WIFI devices

24

Comments

  • kryystkryyst Ontario, Canada
    edited December 2007
    halo2_god wrote:
    Could any one show me a wifi device (USB) aroud 50 - 60 dollars from best buy or around 30 dollars from something like newegg.com, radio shack ect... I Want one with long range atleast 100 feet and atleast 54MBPS i also want it to have WEP if you can find one that supports WPA for like 10-15 dollars more show me.

    Any new off the shelf 54g router will do what you want.
  • the_technocratthe_technocrat IC-MotY1 Indy Icrontian
    edited December 2007
    halo2_god wrote:
    Could any one show me a wifi device (USB) aroud 50 - 60 dollars from best buy or around 30 dollars from something like newegg.com, radio shack ect... I Want one with long range atleast 100 feet and atleast 54MBPS i also want it to have WEP if you can find one that supports WPA for like 10-15 dollars more show me.

    http://www.newegg.com/Product/ProductList.aspx?Submit=ENE&Description=usb%20802%2E11g&bop=And&Order=PRICE
  • the_technocratthe_technocrat IC-MotY1 Indy Icrontian
    edited December 2007
    kryyst wrote:
    Any new off the shelf 54g router will do what you want.

    That's a good point as well, you could easily wire the metal shelf found in most retail stores up to the antenna's BNC connector and have a high-gain antenna.
  • BuddyJBuddyJ Dept. of Propaganda OKC Icrontian
    edited December 2007
  • the_technocratthe_technocrat IC-MotY1 Indy Icrontian
    edited December 2007

    That's not a dish, that's a parabolic antenna.

    The parabolic antenna is a high-gain reflector antenna used for radio, television and data communications, and also for radiolocation (RADAR), on the UHF and SHF parts of the electromagnetic spectrum. The relatively short wavelength of electromagnetic (radio) energy at these frequencies allows reasonably sized reflectors to exhibit the very desirable highly directional response for both receiving and transmitting.
    Parabolic antennas at the Very Large Array Radio Telescope in New Mexico, USA
    Parabolic antennas at the Very Large Array Radio Telescope in New Mexico, USA

    A typical parabolic antenna consists of a parabolic reflector illuminated by a small feed antenna.

    The reflector is a metallic surface formed into a paraboloid of revolution and (usually) truncated in a circular rim that forms the diameter of the antenna. This paraboloid possesses a distinct focal point by virtue of having the reflective property of parabolas in that a point light source at this focus produces a parallel light beam aligned with the axis of revolution.

    The feed antenna is placed at the reflector focus. This antenna is typically a low-gain type such as a half-wave dipole or a small waveguide horn. In more complex designs, such as the Cassegrain antenna, a sub-reflector is used to direct the energy into the parabolic reflector from a feed antenna located away from the primary focal point. The feed antenna is connected to the associated radio-frequency (RF) transmitting or receiving equipment by means of a coaxial cable transmission line or hollow waveguide.
    Main types of parabolic antennas
    Main types of parabolic antennas

    Considering the parabolic antenna as a circular aperture gives the following approximation for the maximum gain:

    G\approx (\pi^2 D^2)/\lambda^2\,

    or

    G\approx (9.87D^2)/\lambda^2\,

    where:

    G \,\! is power gain over isotropic
    D \,\! is reflector diameter in same units as wavelength
    \lambda \,\! is wavelength

    Practical considerations of antenna effective area and sidelobe suppression reduce the actual gain obtained to between 35 and 55 percent of this theoretical value. For theoretical considerations of mutual interference (at frequencies between 2 and c. 30 GHz - typically in the Fixed Satellite Service) where specific antenna performance has not been defined, a reference antenna based on Recommendation ITU-R S.465 is used to calculate the interference, which will include the likely sidelobes for off-axis effects.

    Applying the formula to just one of the 25-meter-diameter VLA antennas shown in the illustration for a wavelength of 21 cm (1.42 GHz, a common radio astronomy frequency) yields an approximate maximum gain of 140,000 times or about 50 dBi (decibels above the isotropic level).

