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| | This article will now explain stretching exercises that will strengthen and thicken your cartilage, which results in an increase of your stature by at least 2-3 inches. Any time you install or adjust an antenna, perform a channel scan with your digital converter to see if you have improved reception and perhaps more channels available to you. Giraffes can go several days without drinking because the leaves they eat contain a lot of water. Although some studies have shown that calcium and vitamins A and D do not help in increasing the height of an adult, they all agree that they are extremely important when it comes to preserving height. You can increase your height with stimulation of the pituitary gland that enables the release of HGH to facilitate natural growth. <br><br>As you perform height gain exercises, maintain an effective breathing technique. You must be confident enough in your looks and always maintain your straight posture, which will help you to become confident. The height of a person plays an essential role in several areas. There are a number of factors that will help determine how tall someone can grow to become. ' 'How much height do I realistically expect to gain from using a growing pill. <br><br>It is becoming increasingly hard to prepare nutritiously balanced food because of the daily demands of living. Do not concentrate only on stretching your legs but make sure that you also stretch your spinal column, arms and neck with the right type of stretching exercises that focus on the particular part of your body. These hormones get sent to our bloodstream after production, and they target certain organs and tissues that aid growth and development. They should practice exercises, or carry out yoga that would help them grow taller. At any given time during the day or night, you can find thousands of people already playing this game online. <br><br>t have an adequate diet or you live in a polluted atmosphere you simply would probably not be in a position to succeed in your tallest height prospective. They're typically created of nutritional supplements and they supply you with bound benefits. Now I can walk with her without having feeling individuals searching at us and saying I'm smaller than her. Are you those types of people who look good enough but could look much better if were a few inches taller. There is some truth on this - height is genetic, however that doesn't explain why so many tall folks, have a lot shorter parents and grandparents. <br><br>We'll start with the simplest method, which unfortunately may also be the most impractical as it involves measuring the shadow of a tall building on the ground. Backward bend- the backward bend is another great exercise to increase height. This exercise works since it makes your oftenly weak muscle in the body that is the muscle in the center of the spine, to work. As a abridge person, I acquire the annual to abound taller so it leads me to amateur the achievement the program. But then all this cannot be permanently overcome because of the short time and effects of every day reversal.<br><br>If you loved this report and you would like to acquire extra info concerning growing taller after 20 - [http://www.estampas.info/ http://www.estampas.info] - kindly visit our page. |
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| [[Image:Paraboloid of Revolution.svg|thumb|right|Paraboloid of revolution]]
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| In [[mathematics]], a '''paraboloid''' is a [[quadric surface]] of special kind. There are two kinds of paraboloids: elliptic and hyperbolic.
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| The ''elliptic paraboloid'' is shaped like an oval cup and can have a [[maximum]] or minimum point. In a suitable coordinate system with three axes <math>x</math>, <math>y</math>, and <math>z</math>, it can be represented by the equation<ref>{{cite book |title=Thomas' Calculus 11th ed. |last=Thomas |first=George B.|coauthors= Maurice D. Weir, [[Joel Hass]], Frank R. Giordiano |year=2005 |publisher= Pearson Education, Inc |isbn=0-321-18558-7 |page=892}}</ref>
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| :<math>
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| \frac{z}{c} = \frac{x^2}{a^2} + \frac{y^2}{b^2}.
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| </math>
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| where <math>a</math> and <math>b</math> are constants that dictate the level of curvature in the <math>x</math>-<math>z</math> and <math>y</math>-<math>z</math> planes respectively. This is an elliptic paraboloid which opens upward.
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| [[Image:HyperbolicParaboloid.png|thumb|right|Hyperbolic paraboloid]]
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| The ''hyperbolic paraboloid'' (not to be confused with a [[hyperboloid]]) is a [[doubly ruled surface]] shaped like a [[saddle]]. In a suitable coordinate system, a hyperbolic paraboloid can be represented by the equation<ref>{{cite book |title=Thomas' Calculus 11th ed. |last=Thomas |first=George B.|coauthors= Maurice D. Weir, Joel Hass, Frank R. Giordiano |year=2005 |publisher= Pearson Education, Inc |isbn=0-321-18558-7 |page=896}}</ref>
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| :<math>
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| \frac{z}{c} = \frac{y^2}{b^2} - \frac{x^2}{a^2}.
