of motion of planets are formulated as point masses located at the centers of mass. The center of mass frame is a frame in which the center of mass of a system is at rest with respect to the origin of the coordinate system. The concept of center of mass in the form of the center of gravity was first introduced by the ancient Greek physicist, mathematician, and engineer Archimedes of Syracuse. He worked with simplified assumptions about gravity that amount to a uniform field, in work on floating bodies he demonstrated that the orientation of a floating object is the one that makes its center of mass as low as possible. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes, Newtons second law is reformulated with respect to the center of mass in Eulers first law. The center of mass is the point at the center of a distribution of mass in space that has the property that the weighted position vectors relative to this point sum to zero. In analogy to statistics, the center of mass is the location of a distribution of mass in space. Solving this equation for R yields the formula R =1 M ∑ i =1 n m i r i, solve this equation for the coordinates R to obtain R =1 M ∭ Q ρ r d V, where M is the total mass in the volume. If a continuous mass distribution has density, which means ρ is constant. The center of mass is not generally the point at which a plane separates the distribution of mass into two equal halves, in analogy with statistics, the median is not the same as the mean. The coordinates R of the center of mass of a system, P1 and P2, with masses m1. The percentages of mass at each point can be viewed as projective coordinates of the point R on this line, another way of interpreting the process here is the mechanical balancing of moments about an arbitrary point

17.
Mass point geometry
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Coincidence - We say that two points m P and n Q coincide if and only if m = n and P = Q. Addition - The sum of two points m P and n Q has mass m + n and point R where R is the point on P Q such that P R, R Q = n, m. In other words, R is the point that perfectly balances the points P and Q. An example of mass point addition is shown at right, mass point addition is closed, commutative, and associative. Scalar Multiplication - Given a mass point m P and a real scalar k. Mass point scalar multiplication is distributive over mass point addition, first, a point is assigned with a mass in the way that other masses are also whole numbers. The principle of calculation is that the foot of a cevian is the addition of the two vertices, for each cevian, the point of concurrency is the sum of the vertex and the foot. Each length ratio may then be calculated from the masses at the points, see Problem One for an example. Splitting masses is the more complicated method necessary when a problem contains transversals in addition to cevians. Any vertex that is on both sides the transversal crosses will have a split mass, see Problem Two for an example. Rouths theorem - Many problems involving triangles with cevians will ask for areas, however, Rouths theorem, which goes hand in hand with mass points, uses ratios of lengths to calculate the ratio of areas between a triangle and a triangle formed by three cevians. Special cevians - When given cevians with special properties, like an angle bisector or an altitude, One very common theorem used likewise is the angle bisector theorem. In triangle A B C, E is on A C so that C E =3 A E and F is on A B so that B F =3 A F. If B E and C F intersect at O and line A O intersects B C at D and we may arbitrarily assign the mass of point A to be 3. By ratios of lengths, the masses at B and C must both be 1, by summing masses, the masses at E and F are both 4. Furthermore, the mass at O is 4 +1 =5, if D E and C F intersect at O, compute O D O E and O C O F. Solution. As this problem involves a transversal, we must use split masses on point C and we may arbitrarily assign the mass of point A to be 15. By ratios of lengths, the mass at B must be 6, by summing masses, we get the masses at D, E, and F to be 15,25, and 21, respectively

