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Semi-minor Axis

Semi-minor axis

In geometry, the semi-minor axis (also semiminor axis) applies to ellipses and hyperbolas.

Ellipse

The semi-minor axis of an ellipse is one half of the minor axis, running from the center, halfway between and perpendicular to the line running between the foci, and to the edge of the ellipse. The minor axis is the longest line that runs perpendicular to the major axis. It is related to the semi-major axis a through the eccentricity e and the semi-latus rectum l, as follows: :b = a \sqrt :al=b^2. A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping l fixed. Thus a and b tend to infinity, a faster than b.

Hyperbola

The semi-minor axis of a hyperbola is the distance from a top, along the tangent line, to each asymptote; if this is in the y-direction it is b in this equation of the hyperbola: \frac - \frac = 1 It is related to the semi-major axis through the eccentricity, as follows: :b = a \sqrt Note that in a hyperbola b can be larger than a! Category:Conic sections Category:Astrodynamics Category:Celestial mechanics

Ellipse

Elliptical redirects here, for the exercise machine, see Elliptical trainer. Elliptical trainer In mathematics, an ellipse (from the Greek for absence) is a plane algebraic curve where the sum of the distances from any point on the curve to two fixed points is constant. The two fixed points are called foci (plural of focus). An ellipse is a type of conic section: if a cone is cut with a plane which does not intersect the cone's base, the intersection of the cone and plane is an ellipse. For a short elementary proof of this, see Dandelin spheres. Algebraically, an ellipse is a curve in the Cartesian plane defined by an equation of the form :A x^2 + B xy + C y^2 + D x + E y + F = 0 such that B^2 < 4 AC, where all of the coefficients are real, and where more than one solution, defining a pair of points (x, y) on the ellipse, exists. An ellipse can be drawn with two pins, a loop of string, and a pencil. The pins are placed at the foci and the pins and pencil are enclosed inside the string. The pencil is placed on the paper inside the string, so the string is taut. The string will form a triangle. If the pencil is moved around so that the string stays taut, the sum of the distances from the pencil to the pins will remain constant, satisfying the definition of an ellipse. triangle The line segment which passes through the foci and terminates on the ellipse is called the major axis. The major axis is along the longest segment that passes through the ellipse. The line which passes through the center (halfway between the foci), at right angles to the major axis, is called the minor axis. A semimajor axis is one half the major axis: the line segment from the center, through a focus, and to the edge of the ellipse. Likewise, the semiminor axis is one half the minor axis. If the two foci coincide, then the ellipse is a circle; in other words, a circle is a special case of an ellipse, one where the eccentricity is zero. An ellipse centred at the origin can be viewed as the image of the unit circle under a linear map associated with a symmetric matrix A = PDP^T, D being a diagonal matrix with the eigenvalues of A, both of which are real positive, along the main diagonal, and P being a real unitary matrix having as columns the eigenvectors of A. Then the axes of the ellipse will lie along the eigenvectors of A, and the squares of the lengths of the axes are the inverses of the eigenvalues.

Parametrisation

The size of an ellipse is determined by two constants, conventionally denoted a and b. The constant a equals the length of the semimajor axis; the constant b equals the length of the semiminor axis. length An ellipse centered at the origin of an x-y coordinate system with its major axis along the x-axis is defined by the equation :\frac + \frac = 1 The derivation of this formula is quite instructive and not overly difficult. The following diagram shows an ellipse demonstrating the Pythagoras equation a² = b² + c² as a special case of the non-parametric equation above (x=0, y=b). center The same ellipse is also represented by the parametric equations: :x = a\,\cos t :y = b\,\sin t :0 \leq t < 2\pi which use the trigonometric functions sine and cosine. If an ellipse is not centered at the origin of an x-y coordinate system, but again has its major axis along the x-axis, it may be specified by the equation :\frac + \frac = 1 where (h,k) is the center. A Gauss-mapped form: :\left(\frac,\frac\right) has normal (\cos\phi,\sin\phi).

Eccentricity

The shape of an ellipse is usually expressed by a number called the eccentricity of the ellipse, conventionally denoted e (not to be confused with the mathematical constant e). The eccentricity is related to a and b by the statement :e = \sqrt or where c (the linear eccentricity of the ellipse) equals the distance from the center to either focus :e = \frac The eccentricity is a positive number less than 1, or 0 in the case of a circle. The greater the eccentricity is, the larger the ratio of a to b is, and therefore the more elongated the ellipse is. The ellipse shown in the image below has an eccentricity of approximately 0.8733. The distance between the foci is 2ae.