    With the advent of TVRO and DBS satellite television, the parabolic antenna became a ubiquitous feature of urban, suburban, and even rural, landscapes. Extensive terrestrial microwave links, such as those between cellphone base stations, and wireless WAN/LAN applications have also proliferated this antenna type. Earlier applications included ground-based and airborne radar and radio astronomy. The largest "dish" antenna in the world is the radio telescope at Arecibo, PR, but, for beam-steering reasons, it is actually a spherical, rather than parabolic, reflector.
    Contents
    [hide]

    * 1 Structure
    * 2 Feeding parabolic antennas.
    * 3 See also
    * 4 External links

    [edit] Structure

    The reflector dish can be solid, mesh or wire in construction and it can be either fully circular or somewhat rectangular depending on the radiation pattern of the feeding element. Solid antennas have more ideal characteristics but are troublesome because of weight and high wind load. Mesh and wire types weigh less, are easier to construct and have nearly ideal characteristics if the holes or gaps are kept under 1/10 of the wavelength.
    Wire-type parabolic antenna (Wi-Fi / WLAN antenna at 2,4Ghz). Oriented to provide horizontal polarization: the reflector wires and the feed element are both horizontal. This antenna has a greater extent in the vertical plane and hence, a narrower beamwidth in that plane. The feed element has a wider beam in the vertical direction than the horizontal and hence matches the reflector by illuminating it fully.
    Wire-type parabolic antenna (Wi-Fi / WLAN antenna at 2,4Ghz). Oriented to provide horizontal polarization: the reflector wires and the feed element are both horizontal. This antenna has a greater extent in the vertical plane and hence, a narrower beamwidth in that plane. The feed element has a wider beam in the vertical direction than the horizontal and hence matches the reflector by illuminating it fully.

    More exotic types include the off-set parabolic antenna, Gregorian and Cassegrain types. In the off-set, the feed element is still located at the focal point, which because of the angles utilized, is usually located below the reflector so that the feed element and support do not interfere with the the main beam. This also allows for easier maintenance of the feed, but is usually only found in smaller antennas.

    The Gregorian and Cassegrain types, sometimes generically referred to as "dual optics" antennas, utilize a secondary reflector, or "sub-reflector", allowing for better control over the colimnated beam as well as allowing the antenna feed system to be more compact. These antennas are usually much larger where prime focus and off-set construction are not as practical. The feed element is usually located in a "feed horn" which protrudes out from the main reflector. This setup is used when the feed element is bulky or heavy such as when it contains a pre-amplifier or even the actual receiver or transmitter. Parabolic antenna theory closely follows optics theory. So a Gregorian antenna can be identified by the fact that it uses a concave sub-reflector, while a Cassegrain antenna uses a convex sub-reflector.

    Antenna feeders

    The actual 'antenna' in a parabolic antenna, that is, the device that interfaces the transmission line or waveguide containing the radio-frequency energy to free space, is the feed element. The reflector surface is entirely passive. This feed element should usually be at the center of the reflector at the focal point of that dish. The focal point is the point where all reflected waves will be concentrated. The focal length f (distance of focal point from the center of the reflector) is calculated with the following equation:

    f = \frac{D^2}{16d}


    where:

    f \,\! is the focal length of the reflector
    D \,\! is reflector diameter in same units as wavelength
    d \,\! is depth of the reflector

    The radiation from the feed element induces a current flow in the conductive reflector surface which, in turn, re-radiates in the desired direction, perpendicular to the directrix plane of the paraboloid. The feed element can be any one of a multitude of antenna types. Whichever type is used, it must exhibit a directivity that efficiently illuminates the reflector and must have the correct polarization for the application -- the polarization of the feed determining the polarization of the entire antenna system. The simplest feed is a half-wave dipole which is commonly used at lower frequencies, sometimes in conjunction with a closely coupled parasitic reflector or "splash plate". At higher frequencies a horn-type becomes more feasible and efficient. To adapt the horn to a coaxial antenna cable, a length of waveguide is used to effect the transition.
  • the_technocratthe_technocrat IC-MotY1 Indy Icrontian
    edited December 2007
    Analytic geometry equations

    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 with axis parallel to the y-axis

    (x - h)^2 = 4p(y - k) \,

    or, alternatively with axis parallel to the x-axis

    (y - k)^2 = 4p(x - h) \,

    More generally, 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, where A \not= 0 \, or C \not= 0 \,, 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 equations.