18.
Metacentric height
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The metacentric height is a measurement of the initial static stability of a floating body. It is calculated as the distance between the centre of gravity of a ship and its metacentre, a larger metacentric height implies greater initial stability against overturning. Hence, a sufficiently, but not excessively, high metacentric height is considered ideal for passenger ships, when a ship heels, the centre of buoyancy of the ship moves laterally. It might also move up or down with respect to the water line, the point at which a vertical line through the heeled centre of buoyancy crosses the line through the original, vertical centre of buoyancy is the metacentre. The metacentre remains directly above the centre of buoyancy by definition, in the diagram, the two Bs show the centres of buoyancy of a ship in the upright and heeled conditions, and M is the metacentre. KM is the distance from the keel to the metacentre, stable floating objects have a natural rolling frequency, just like a weight on a spring, where the frequency is increased as the spring gets stiffer. Metacentre is determined by the ratio between the resistance of the boat and the volume of the boat. Wide and shallow or narrow and deep hulls have high transverse metacenters, and the opposite have low metacenters, ignoring the ballast, wide and shallow or narrow and deep means that the ship is very quick to roll and very hard to overturn and is stiff. A log shaped round bottomed means that it is slow to roll and easy to overturn, G, is the center of gravity. GM, the parameter of a boat, can be lengthened by lowering the center of gravity or changing the hull form or both. An ideal boat strikes a balance, very tender boats with very slow roll periods are at risk of overturning, but are comfortable for passengers. However, vessels with a metacentric height are excessively stable with a short roll period resulting in high accelerations at the deck level. In such vessels, the motion is not uncomfortable because of the moment of inertia of the tall mast. The centre of buoyancy is at the centre of mass of the volume of water that the hull displaces and this point is referred to as B in naval architecture. The centre of gravity of the ship is commonly denoted as point G or VCG, when a ship is at equilibrium, the centre of buoyancy is vertically in line with the centre of gravity of the ship. The metacentre is the point where the lines intersect of the force of buoyancy of φ ± dφ. When the ship is vertical, the metacentre lies above the centre of gravity and this distance is also abbreviated as GM. Work must be done to roll a stable hull and this is converted to potential energy by raising the centre of mass of the hull with respect to the water level or by lowering the centre of buoyancy or both

19.
Center of population
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In demographics, the center of population of a region is a geographical point that describes a centerpoint of the regions population. There are several different ways of defining such a point, leading to different geographical locations. A further complication is caused by the shape of the Earth. Different center points are obtained depending on whether the center is computed in three-dimensional space, or restricted to the curved surface, or computed using a flat map projection. The mean center, or centroid, is the point on which a rigid, weightless map would balance perfectly, mathematically, the centroid is the point to which the population has the smallest possible sum of squared distances. It is easily found by taking the mean of each coordinate. If defined in the space, the centroid of points on the Earths surface is actually inside the Earth. This point could then be projected back to the surface, alternatively, one could define the centroid directly on a flat map projection, this is, for example, the definition that the US Census Bureau uses. Contrary to a misconception, the centroid does not minimize the average distance to the population. That property belongs to the geometric median, the median center is the intersection of two perpendicular lines, each of which divides the population into two equal halves. Typically these two lines are chosen to be a parallel and a meridian, in that case, this center is easily found by taking separately the medians of the populations latitude and longitude coordinates. Tukey called this the cross median, the geometric median is the point to which the population has the smallest possible sum of distances. Because of this property, it is known as the point of minimum aggregate travel. Unfortunately, there is no direct closed-form expression for the geometric median, in practical computation, decisions are also made on the granularity of the population data, depending on population density patterns or other factors. For instance, the center of population of all the cities in a country may be different from the center of population of all the states in the same country, different methods may yield different results. Practical uses for finding the center of population include locating possible sites for forward capitals, such as Brasília, practical selection of a new site for a capital is a complex problem that depends also on population density patterns and transportation networks. It is important to use a method that does not depend on a projection when dealing with the entire world. As described in INED, the solution methodology deals only with the globe, as a result, the answer is independent of which map projection is used or where it is centered

20.
Power center (geometry)
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In geometry, the power center of three circles, also called the radical center, is the intersection point of the three radical axes of the pairs of circles. This is a case of the three conics theorem. The three radical axes meet in a point, the radical center, for the following reason. The radical axis of a pair of circles is defined as the set of points that have equal power h with respect to both circles. For example, for every point P on the axis of circles 1 and 2. Similarly, for every point on the axis of circles 2 and 3. Therefore, at the point of these two lines, all three powers must be equal, h1 = h2 = h3. Since this implies that h1 = h3, this point must also lie on the axis of circles 1 and 3. Hence, all three radical axes pass through the point, the radical center. The radical center has several applications in geometry and it has an important role in a solution to Apollonius problem published by Joseph Diaz Gergonne in 1814. In the power diagram of a system of circles, all of the vertices of the diagram are located at centers of triples of circles. The Spieker center of a triangle is the center of its excircles. Several types of radical circles have been defined as well, such as the circle of the Lucas circles. Advanced Euclidean Geometry, An elementary treatise on the geometry of the triangle, the Penguin Dictionary of Curious and Interesting Geometry. 100 Great Problems of Elementary Mathematics, Their History and Solutions, an elementary treatise on modern pure geometry. Radical Center at Cut-the-Knot Radical Axis and Center at Cut-the-Knot