Semi-latus rectum and polar coordinates

The semi-latus rectum of an ellipse, usually denoted l\,\! (lowercase L), is the distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular to the major axis. It is related to a\,\! and b\,\! (the ellipse's semi-axes) by the formula al=b^2\,\! or, if using the eccentricity, l=a(1-e^2)\,\!. perpendicular In polar coordinates, an ellipse with one focus at the origin and the other on the negative x-axis is given by the equation : r (1 + e \cos \theta) = l \,\! An ellipse can also be thought of as a projection of a circle: a circle on a plane at angle φ to the horizontal projected vertically onto a horizontal plane gives an ellipse of eccentricity sin φ, provided φ is not 90°.

Area

The area enclosed by an ellipse is \pi ab\,\!, where \pi is Archimedes' constant.

Circumference

The circumference of an ellipse is 4aE(e), where the function E is the complete elliptic integral of the second kind. The exact infinite series is: :c = 2\pi a \left[\right]\!\, A good approximation is Ramanujan's: :c \approx \pi \left[3(a+b) - \sqrt\right]\!\, which can also be written as: :c \approx \pi a \left[ 3 (1+\sqrt) - \sqrt \right] \!\, More generally, the arc length of a portion of the circumference, as a function of the angle subtended, is given by an incomplete elliptic integral. The inverse function, the angle subtended as a function of the arc length, is given by the elliptic functions.

Stretching and Projection

An ellipse may be uniformly stretched along any axis, in or out of the plane of the ellipse, and it will still be an ellipse. The stretched ellipse will have different properties (perhaps changed eccentricity and semi-major axis length, for instance), but it will still be an ellipse (or a degenerate ellipse: a circle or a point). Similarly, any oblique projection onto a plane results in a conic section. If the projection is a closed curve on the plane, then the curve is an ellipse or a degenerate ellipse.

Reflection property

Assume an elliptic mirror with a light source at one of the foci. Then all rays are reflected to a single point — the second focus. Since no other curve has such a property, it can be used as an alternative definition of an ellipse.

Ellipses in physics

Indian astronomer Aryabhata discovered that the orbits of the planets around the sun are ellipses in 499, which he described in his book, the Aryabhatiya [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Aryabhata_I.html]. In the 17th century, Johannes Kepler explained that the orbits along which the planets travel around the Sun are ellipses, which is Kepler's first law. Later, Isaac Newton explained this as a corollary of his law of universal gravitation. More generally, in the gravitational two-body problem, if the two bodies are bound to each other (i.e., the total energy is negative), their orbits are similar ellipses with the common barycenter being one of the foci of each ellipse. Interestingly, the orbit of either body in the reference frame of the other is also an ellipse, with the other body at one focus. The general solution for a harmonic oscillator in two or more dimensions is also an ellipse, but this time with the origin of the force located at the center of the ellipse. Albert Einstein also used the ellipse to prove his theory of relativity by using an elliptical shaped mass. Einstein's contributions to modern physics may not have been discovered if it were not for ellipses.

Ellipses in computer graphics

Drawing an ellipse is a common graphics primitive in standard display libraries, such as the QuickDraw and GDI interfaces on the Macintosh and Windows systems. Often such libraries are limited and can only draw an ellipse with either the major axis or the minor axis horizontal. Jack Bresenham at IBM is most famous for the invention of 2D drawing primitives, including line and circle drawing, using only fast integer operations such as addition and branch on carry bit. An efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken (IEEE CG&A, Sept. 1984). A more challenging task is to perform these drawing operations with antialiasing, to create a smooth-looking curve. The curve drawing algorithms of Xiaolin Wu (SIGGRAPH 91) are an example.

See also


- Ellipsoid, a higher dimensional analog of an ellipse
- Spheroid, the ellipsoids obtained by rotating an ellipse about its major or minor axis.
- Super ellipse, a generalization of an ellipse that can look more rectangular
- Hyperbola
- Parabola
- Orbit
- Oval (geometry)
- True, eccentric, and mean anomalies Category:Conic sections ko:타원 ja:楕円

Semi-major axis

In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolas.