    [edit] Other geometric definitions
    Parabolas are conic sections.
    Parabolas are conic sections.

    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).

    [edit] Equations

    (with vertex (h, k) and distance p between vertex and focus - note that if the vertex is below the focus, or equivalently above the directrix, p is positive, otherwise p is negative; similarly with horizontal axis of symmetry p is positive if vertex is to the left of the focus, or equivalently to the right of the directrix)

    [edit] Cartesian

    [edit] Vertical axis of symmetry

    (x - h)^2 = 4p(y - k) \,

    y = a(x-h)^2 + k \,

    y = ax^2 + bx + c \,

    \mbox{where }a = \frac{1}{4p}; \ \ b = \frac{-h}{2p}; \ \ c = \frac{h^2}{4p} + k; \ \
    h = \frac{-b}{2a}; \ \ k = \frac{4ac - b^2}{4a}.

    x(t) = 2pt + h; \ \ y(t) = pt^2 + k \,

    [edit] Horizontal axis of symmetry

    (y - k)^2 = 4p(x - h) \,

    x = a(y - k)^2 + h \,

    x = ay^2 + by + c \,

    \mbox{where }a = \frac{1}{4p}; \ \ b = \frac{-k}{2p}; \ \ c = \frac{k^2}{4p} + h; \ \
    h = \frac{4ac - b^2}{4a}; \ \ k = \frac{-b}{2a}.

    x(t) = pt^2 + h; \ \ y(t) = 2pt + k \,

    [edit] Semi-latus rectum and polar coordinates

    In polar coordinates, a parabola with the focus at the origin and the directrix on the positive x-axis, is given by the equation

    r (1 + \cos \theta) = l \,

    where l is the semilatus 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.

    [edit] Gauss-mapped form

    A Gauss-mapped form: (tan2φ,2tanφ) has normal (cosφ,sinφ).

    [edit] Derivation of the focus
    Parabolic curve showing directrix (L) and focus (F). The distance from a given point Pn to the focus is always the same as the distance from Pn to a point Qn directly below, on the directrix.
    Parabolic curve showing directrix (L) and focus (F). The distance from a given point Pn to the focus is always the same as the distance from Pn to a point Qn directly below, on the directrix.

    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{ x^2 + (y - f)^2 },
    \| QP \| = y + f.
    \| FP \| = \| QP \|
    \sqrt{x^2 + (a x^2 - f)^2 } = 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 = {1 \over 4 a }

    Now let p=f and the equation for the parabola becomes

    x^2 = 4 p y \quad

    Q.E.D.

    All this was for a parabola centered at the origin. For any generalized parabola, with its equation given in the standard form

    y = ax2 + bx + c,

    the focus is located at the point

    \left (\frac{-b}{2a},\frac{-b^2}{4a}+c+\frac{1}{4a} \right)

    and the directrix is designated by the equation

    y=\frac{-b^2}{4a}+c-\frac{1}{4a}

    [edit] Reflective property of the tangent

    The tangent of the parabola described by equation (1) has slope

    {dy \over dx} = 2 a x = {2 y \over 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
    {F + Q \over 2} = {(0,f) + (x,-f) \over 2} = {(x,0) \over 2} = ({x \over 2}, 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.)

    [edit] What happens to a parabola when "b" varies?

