Line segment
In geometry, a line segment is a part of a line, bounded by two distinct end points, contains every point on the line between its endpoints. A closed line segment includes both endpoints. Examples of line segments include the sides of a square. More when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge if they are adjacent vertices, or otherwise a diagonal; when the end points both lie on a curve such as a circle, a line segment is called a chord. If V is a vector space over R or C, L is a subset of V L is a line segment if L can be parameterized as L = for some vectors u, v ∈ V, in which case the vectors u and u + v are called the end points of L. Sometimes one needs to distinguish between "open" and "closed" line segments. One defines a closed line segment as above, an open line segment as a subset L that can be parametrized as L = for some vectors u, v ∈ V. Equivalently, a line segment is the convex hull of two points. Thus, the line segment can be expressed as a convex combination of the segment's two end points.
In geometry, it is sometimes defined that a point B is between two other points A and C, if the distance AB added to the distance BC is equal to the distance AC. Thus in R 2 the line segment with endpoints A = and C = is the following collection of points:. A line segment is a non-empty set. If V is a topological vector space a closed line segment is a closed set in V. However, an open line segment is an open set in V if and only if V is one-dimensional. More than above, the concept of a line segment can be defined in an ordered geometry. A pair of line segments can be any one of the following: intersecting, skew, or none of these; the last possibility is a way that line segments differ from lines: if two nonparallel lines are in the same Euclidean plane they must cross each other, but that need not be true of segments. In an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else be defined in terms of an isometry of a line.
Segments play an important role in other theories. For example, a set is convex if the segment that joins any two points of the set is contained in the set; this is important because it transforms some of the analysis of convex sets to the analysis of a line segment. The Segment Addition Postulate can be used to add congruent segment or segments with equal lengths and substitute other segments into another statement to make segments congruent. A line segment can be viewed as a degenerate case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints, the eccentricity goes to one. A standard definition of an ellipse is the set of points for which the sum of a point's distances to two foci is a constant. A complete orbit of this ellipse traverses the line segment twice; as a degenerate orbit this is a radial elliptic trajectory. In addition to appearing as the edges and diagonals of polygons and polyhedra, line segments appear in numerous other locations relative to other geometric shapes.
Some frequently considered segments in a triangle include the three altitudes, the three medians, the perpendicular bisectors of the sides, the internal angle bisectors. In each case there are various equalities relating these segment lengths to others as well as various inequalities. Other segment
Azimuth
An azimuth is an angular measurement in a spherical coordinate system. The vector from an observer to a point of interest is projected perpendicularly onto a reference plane; when used as a celestial coordinate, the azimuth is the horizontal direction of a star or other astronomical object in the sky. The star is the point of interest, the reference plane is the local area around an observer on Earth's surface, the reference vector points to true north; the azimuth is the star's vector on the horizontal plane. Azimuth is measured in degrees; the concept is used in navigation, engineering, mapping and ballistics. In land navigation, azimuth is denoted alpha, α, defined as a horizontal angle measured clockwise from a north base line or meridian. Azimuth has been more defined as a horizontal angle measured clockwise from any fixed reference plane or established base direction line. Today, the reference plane for an azimuth is true north, measured as a 0° azimuth, though other angular units can be used.
Moving clockwise on a 360 degree circle, east has azimuth 90°, south 180°, west 270°. There are exceptions: some navigation systems use south as the reference vector. Any direction can be the reference vector, as long as it is defined. Quite azimuths or compass bearings are stated in a system in which either north or south can be the zero, the angle may be measured clockwise or anticlockwise from the zero. For example, a bearing might be described as " south, thirty degrees east", abbreviated "S30°E", the bearing 30 degrees in the eastward direction from south, i.e. the bearing 150 degrees clockwise from north. The reference direction, stated first, is always north or south, the turning direction, stated last, is east or west; the directions are chosen so that the angle, stated between them, is positive, between zero and 90 degrees. If the bearing happens to be in the direction of one of the cardinal points, a different notation, e.g. "due east", is used instead. The cartographical azimuth can be calculated when the coordinates of 2 points are known in a flat plane: α = 180 π atan2 Remark that the reference axes are swapped relative to the mathematical polar coordinate system and that the azimuth is clockwise relative to the north.