Ellipse

The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape. The semi-major axis is one half of the major axis, and thus runs from the centre, through a focus, and to the edge of the ellipse. It is related to the semi-minor axis b\,\! through the eccentricity e\,\! and the semi-latus rectum \ell\,\!, as follows: :b = a \sqrt\,\! :\ell=a(1-e^2)\,\!. :a\ell=b^2\,\!. A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping \ell\,\! fixed. Thus a\,\! and b\,\! tend to infinity, a\,\! faster than b\,\!. The semi-major axis is the mean value of the smallest and largest distances from one focus to the points on the ellipse. Now consider the equation in polar coordinates, with one focus at the origin and the other on the positive x-axis, : r(1-e\cos\theta)=l\,\! The mean value of r=\,\! and r=\,\!, is a=\,\!.

Hyperbola

The semi-major axis of a hyperbola is one half of the distance between the two branches; if this is in the x-direction the equation is: \frac - \frac = 1 In terms of the semi-latus rectum and the eccentricity we have a=

Astronomy

Orbital period

In astrodynamics the orbital period T\, of a small body orbiting a central body in a circular or elliptical orbit is: :T = 2\pi\sqrt where: :a\, is the length of the orbit's semi-major axis : \mu is the standard gravitational parameter Note that for all ellipses with a given semi-major axis, the orbital period is the same, regardless of eccentricity. In astronomy, the semi-major axis is one of the most important orbital elements of an orbit, along with its orbital period. For solar system objects, the semi-major axis is related to the period of the orbit by Kepler's third law (originally empirically derived), :P^2=a^3\, where P is the period in years, and a is the semimajor axis in astronomical units. This form turns out to be a simplification of the general form for the two-body problem, as determined by Newton: :P^2= \fraca^3\, where G is the gravitational constant, and M is the mass of the central body, and m is the mass of the orbiting body. Typically, the central body's mass is so much greater than the orbiting body's, that m may be ignored. Making that assumption and using typical astronomy units results in the simpler form Kepler discovered. Remarkably, the orbiting body's path around the barycentre and its path relative to its primary are both ellipses. The semi-major axis used in astronomy is always the primary-to-secondary distance; thus, the orbital parameters of the planets are given in heliocentric terms. The difference between the primocentric and "absolute" orbits may best be illustrated by looking at the Earth-Moon system. The mass ratio in this case is 81.30059. The Earth-Moon characteristic distance, the semi-major axis of the geocentric lunar orbit, is 384,400 km. The barycentric lunar orbit, on the other hand, has a semi-major axis of 379,700 km, the Earth's counter-orbit taking up the difference, 4,700 km. The Moon's average barycentric orbital speed is 1.010 km/s, whilst the Earth's is 0.012 km/s. The total of these speeds gives the geocentric lunar average orbital speed, 1.022 km/s; the same value may be obtained by considering just the geocentric semi-major axis value.

Average distance

It is often said that the semi-major axis is the "average" distance between the primary (the focus of the ellipse) and the orbiting body. This is not quite accurate, as it depends over what the average is taken.
- averaging the distance over the eccentric anomaly (q.v.) indeed results in the semi-major axis.
- averaging over the true anomaly (the true orbital angle, measured at the focus) results, oddly enough, in the semi-minor axis b = a \sqrt\,\!.
- averaging over the mean anomaly (the fraction of the orbital period that has elapsed since pericentre, expressed as an angle), finally, gives the time-average (which is what "average" usually means to the layman): a (1 + \frac)\,\!. The time-average of the inverse of the radius, r^\,\!, is a^\,\!.

Energy; calculation of semi-major axis from state vectors

In astrodynamics semi-major axis a \, can be calculated from orbital state vectors: a = \, for an elliptical orbit and a = \, for a hyperbolic trajectory and \epsilon = - (specific orbital energy) and \mu = GM \, (standard gravitational parameter), where:
- v\, is orbital velocity from velocity vector of an orbiting object,
- \mathbf\, is cartesian position vector of an orbiting object in coordinates of a reference frame with respect to which the elements of the orbit are to be calculated (e.g. geocentric equatorial for an orbit around Earth, or heliocentric ecliptic for an orbit around the Sun),
- G \, is the gravitational constant,
- M \, the mass of the central body. Note that for a given central body and total specific energy, the semi-major axis is always the same, regardless of eccentricity. Conversely, for a given central body and semi-major axis, the total specific energy is always the same.