    Vertex of a parabola: Finding the y-coordinate

    We know the x-coordinate at the vertex is x=-\frac{b}{2a}, so substitute it into the equation ax2 + bx + c

    y=a\left (-\frac{b}{2a}\right )^2 + b \left ( -\frac{b}{2a} \right ) + c\qquad\textrm{Then~simplify\ldots}
    =\frac{ab^2}{4a^2} -\frac{b^2}{2a} + c
    =\frac{b^2}{4a} -\frac{2\cdot b^2}{2\cdot 2a} + c\cdot\frac{4a}{4a}
    =\frac{-b^2+4ac}{4a}
    =-\frac{b^2-4ac}{4a}=-\frac{D}{4a}

    Thus, the vertex is at point…

    \left (-\frac{b}{2a},-\frac{D}{4a}\right )

    [edit] Parabolas in the physical world
    A bouncing ball captured with a stroboscopic flash at 25 images per second. Note that the ball becomes significantly non-spherical after each bounce, especially after the first. That, along with spin and air resistance, causes the curve swept out to deviate slightly from the expected perfect parabola
    A bouncing ball captured with a stroboscopic flash at 25 images per second. Note that the ball becomes significantly non-spherical after each bounce, especially after the first. That, along with spin and air resistance, causes the curve swept out to deviate slightly from the expected perfect parabola

    In nature, approximations of parabolas 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 a 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, for example, always distorts the shape, 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.
    Parabolic shape formed by the surface of a Newtonian liquid under rotation
    Parabolic shape formed by the surface of a Newtonian liquid under rotation

    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. Such parabolic orbits are a special case that are rarely found 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. An object following a parabolic orbit moves at the exact escape velocity of the object it is orbiting, while elliptical orbits are slower and hyperbolic orbits are faster.

    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 toward a parabola.

    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 may have been discovered in the 3rd century BC by the geometer Archimedes, who, according to a legend of debatable veracity,[1] 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.

    Paraboloids 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 liquid to climb the walls of the container, forming a parabolic surface. This is the principle behind the liquid mirror telescope.

    Aircraft used to create a weightless state for purposes of experimentation, such as NASA's “vomit comet,” follow a vertically parabolic trajectory for brief periods in order to trace the course of an object in free fall, which produces the same effect as zero gravity for most purposes.
  • BuddyJBuddyJ Dept. of Propaganda OKC Icrontian
    edited December 2007
    Analytic geometry equations

    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 with axis parallel to the y-axis

    (x - h)^2 = 4p(y - k) \,

    or, alternatively with axis parallel to the x-axis

    (y - k)^2 = 4p(x - h) \,

    More generally, 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, where A \not= 0 \, or C \not= 0 \,, 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 equations.

    [edit] Other geometric definitions
    Parabolas are conic sections.
    Parabolas are conic sections.

    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).

    [edit] Equations

    (with vertex (h, k) and distance p between vertex and focus - note that if the vertex is below the focus, or equivalently above the directrix, p is positive, otherwise p is negative; similarly with horizontal axis of symmetry p is positive if vertex is to the left of the focus, or equivalently to the right of the directrix)

    [edit] Cartesian

    [edit] Vertical axis of symmetry

    (x - h)^2 = 4p(y - k) \,

    y = a(x-h)^2 + k \,

    y = ax^2 + bx + c \,

    \mbox{where }a = \frac{1}{4p}; \ \ b = \frac{-h}{2p}; \ \ c = \frac{h^2}{4p} + k; \ \
    h = \frac{-b}{2a}; \ \ k = \frac{4ac - b^2}{4a}.

    x(t) = 2pt + h; \ \ y(t) = pt^2 + k \,

    [edit] Horizontal axis of symmetry

    (y - k)^2 = 4p(x - h) \,

    x = a(y - k)^2 + h \,

    x = ay^2 + by + c \,

    \mbox{where }a = \frac{1}{4p}; \ \ b = \frac{-k}{2p}; \ \ c = \frac{k^2}{4p} + h; \ \
    h = \frac{4ac - b^2}{4a}; \ \ k = \frac{-b}{2a}.

    x(t) = pt^2 + h; \ \ y(t) = 2pt + k \,

    [edit] Semi-latus rectum and polar coordinates

    In polar coordinates, a parabola with the focus at the origin and the directrix on the positive x-axis, is given by the equation

    r (1 + \cos \theta) = l \,

    where l is the semilatus 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.