This is the reason why the Y axis in the above formula are swapped. If the azimuth becomes negative, one can always add 360°; the formula in radians would be easier: α = atan2 Caveat: Most computer libraries reverse the order of the atan2 parameters. When the coordinates of one point, the distance L, the azimuth α to another point are known, one can calculate its coordinates: X 2 = X 1 + L sin α Y 2 = Y 1 + L cos α This is used in triangulation. We are standing at latitude φ 1, longitude zero. We can get a fair approximation by assuming the Earth is a sphere, in which case the azimuth α is given by tan α = sin L cos φ 1 tan φ 2 − sin φ 1 cos L A better approximation assumes the Earth is a slightly-squashed sphere. Normal-section azimuth is the angle measured at our viewpoint by a theodolite whose axis is perpendicular to the surface of the spheroid; the difference is immeasurably small. Various websites will calculate geodetic azimuth. Formulas for calculating geodetic azimuth are linked in the distance
Sphere of influence (astrodynamics)
A sphere of influence in astrodynamics and astronomy is the oblate-spheroid-shaped region around a celestial body where the primary gravitational influence on an orbiting object is that body. This is used to describe the areas in the Solar System where planets dominate the orbits of surrounding objects such as moons, despite the presence of the much more massive but distant Sun. In the patched conic approximation, used in estimating the trajectories of bodies moving between the neighbourhoods of different masses using a two body approximation and hyperbolae, the SOI is taken as the boundary where the trajectory switches which mass field it is influenced by; the general equation describing the radius of the sphere r S O I of a planet: r S O I ≈ a 2 / 5 where a is the semimajor axis of the smaller object's orbit around the larger body. M and M are the masses of the larger object, respectively. In the patched conic approximation, once an object leaves the planet's SOI, the primary/only gravitational influence is the Sun.
Because the definition of rSOI relies on the presence of the Sun and a planet, the term is only applicable in a three-body or greater system and requires the mass of the primary body to be much greater than the mass of the secondary body. This changes the three-body problem into a restricted two-body problem; the table shows the values of the sphere of gravity of the bodies of the solar system in relation to the Sun.: The Sphere of influence is, in fact, not quite a sphere. The distance to the SOI depends on the angular distance θ from the massive body. A more accurate formula is given by r S O I ≈ a 2 / 5 1 1 + 3 cos 2 10 Averaging over all possible directions we get r S O I ¯ = 0.9431 a 2 / 5 Consider two point masses A and B at locations r A and r B, with mass m A and m B respectively. The distance R = | r B − r A | separates the two objects. Given a massless third point C at location r C, one can ask whether to use a frame centered on A or on B to analyse the dynamics of C. Let's consider a frame centered on A.
The gravity of B is denoted as g B and will be treated as a perturbation to the dynamics of C due to the gravity g A of body A. Due their gravitational interactions, point A is attracted to point B with acceleration a A = G m B R 3, this frame is therefore non-inertial. To quantify the effects of the perturbations in this frame, one should consider the ratio of the perturbations to the main body gravity i.e. χ A = | g B − a A | | g A |. The perturbation g B − a A is known as the tidal forces due to body B, it is possible to construct the perturbation ratio χ B for the frame centered on B by interchanging A ↔ B. As C gets close to A, χ A →
Apsis
The term apsis refers to an extreme point in the orbit of an object. It denotes either the respective distance of the bodies; the word comes via Latin from Greek, there denoting a whole orbit, is cognate with apse. Except for the theoretical possibility of one common circular orbit for two bodies of equal mass at diametral positions, there are two apsides for any elliptic orbit, named with the prefixes peri- and ap-/apo-, added in reference to the body being orbited. All periodic orbits are, according to Newton's Laws of motion, ellipses: either the two individual ellipses of both bodies, with the center of mass of this two-body system at the one common focus of the ellipses, or the orbital ellipses, with one body taken as fixed at one focus, the other body orbiting this focus. All these ellipses share a straight line, the line of apsides, that contains their major axes, the foci, the vertices, thus the periapsis and the apoapsis; the major axis of the orbital ellipse is the distance of the apsides, when taken as points on the orbit, or their sum, when taken as distances.