Example

The International Space Station has an orbital period of 91.74 minutes, hence the semi-major axis is 6738 km [http://www.google.com/search?num=100&hl=en&lr=&newwindow=1&safe=off&q=%28%2891.74
- 60%2F2%2Fpi%29%5E2
- 398600%29%5E%281%2F3%29]. Every minute more corresponds to ca. 50 km more: the extra 300 km of orbit length takes 40 seconds, the lower speed accounts for an additional 20 seconds.

References


- [http://orca.phys.uvic.ca/~tatum/celmechs/celm9.pdf Jeremy B. Tatum, Celestial Mechanics, Chapter 9 - The Two Body Problem in Two Dimensions (2004)]
- [http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000071000011001198000001&idtype=cvips&gifs=yes Darren M. Williams, Average distance between a star and planet in an eccentric orbit, American Journal of Physics, November 2003, Volume 71, Issue 11, pp. 1198-1200] Category:Conic sections Category:Astrodynamics Category:Celestial mechanics

Eccentricity (mathematics)

(This page refers to eccentricity in mathematics. For other uses, see the disambiguation page eccentricity.) In mathematics, eccentricity is a parameter associated with every conic section, see Conic section#Eccentricity. It can be thought of as a measure of how much the conic section deviates from being circular. In particular,
- The eccentricity of a circle is zero.
- The eccentricity of an ellipse is greater than zero and less than 1.
- The eccentricity of a parabola is 1.
- The eccentricity of a hyperbola is greater than 1 and less than infinity.
- The eccentricity of a straight line is infinity. It is given by: :e = \sqrt Where a is the length of the semimajor axis of the section, b the length of the semiminor axis, and k is equal to +1 for an ellipse, 0 for a parabola, and -1 for a hyperbola. It is also called the first eccentricity when necessary to distinguish it from the second eccentricity, e', which is sometimes used for algebraic convenience. The second eccentricity is defined as: :e' = \sqrt And is related to the first eccentricity by the equation: :1 = (1 - e^2)(1 + e'^2)\,\!

Ellipse

semiminor axis For any ellipse, where the length of the semi-major axis is a, and where the same of the semi-minor axis is b, the eccentricity is given by: :e = \sqrt The eccentricity is the ratio of the distance between the foci (F_1 and F_2) to the major axis; i.e. \left ( \frac \right ). The term linear eccentricity is used for .

Hyperbola

For any hyperbola, where the length of the semi-major axis is a, and where the same of the semi-minor axis is b, eccentricity is given by: :e = \sqrt

Surfaces

The eccentricity of a surface is the eccentricity of a designated section of the surface. For example, on a triaxial ellipsoid, the meridional eccentricity is that of the ellipse formed by a section containing both the longest and the shortest axes (one of which will be the polar axis), and the equatorial eccentricity is the eccentricity of the ellipse formed by a section through the centre, perpendicular to the polar axis (i.e. in the equatorial plane).

External links


- [http://mathworld.wolfram.com/Eccentricity.html MathWorld: Eccentricity] Category:Conic sections als:Exzentrizität (Mathematik)

Parabola

The parabola (from the Greek: παραβολή) is a conic section generated by the intersection of a cone and a plane tangent to the cone or parallel to some plane tangent to the cone. If the plane is itself tangent to the cone, one would obtain a degenerate parabola, a line. A parabola can also be defined as locus of points which are equidistant from a given point (the focus) and a given line (the directrix). Algebraically, a parabola is a curve in the Cartesian plane defined by an irreducible equation of the form :A x^2 + B xy + C y^2 + D x + E y + F = 0 such that B^2 = 4 AC, where all of the coefficients are real, and where more than one solution, defining a pair of points (x, y) on the parabola, exists. That the equation is irreducible means it does not factor as a product of two not necessarily distinct linear factors.

Definitions and overview

Cartesian plane In Cartesian coordinates, a parabola with an axis parallel to the y axis with vertex (h, k), focus (h, k + p), and directrix y = k - p, with p being the distance from the vertex to the focus, has the equation :(x - h)^2 = 4p(y - k) \, or, alternatively :(y - k) = \frac(x-h)^2 \, A parabola may also be characterized as a conic section with an eccentricity of 1. As a consequence of this, all parabolas are similar. A parabola can also be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction. In this sense, a parabola may be considered an ellipse that has one focus at infinity. The parabola is an inverse transform of a cardioid. A parabola has a single axis of reflective symmetry, which passes through its focus and is perpendicular to its directrix. The point of intersection of this axis and the parabola is called the vertex. A parabola spun about this axis in three dimensions traces out a shape known as a paraboloid of revolution. The parabola is found in numerous situations in the physical world (see below).