    [edit] Gauss-mapped form

    A Gauss-mapped form: (tan2φ,2tanφ) has normal (cosφ,sinφ).

    [edit] Derivation of the focus
    Parabolic curve showing directrix (L) and focus (F). The distance from a given point Pn to the focus is always the same as the distance from Pn to a point Qn directly below, on the directrix.
    Parabolic curve showing directrix (L) and focus (F). The distance from a given point Pn to the focus is always the same as the distance from Pn to a point Qn directly below, on the directrix.

    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{ x^2 + (y - f)^2 },
    \| QP \| = y + f.
    \| FP \| = \| QP \|
    \sqrt{x^2 + (a x^2 - f)^2 } = 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 = {1 \over 4 a }

    Now let p=f and the equation for the parabola becomes

    x^2 = 4 p y \quad

    Q.E.D.

    All this was for a parabola centered at the origin. For any generalized parabola, with its equation given in the standard form

    y = ax2 + bx + c,

    the focus is located at the point

    \left (\frac{-b}{2a},\frac{-b^2}{4a}+c+\frac{1}{4a} \right)

    and the directrix is designated by the equation

    y=\frac{-b^2}{4a}+c-\frac{1}{4a}

    [edit] Reflective property of the tangent

    The tangent of the parabola described by equation (1) has slope

    {dy \over dx} = 2 a x = {2 y \over 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
    {F + Q \over 2} = {(0,f) + (x,-f) \over 2} = {(x,0) \over 2} = ({x \over 2}, 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.)

    [edit] What happens to a parabola when "b" varies?

    Vertex of a parabola: Finding the y-coordinate

    We know the x-coordinate at the vertex is x=-\frac{b}{2a}, so substitute it into the equation ax2 + bx + c

    y=a\left (-\frac{b}{2a}\right )^2 + b \left ( -\frac{b}{2a} \right ) + c\qquad\textrm{Then~simplify\ldots}
    =\frac{ab^2}{4a^2} -\frac{b^2}{2a} + c
    =\frac{b^2}{4a} -\frac{2\cdot b^2}{2\cdot 2a} + c\cdot\frac{4a}{4a}
    =\frac{-b^2+4ac}{4a}
    =-\frac{b^2-4ac}{4a}=-\frac{D}{4a}

    Thus, the vertex is at point…

    \left (-\frac{b}{2a},-\frac{D}{4a}\right )

    [edit] Parabolas in the physical world
    A bouncing ball captured with a stroboscopic flash at 25 images per second. Note that the ball becomes significantly non-spherical after each bounce, especially after the first. That, along with spin and air resistance, causes the curve swept out to deviate slightly from the expected perfect parabola
    A bouncing ball captured with a stroboscopic flash at 25 images per second. Note that the ball becomes significantly non-spherical after each bounce, especially after the first. That, along with spin and air resistance, causes the curve swept out to deviate slightly from the expected perfect parabola

    In nature, approximations of parabolas 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 a 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, for example, always distorts the shape, 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.
    Parabolic shape formed by the surface of a Newtonian liquid under rotation
    Parabolic shape formed by the surface of a Newtonian liquid under rotation

    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. Such parabolic orbits are a special case that are rarely found 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. An object following a parabolic orbit moves at the exact escape velocity of the object it is orbiting, while elliptical orbits are slower and hyperbolic orbits are faster.

    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 toward a parabola.

    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 may have been discovered in the 3rd century BC by the geometer Archimedes, who, according to a legend of debatable veracity,[1] 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.

    Paraboloids 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 liquid to climb the walls of the container, forming a parabolic surface. This is the principle behind the liquid mirror telescope.

    Aircraft used to create a weightless state for purposes of experimentation, such as NASA's “vomit comet,” follow a vertically parabolic trajectory for brief periods in order to trace the course of an object in free fall, which produces the same effect as zero gravity for most purposes.