The major axes of the individual ellipses around the barycenter the contributions to the major axis of the orbital ellipses are inverse proportional to the masses of the bodies, i.e. a bigger mass implies a smaller axis/contribution. Only when one mass is sufficiently larger than the other, the individual ellipse of the smaller body around the barycenter comprises the individual ellipse of the larger body as shown in the second figure. For remarkable asymmetry, the barycenter of the two bodies may lie well within the bigger body, e.g. the Earth–Moon barycenter is about 75% of the way from Earth's center to its surface. If the smaller mass is negligible compared to the larger the orbital parameters are independent of the smaller mass. For general orbits, the terms periapsis and apoapsis are used. Pericenter and apocenter are equivalent alternatives, referring explicitly to the respective points on the orbits, whereas periapsis and apoapsis may refer to the smallest and largest distances of the orbiter and its host.
For a body orbiting the Sun, the point of least distance is the perihelion, the point of greatest distance is the aphelion. The terms become apastron when discussing orbits around other stars. For any satellite of Earth, including the Moon, the point of least distance is the perigee and greatest distance the apogee, from Ancient Greek Γῆ, "land" or "earth". For objects in lunar orbit, the point of least distance is sometimes called the pericynthion and the greatest distance the apocynthion. Perilune and apolune are used. In orbital mechanics, the apsides technically refer to the distance measured between the barycenters of the central body and orbiting body. However, in the case of a spacecraft, the terms are used to refer to the orbital altitude of the spacecraft above the surface of the central body; these formulae characterize the pericenter and apocenter of an orbit: Pericenter Maximum speed, v per = μ a, at minimum distance, r per = a. Apocenter Minimum speed, v ap = μ a, at maximum distance, r ap = a.
While, in accordance with Kepler's laws of planetary motion and the conservation of energy, these two quantities are constant for a given orbit: Specific relative angular momentum h = μ a Specific orbital energy ε = − μ 2 a where: a is the semi-major axis: a = r per + r ap 2 μ is the standard gravitational parameter e is the eccentricity, defined as e = r ap − r per r ap + r per = 1 − 2 r ap r per + 1 Note t
Parabola
In mathematics, a parabola is a plane curve, mirror-symmetrical and is U-shaped. It fits several superficially different other mathematical descriptions, which can all be proved to define the same curves. One description of a parabola involves a line; the focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane, parallel to another plane, tangential to the conical surface; the line perpendicular to the directrix and passing through the focus is called the "axis of symmetry". The point on the parabola that intersects the axis of symmetry is called the "vertex", is the point where the parabola is most curved; the distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The "latus rectum" is the chord of the parabola, parallel to the directrix and passes through the focus.
Parabolas can open up, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit on any other parabola—that is, all parabolas are geometrically similar. Parabolas have the property that, if they are made of material that reflects light light which travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected into a parallel beam, leaving the parabola parallel to the axis of symmetry; the same effects occur with other forms of energy. This reflective property is the basis of many practical uses of parabolas; the parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors to the design of ballistic missiles. They are used in physics and many other areas; the earliest known work on conic sections was by Menaechmus in the fourth century BC.
He discovered a way to solve the problem of doubling the cube using parabolas. The area enclosed by a parabola and a line segment, the so-called "parabola segment", was computed by Archimedes via the method of exhaustion in the third century BC, in his The Quadrature of the Parabola; the name "parabola" is due to Apollonius. It means "application", referring to "application of areas" concept, that has a connection with this curve, as Apollonius had proved; the focus–directrix property of the parabola and other conic sections is due to Pappus. Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity; the idea that a parabolic reflector could produce an image was well known before the invention of the reflecting telescope. Designs were proposed in the early to mid seventeenth century by many mathematicians including René Descartes, Marin Mersenne, James Gregory; when Isaac Newton built the first reflecting telescope in 1668, he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror.
Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes and radar receivers. A parabola can be defined geometrically as a set of points in the Euclidean plane: A parabola is a set of points, such that for any point P of the set the distance | P F | to a fixed point F, the focus, is equal to the distance | P l | to a fixed line l, the directrix: The midpoint V of the perpendicular from the focus F onto the directrix l is called vertex and the line F V the axis of symmetry of the parabola. If one introduces cartesian coordinates, such that F =, f > 0, the directrix has the equation y = − f one obtains for a point P = from | P F | 2 = | P l | 2 the equation x 2 + 2 = 2. Solving for y yields y = 1 4 f x 2; the parabola is U-shaped. The horizontal chord through the focus is called the latus rectum; the latus rectum is parallel to the directrix. The semi-latus
Lagrangian point
In celestial mechanics, the Lagrangian points are the points near two large bodies in orbit where a smaller object will maintain its position relative to the large orbiting bodies. At other locations, a small object would go into its own orbit around one of the large bodies, but at the Lagrangian points the gravitational forces of the two large bodies, the centripetal force of orbital motion, the Coriolis acceleration all match up in a way that cause the small object to maintain a stable or nearly stable position relative to the large bodies. There are five such points, labeled L1 to L5, all in the orbital plane of the two large bodies, for each given combination of two orbital bodies. For instance, there are five Lagrangian points L1 to L5 for the Sun-Earth system, in a similar way there are five different Lagrangian points for the Earth-Moon system. L1, L2, L3 are on the line through the centers of the two large bodies. L4 and L5 each form an equilateral triangle with the centers of the large bodies.