Equations

Cartesian


- Vertical axis of symmetry: :(x - h)^2 = 4p(y - k) \quad
- Horizontal axis of symmetry: :(y - h)^2 = 4p(x - k) \quad
- Quadratic (vertical axis of symmetry): :y = ax^2 + bx + c \, ::\mboxa = \frac; \ \ b = \frac; \ \ c = \frac + k and the vertex \left( \frac,\ \frac \right).
- Quadratic (horizontal axis of symmetry): :x = ay^2 + by + c \; :a, b, and c are the same as above. The coordinates of the vertex are reversed.

Parametric

:x = 2pt + h \, :y = pt^2 + k \,

Semi-latus rectum and polar coordinates

In polar coordinates, a parabola with the focus at the origin and the top on the negative x-axis, is given by the equation : r (1 - \cos \theta) = l \, where l is the semi-latus rectum: the distance from the focus to the parabola itself, measured along a line perpendicular to the axis. Note that this is twice the distance from the focus to the apex of the parabola or the perpendicular distance from the focus to the latus rectum.

Gauss-mapped form

A Gauss-mapped form: (\tan^2\phi,2\tan\phi) has normal (\cos\phi,\sin\phi).

See also


- Paraboloid
- Ellipse
- Hyperbola
- conic section

Derivation of the focus

Given a parabola parallel to the y-axis with vertex (0,0) and with equation : y = a x^2, \qquad \qquad \qquad (1) then there is a point (0,f) — the focus — such that any point P on the parabola will be equidistant from both the focus and a line perpendicular to the axis of symmetry of the parabola (the linea directrix), in this case parallel to the x axis. Since the vertex is one of the possible points P, it follows that the linea directrix passes through the point (0,-f). So for any point P=(x,y), it will be equidistant from (0,f) and (x,-f). It is desired to find the value of f which has this property. Let F denote the focus, and let Q denote the point at (x,-f). Line FP has the same length as line QP. : \| FP \| = \sqrt, : \| QP \| = y + f. : \| FP \| = \| QP \| : \sqrt = a x^2 + f \qquad Square both sides, : x^2 + (a x^2 - f)^2 = (a x^2 + f)^2 \qquad ::: = a^2 x^4 + f^2 + 2 a x^2 f \quad : x^2 + a^2 x^4 + f^2 - 2 a x^2 f = a^2 x^4 + f^2 + 2 a x^2 f \quad Cancel out terms from both sides, : x^2 - 2 a x^2 f = 2 a x^2 f, \quad : x^2 = 4 a x^2 f. \quad Cancel out the x2 from both sides (x is generally not zero), : 1 = 4 a f \quad : f = Now let p=f and the equation for the parabola becomes : x^2 = 4 p y \quad Q.E.D.

Reflective property of the tangent

The tangent of the parabola described by equation (1) has slope : = 2 a x = This line intersects the y-axis at the point (0,-y) = (0, - a x2), and the x-axis at the point (x/2,0). Let this point be called G. Point G is also the midpoint of points F and Q: : F = (0,f), \quad : Q = (x,-f), \quad : = = = (, 0). Since G is the midpoint of line FQ, this means that : \| FG \| \cong \| GQ \|, and it is already known that P is equidistant from both F and Q: : \| PF \| \cong \| PQ \|, and, thirdly, line GP is equal to itself, therefore: :\Delta FGP \cong \Delta QGP It follows that \angle FPG \cong \angle GPQ . Line QP can be extended beyond P to some point T, and line GP can be extended beyond P to some point R. Then \angle RPT and \angle GPQ are vertical, so they are equal (congruent). But \angle GPQ is equal to \angle FPG . Therefore \angle RPT is equal to \angle FPG . The line RG is tangent to the parabola at P, so any light beam bouncing off point P will behave as if line RG were a mirror and it were bouncing off that mirror. Let a light beam travel down the vertical line TP and bounce off from P. The beam's angle of inclination from the mirror is \angle RPT , so when it bounces off, its angle of inclination must be equal to \angle RPT . But \angle FPG has been shown to be equal to \angle RPT . Therefore the beam bounces off along the line FP: directly towards the focus. Conclusion: Any light beam moving vertically downwards in the concavity of the parabola (parallel to the axis of symmetry) will bounce off the parabola moving directly towards the focus. (See parabolic reflector.)