    TT is right, but I think what he's forgetting and what would really fix your problem is to just get an isotropic radiator from Radio Shack. They should have a USB 2.0 model for under $50.
  • the_technocratthe_technocrat IC-MotY1 Indy Icrontian
    edited December 2007
    No he's looking for a WIRELESS antenna, and besides, you can run Cat5e up a central HVAC's intake vent, not in a heated water pipe, you sill knob-tobbler!
  • the_technocratthe_technocrat IC-MotY1 Indy Icrontian
    edited December 2007
    although an isotropic radiator is a good idea. Hold on, I have to go get a drink of water, brb
  • ThraxThrax 🐌 Austin, TX Icrontian
    edited December 2007
    If you attach a shoelace to your wifi card you should get an awesome signal but a narrow projection of signal. It reminds me of the one time I attached my cell phone to a can of progresso soup and a sock and got over 9000 bars and the FCC came and were all like "what what" and I was like. :( also because its super easy if you made your own parabolic superantenna out of aluminum foil.

    DON'T HARASS ME I KNOW WHAT I'M FRIGGIN DOING GRRRRRRRANGRYGRRRRRRRR

    attachment.php?attachmentid=24458&stc=1&d=1197573381

    ASSCLOWNS
  • the_technocratthe_technocrat IC-MotY1 Indy Icrontian
    edited December 2007
    OK, so anyway, you'd have to find a solution to the Hairy Ball Theorem to get the radiator to put out a wireless signal. Not a bad idea at all, and I think Radio Shack sells an adapter. Although it's one of those crappy adapter where its one common end and you have to bring your radiator in to find the end that fits yours, etc.
  • the_technocratthe_technocrat IC-MotY1 Indy Icrontian
    edited December 2007
    Thrax wrote:

    DON'T HARASS ME I KNOW WHAT I'M FRIGGIN DOING GRRRRRRRANGRYGRRRRRRRR

    attachment.php?attachmentid=24458&stc=1&d=1197573381

    ASSCLOWNS

    Obviously you do not. The nonuniform consistency of progresso soup will BLOCK any signal instead of AMPLIFYING it. Any first-year EE or art major can tell you that. If you had gone with a more traditional, non-facist soup like Cream of Tomato, you wouldn't have gotten caught.
  • the_technocratthe_technocrat IC-MotY1 Indy Icrontian
    edited December 2007
    AND ANOTHER THING oops caps I don't appreciate being called names. you have no idea who i am. I've never met you. You weren't there when I was molested at the circus, so you couldn't possibly KNOW WHAT I AM oops caps going through!!!11!

    and I have more to say but I have to go make some soup, THANKS A LOT
  • BuddyJBuddyJ Dept. of Propaganda OKC Icrontian
    edited December 2007
    <object width="638" height="533"><param name="movie" value="http://www.youtube.com/v/XbgvvzVvNSI&rel=1&autoplay=1"></param><param name="wmode" value="transparent"></param><embed src="http://www.youtube.com/v/XbgvvzVvNSI&rel=1&autoplay=1&quot; type="application/x-shockwave-flash" wmode="transparent" width="638" height="533"></embed></object>
  • the_technocratthe_technocrat IC-MotY1 Indy Icrontian
    edited December 2007
    <object width="638" height="533"><param name="movie" value="http://www.youtube.com/v/XbgvvzVvNSI&rel=1&autoplay=1"></param><param name="wmode" value="transparent"></param><embed src="http://www.youtube.com/v/XbgvvzVvNSI&rel=1&autoplay=1&quot; type="application/x-shockwave-flash" wmode="transparent" width="638" height="533"></embed></object>

    see, i told you it works, look, you can see the signal going in right there.
  • ThraxThrax 🐌 Austin, TX Icrontian
    edited December 2007
    attachment.php?attachmentid=12669&d=1098763559
  • GrayFoxGrayFox /dev/urandom Member
    edited December 2007
    When did Icrontic turn into 4chan ......

    edit: Might as well get with the flow.