L4 and L5 are stable, which implies that objects can orbit around them in a rotating coordinate system tied to the two large bodies. Several planets have trojan satellites near their L5 points with respect to the Sun. Jupiter has more than a million of these trojans. Artificial satellites have been placed at L1 and L2 with respect to the Sun and Earth, with respect to the Earth and the Moon; the Lagrangian points have been proposed for uses in space exploration. The three collinear Lagrange points were discovered by Leonhard Euler a few years before Joseph-Louis Lagrange discovered the remaining two. In 1772, Lagrange published an "Essay on the three-body problem". In the first chapter he considered the general three-body problem. From that, in the second chapter, he demonstrated two special constant-pattern solutions, the collinear and the equilateral, for any three masses, with circular orbits; the five Lagrangian points are labeled and defined as follows: The L1 point lies on the line defined by the two large masses M1 and M2, between them.
It is the most intuitively understood of the Lagrangian points: the one where the gravitational attraction of M2 cancels M1's gravitational attraction. Explanation An object that orbits the Sun more than Earth would have a shorter orbital period than Earth, but that ignores the effect of Earth's own gravitational pull. If the object is directly between Earth and the Sun Earth's gravity counteracts some of the Sun's pull on the object, therefore increases the orbital period of the object; the closer to Earth the object is, the greater this effect is. At the L1 point, the orbital period of the object becomes equal to Earth's orbital period. L1 is 0.01 au, 1/100th the distance to the Sun. The L2 point lies on the line beyond the smaller of the two. Here, the gravitational forces of the two large masses balance the centrifugal effect on a body at L2. Explanation On the opposite side of Earth from the Sun, the orbital period of an object would be greater than that of Earth; the extra pull of Earth's gravity decreases the orbital period of the object, at the L2 point that orbital period becomes equal to Earth's.
Like L1, L2 is 0.01 au from Earth. The L3 point lies on the line defined beyond the larger of the two. Explanation Within the Sun-Earth system, the L3 point exists on the opposite side of the Sun, a little outside Earth's orbit and further from the Sun than Earth is; this placement occurs because the Sun is affected by Earth's gravity and so orbits around the two bodies' barycenter, well inside the body of the Sun. At the L3 point, the combined pull of Earth and Sun cause the object to orbit with the same period as Earth; the L4 and L5 points lie at the third corners of the two equilateral triangles in the plane of orbit whose common base is the line between the centers of the two masses, such that the point lies behind or ahead of the smaller mass with regard to its orbit around the larger mass. The triangular points are stable equilibria, provided that the ratio of M1/M2 is greater than 24.96. This is the case for the Sun–Earth system, the Sun–Jupiter system, and, by a smaller margin, the Earth–Moon system.
When a body at these points is perturbed, it moves away from the point, but the factor opposite of that, increased or decreased by the perturbation will increase or decrease, bending the object's path into a stable, kidney bean-shaped orbit around the point. In contrast to L4 and L5, where stable equilibrium exists, the points L1, L2, L3 are positions of unstable equilibrium. Any object orbiting at L1, L2, or L3 will tend to fall out of orbit, it is common to orbiting the L4 and L5 points of natural orbital systems. These are called "trojans". In the 20th century, asteroids discovered orbiting at the Sun–Jupiter L4 and L5 points were named after characters from Homer's Iliad. Asteroids at the L4 point, which leads Jupiter, are referred to as the "Greek camp", whereas those at the L5 point are referred to as the "Trojan camp". Other examples of natural objects orbiting at Lagrange points: The Sun–Earth L4 and L5 points contain interplanetary dust and at least one asteroid, 2010 TK7, detected in October 2010 by Wide-field Infrared Survey Explorer and announced during July 2011.
The Earth–Moon L4 and
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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