Parabolae in the physical world

In nature, approximations of parabolae and paraboloids are found in many diverse situations. The most well-known instance of the parabola in the history of physics is the trajectory of particle or body in motion under the influence of a uniform gravitational field without air resistance (for instance, a baseball flying through the air, neglecting air friction). The parabolic trajectory of projectiles was discovered experimentally by Galileo in the early 17th century, who performed experiments with balls rolling on inclined planes. The parabolic shape for projectiles was later proven mathematically by Isaac Newton. For objects extended in space, such as a diver jumping from a diving board, the object itself follows a complex motion as it rotates, but the center of mass of the object nevertheless forms a parabola. As in all cases in the physical world, the trajectory is always an approximation of a parabola. The presence of air resistance always distorts the shape, for example, although at low speeds, the shape is a good approximation of a parabola. At higher speeds, such as in ballistics, the shape is highly distorted and does not resemble a parabola. center of mass Another situation in which parabola may arise in nature is in two-body orbits, for example, of a small planetoid or other object under the influence of the gravitation of the sun. In such a case, parabolic orbits are a special case that are in nature. Orbits that form a hyperbola or an ellipse are much more common. In fact, the parabolic orbit is the borderline case between those two types of orbit. Approximations of parabolas are also found in the shape of cables of suspension bridges. Freely hanging cables do not describe parabolas, but rather catenary curves. Under the influence of a uniform load (for example, the deck of bridge), however, the cable is deformed towards a parabola. catenary's Casa Milà.]] Paraboloids arise in several physical situations as well. The most well-known instance is the parabolic reflector, which is a mirror or similar reflective device that concentrates light or other forms of electromagnetic radiation to a common focal point. The principle of the parabolic reflector was discovered by the geometer Archimedes in the 3rd century B.C., who constructed parabolic mirrors to defend Syracuse against the Roman fleet, by concentrating the sun's rays to set fire to the decks of the Roman ships. The principle was applied to telescopes in the 17th century. Today, paraboloid reflectors can be commonly observed throughout much of the world in microwave and satellite dish antennas. Paraboloid are also observed in the surface of a liquid confined to a container and rotated around the central axis. In this case, the centrifugal force, causes the surface of the liquid to climb the walls of the container, forming a parabolic surface.

Constructing a parabola

A parabola can be constructed geometrically as follows: Draw point F where you want the focus of the parabola. Draw point V where you want the vertex of the parabola. Draw line AA through points F and V this is the axis of the parabola. Draw line CC perpendicular to AA, through V this is a construction line. Draw line DD parallel to line CC on the opposite side and as far away as point F is from point V. This is the directrix. The following steps are repeated for however many points you want: Choose any point Q1 on line DD Draw the line FQ1. Label the point where FQ1 crosses line CC as R1. Draw line LR1 perpendicular to FQ1 at point R1. Draw line LQ1 perpendicular to line DD at point Q1. Label point P1 where lines LR1 and LQ1 cross. This point is on the parabola. Repeat by incrementing the point number until all points are constructed. Draw the parabola by drawing a smooth line through all the points V, P1,P2, ... ,Pn This Image shows construction for two points P1 and P2. Image:HowtoGraphiclyConstructParabola.png

By paper folding

Draw a straight line (the directrix) on a piece of paper, and a point (the focus) somewhere not on the line. Then fold the paper over so that the focus point touches the directrix line and crease the fold. Also crease the paper vertically where the focus point touches the directrix line. The point where these two lines intersect is a point on the parabola. Do this several times to get more points on the parabola. The envelope formed by the creases will make a nice parabola. One can make an ellipse or hyperbola similarly by using a circle and a point.