    PoolsClosed.jpg

    edit: Just noticed theres 2 pages before this.
  • the_technocratthe_technocrat IC-MotY1 Indy Icrontian
    edited December 2007
    This is a discussion about wireless. What does that have to do with a pool? Unless you're talking about a pool of leasable IP addresses, in which case, yes, it should be closed with some sort of MAC filtering. ???
  • ThraxThrax 🐌 Austin, TX Icrontian
    edited December 2007
    noob you can mac spoof and get in np. If you use Cain & Abel you can sniff the MAC over wifi. I hate morons like you.
  • GrayFoxGrayFox /dev/urandom Member
    edited December 2007
    Thrax wrote:
    noob you can mac spoof and get in np. If you use Cain & Abel you can sniff the MAC over wifi. I hate morons like you.
    I thought you could only do this on linux and windows drivers are too crippled ?. (Wouldn't surprise me if Cain's dev team figured it out tho).

    The Cain & Abel development team are always doing more and more things that scare me with how easy they make it. I remember when cracking wep at a reasonable speed required linux/bsd/mac os......
  • the_technocratthe_technocrat IC-MotY1 Indy Icrontian
    edited December 2007
    Thrax wrote:
    noob you can mac spoof and get in np. If you use Cain & Abel you can sniff the MAC over wifi. I hate morons like you.

    you're right! (/subscription to internet)

    I used to know this gay friend of mine that could explain about the plenum cable, I'll try to get him on here.
  • primesuspectprimesuspect Beepin n' Boopin Detroit, MI Icrontian
    edited December 2007
    No he's looking for a WIRELESS antenna, and besides, you can run Cat5e up a central HVAC's intake vent, not in a heated water pipe, you sill knob-tobbler!

    You have to be sure to use Plenum Cat5E when running network cable through any forced-air ductwork.

    PVC (or Polyvinyl Chloride), which is illustrated here:


    PVC_formula.gif

    Has a few characteristics which make it a code violation to install through any ductwork, notably toxic fumes being given off during combustion. Being that PVC is a thermoplastic material, it is much more flexible however.

    Plenum, while having negative traits such as higher difficulty of crimping, stripping, and less flexibility is nonetheless safer (and sometimes the only legal choice) to use when installing cable through ductwork.

    Plenum can be made with a variety of materials - DuPont Teflon is a commonly used one. It is important to note that "Plenum" is not a material or form of construction, but a rating based on the cable's ability to be used in the "plenum space" of commercial and residential construction projects that include space for HVAC ducting.

    In an interesting bit of controversy, in 2006, significant concern developed over the potential toxicity of FEP and related fluorochemicals used in the construction of jacketing rated "plenum". The NFPA Technical Committee on Air Conditioning, in response to public comment, has referred the issue of toxicity of cabling materials to the NFPA Committee on Toxicity for review before 2008.
  • the_technocratthe_technocrat IC-MotY1 Indy Icrontian
    edited December 2007
    oh hi!
  • primesuspectprimesuspect Beepin n' Boopin Detroit, MI Icrontian
    edited December 2007
    Christ. Worst timing EVER.
  • LincLinc Owner Detroit Icrontian
    edited December 2007
    HAHAHhahaha. OMG. Brian just looked up from his computer with this horrified/laughing face and sputters "oh my god! oh man! I just owned myself BAD!" ;D;D;D
  • primesuspectprimesuspect Beepin n' Boopin Detroit, MI Icrontian
    edited December 2007
    We couldn't have coordinated that if we had tried.
  • ThraxThrax 🐌 Austin, TX Icrontian
    edited December 2007
    this is srs thread, brian is srs
  • primesuspectprimesuspect Beepin n' Boopin Detroit, MI Icrontian
    edited December 2007
    WAITAMINNIT

    I see what you did there. My hat's off to you, sir. Well played.
  • LincLinc Owner Detroit Icrontian
    edited December 2007
    No wait! It's even better! TT edited his post AFTER Brian posted, making him THINK he had owned HIMSELF. Hahahaha.
  • BuddyJBuddyJ Dept. of Propaganda OKC Icrontian
    edited December 2007
    hilarity ensues
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