See also


- Catenary

External links


- [http://mathworld.wolfram.com/Parabola.html MathWorld: Parabola]
- [http://www.cut-the-knot.org/Curriculum/Geometry/ArchimedesTriangle.shtml Archimedes Triangle and Squaring of Parabola] at cut-the-knot
- [http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaLambert.shtml Two Tangents to Parabola] at cut-the-knot
- [http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaEnvelope.shtml Parabola As Envelope of Straight Lines] at cut-the-knot
- [http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaMirror.shtml Parabolic Mirror] at cut-the-knot
- [http://www.cut-the-knot.org/Curriculum/Geometry/ThreeParabolaTangents.shtml Three Parabola Tangents] at cut-the-knot
- [http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaLambert.shtml Two Tangents to Parabola] at cut-the-knot
- [http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaFocal.shtml Focal Properties of Parabola] at cut-the-knot
- [http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaMesh.shtml Parabola As Envelope II] at cut-the-knot Category:Conic sections ko:포물선 ja:放物線

Semi-major axis

In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolas.

Ellipse

The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape. The semi-major axis is one half of the major axis, and thus runs from the centre, through a focus, and to the edge of the ellipse. It is related to the semi-minor axis b\,\! through the eccentricity e\,\! and the semi-latus rectum \ell\,\!, as follows: :b = a \sqrt\,\! :\ell=a(1-e^2)\,\!. :a\ell=b^2\,\!. A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping \ell\,\! fixed. Thus a\,\! and b\,\! tend to infinity, a\,\! faster than b\,\!. The semi-major axis is the mean value of the smallest and largest distances from one focus to the points on the ellipse. Now consider the equation in polar coordinates, with one focus at the origin and the other on the positive x-axis, : r(1-e\cos\theta)=l\,\! The mean value of r=\,\! and r=\,\!, is a=\,\!.

Hyperbola

The semi-major axis of a hyperbola is one half of the distance between the two branches; if this is in the x-direction the equation is: \frac - \frac = 1 In terms of the semi-latus rectum and the eccentricity we have a=

Astronomy

Orbital period

In astrodynamics the orbital period T\, of a small body orbiting a central body in a circular or elliptical orbit is: :T = 2\pi\sqrt where: :a\, is the length of the orbit's semi-major axis : \mu is the standard gravitational parameter Note that for all ellipses with a given semi-major axis, the orbital period is the same, regardless of eccentricity. In astronomy, the semi-major axis is one of the most important orbital elements of an orbit, along with its orbital period. For solar system objects, the semi-major axis is related to the period of the orbit by Kepler's third law (originally empirically derived), :P^2=a^3\, where P is the period in years, and a is the semimajor axis in astronomical units. This form turns out to be a simplification of the general form for the two-body problem, as determined by Newton: :P^2= \fraca^3\, where G is the gravitational constant, and M is the mass of the central body, and m is the mass of the orbiting body. Typically, the central body's mass is so much greater than the orbiting body's, that m may be ignored. Making that assumption and using typical astronomy units results in the simpler form Kepler discovered. Remarkably, the orbiting body's path around the barycentre and its path relative to its primary are both ellipses. The semi-major axis used in astronomy is always the primary-to-secondary distance; thus, the orbital parameters of the planets are given in heliocentric terms. The difference between the primocentric and "absolute" orbits may best be illustrated by looking at the Earth-Moon system. The mass ratio in this case is 81.30059. The Earth-Moon characteristic distance, the semi-major axis of the geocentric lunar orbit, is 384,400 km. The barycentric lunar orbit, on the other hand, has a semi-major axis of 379,700 km, the Earth's counter-orbit taking up the difference, 4,700 km. The Moon's average barycentric orbital speed is 1.010 km/s, whilst the Earth's is 0.012 km/s. The total of these speeds gives the geocentric lunar average orbital speed, 1.022 km/s; the same value may be obtained by considering just the geocentric semi-major axis value.

Average distance

It is often said that the semi-major axis is the "average" distance between the primary (the focus of the ellipse) and the orbiting body. This is not quite accurate, as it depends over what the average is taken.
- averaging the distance over the eccentric anomaly (q.v.) indeed results in the semi-major axis.
- averaging over the true anomaly (the true orbital angle, measured at the focus) results, oddly enough, in the semi-minor axis b = a \sqrt\,\!.
- averaging over the mean anomaly (the fraction of the orbital period that has elapsed since pericentre, expressed as an angle), finally, gives the time-average (which is what "average" usually means to the layman): a (1 + \frac)\,\!. The time-average of the inverse of the radius, r^\,\!, is a^\,\!.

Energy; calculation of semi-major axis from state vectors

In astrodynamics semi-major axis a \, can be calculated from orbital state vectors: a = \, for an elliptical orbit and a = \, for a hyperbolic trajectory and \epsilon = - (specific orbital energy) and \mu = GM \, (standard gravitational parameter), where:
- v\, is orbital velocity from velocity vector of an orbiting object,
- \mathbf\, is cartesian position vector of an orbiting object in coordinates of a reference frame with respect to which the elements of the orbit are to be calculated (e.g. geocentric equatorial for an orbit around Earth, or heliocentric ecliptic for an orbit around the Sun),
- G \, is the gravitational constant,
- M \, the mass of the central body. Note that for a given central body and total specific energy, the semi-major axis is always the same, regardless of eccentricity. Conversely, for a given central body and semi-major axis, the total specific energy is always the same.

Example

The International Space Station has an orbital period of 91.74 minutes, hence the semi-major axis is 6738 km [http://www.google.com/search?num=100&hl=en&lr=&newwindow=1&safe=off&q=%28%2891.74
- 60%2F2%2Fpi%29%5E2
- 398600%29%5E%281%2F3%29]. Every minute more corresponds to ca. 50 km more: the extra 300 km of orbit length takes 40 seconds, the lower speed accounts for an additional 20 seconds.

References


- [http://orca.phys.uvic.ca/~tatum/celmechs/celm9.pdf Jeremy B. Tatum, Celestial Mechanics, Chapter 9 - The Two Body Problem in Two Dimensions (2004)]
- [http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000071000011001198000001&idtype=cvips&gifs=yes Darren M. Williams, Average distance between a star and planet in an eccentric orbit, American Journal of Physics, November 2003, Volume 71, Issue 11, pp. 1198-1200] Category:Conic sections Category:Astrodynamics Category:Celestial mechanics

Eccentricity (mathematics)

(This page refers to eccentricity in mathematics. For other uses, see the disambiguation page eccentricity.) In mathematics, eccentricity is a parameter associated with every conic section, see Conic section#Eccentricity. It can be thought of as a measure of how much the conic section deviates from being circular. In particular,
- The eccentricity of a circle is zero.
- The eccentricity of an ellipse is greater than zero and less than 1.
- The eccentricity of a parabola is 1.
- The eccentricity of a hyperbola is greater than 1 and less than infinity.
- The eccentricity of a straight line is infinity. It is given by: :e = \sqrt Where a is the length of the semimajor axis of the section, b the length of the semiminor axis, and k is equal to +1 for an ellipse, 0 for a parabola, and -1 for a hyperbola. It is also called the first eccentricity when necessary to distinguish it from the second eccentricity, e', which is sometimes used for algebraic convenience. The second eccentricity is defined as: :e' = \sqrt And is related to the first eccentricity by the equation: :1 = (1 - e^2)(1 + e'^2)\,\!

Ellipse

semiminor axis For any ellipse, where the length of the semi-major axis is a, and where the same of the semi-minor axis is b, the eccentricity is given by: :e = \sqrt The eccentricity is the ratio of the distance between the foci (F_1 and F_2) to the major axis; i.e. \left ( \frac \right ). The term linear eccentricity is used for .

Hyperbola

For any hyperbola, where the length of the semi-major axis is a, and where the same of the semi-minor axis is b, eccentricity is given by: :e = \sqrt

Surfaces

The eccentricity of a surface is the eccentricity of a designated section of the surface. For example, on a triaxial ellipsoid, the meridional eccentricity is that of the ellipse formed by a section containing both the longest and the shortest axes (one of which will be the polar axis), and the equatorial eccentricity is the eccentricity of the ellipse formed by a section through the centre, perpendicular to the polar axis (i.e. in the equatorial plane).

External links


- [http://mathworld.wolfram.com/Eccentricity.html MathWorld: Eccentricity] Category:Conic sections als:Exzentrizität (Mathematik)

Category:Conic sections

Category:Geometric shapes Category:Curves Category:Algebraic curves

Category:Astrodynamics

Astrodynamics is a term application of physics, particularly Newtonian mechanics, to space objects such as stars, planets and satellites. Category: Celestial mechanics Category: Astronomy

Category:Celestial mechanics

Celestial mechanics is an application of physics, particularly Newtonian mechanics, to astronomical objects such as stars and planets. Category:Classical mechanics Category:Astronomy ko:분류:천체역학

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