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| </math>
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| For c>0, this is a hyperbolic paraboloid that opens down along the x-axis and up along the y-axis (i.e., the parabola in the plane x=0 opens upward and the parabola in the plane y=0 opens downward).
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| == Properties ==
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| [[File:Parabola with focus and arbitrary line.svg|thumb|200px|Parallel rays coming into a circular paraboloidal mirror are reflected to the focal point, F, or ''vice versa''.]]
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| [[File:Hyperbolic-paraboloid.svg|thumb|The hyperbolic paraboloid is a [[doubly ruled surface]], and thus can be used to construct a [[saddle roof]] from straight beams.]]
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| [[Image:W-wa Ochota PKP-WKD.jpg|thumb|right|[[Warszawa Ochota railway station]], an example of a hyperbolic paraboloid structure.]]
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| With ''a = b'' an elliptic paraboloid is a ''paraboloid of revolution'': a surface obtained by revolving a [[parabola]] around its axis. It is the shape of the [[parabolic reflector]]s used in [[mirror]]s, [[antenna (electronics)|antenna]] dishes, and the like; and is also the shape of the surface of a rotating liquid, a principle used in [[liquid mirror telescope]]s and in making solid telescope mirrors (see [[Rotating furnace]]). This shape is also called a ''circular paraboloid''.
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| There is a point called the [[Focus (optics)|''focus'']] (or ''focal point'') on the axis of a circular paraboloid such that, if the paraboloid is a mirror, light from a point source at the focus is reflected into a parallel beam, parallel to the axis of the paraboloid. This also works the other way around: a parallel beam of light incident on the paraboloid parallel to its axis is concentrated at the focal point. This applies also for other waves, hence [[parabolic antenna]]s. For a geometrical proof, click [[Parabola#Proof of the reflective property|here]].
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| The hyperbolic paraboloid is a [[doubly ruled surface]]: it contains two families of mutually [[skew lines]]. The lines in each family are parallel to a common plane, but not to each other.
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| ==Curvature==
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| The elliptic paraboloid, parametrized simply as
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| :<math> \vec \sigma(u,v) = \left(u, v, {u^2 \over a^2} + {v^2 \over b^2}\right) </math>
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| has [[Gaussian curvature]]
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| :<math> K(u,v) = {4 \over a^2 b^2 \left(1 + {4 u^2 \over a^4} + {4 v^2 \over b^4}\right)^2} </math>
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| and [[mean curvature]] | |
| :<math> H(u,v) = {a^2 + b^2 + {4 u^2 \over a^2} + {4 v^2 \over b^2} \over a^2 b^2 \left(1 + {4 u^2 \over a^4} + {4 v^2 \over b^4}\right)^{3/2}} </math>
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| which are both always positive, have their maximum at the origin, become smaller as a point on the surface moves further away from the origin, and tend asymptotically to zero as the said point moves infinitely away from the origin. | |
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| The hyperbolic paraboloid, when parametrized as | |
| :<math> \vec \sigma (u,v) = \left(u, v, {u^2 \over a^2} - {v^2 \over b^2}\right) </math>
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| has Gaussian curvature
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| :<math> K(u,v) = {-4 \over a^2 b^2 \left(1 + {4 u^2 \over a^4} + {4 v^2 \over b^4}\right)^2} </math>
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| and mean curvature
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| :<math> H(u,v) = {-a^2 + b^2 - {4 u^2 \over a^2} + {4 v^2 \over b^2} \over a^2 b^2 \left(1 + {4 u^2 \over a^4} + {4 v^2 \over b^4}\right)^{3/2}}. </math>
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| ==Multiplication table==
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| If the hyperbolic paraboloid
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| :<math> z = {x^2 \over a^2} - {y^2 \over b^2} </math>
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| is rotated by an angle of π/4 in the +''z'' direction (according to the [[right hand rule]]), the result is the surface
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| :<math> z = {1\over 2} (x^2 + y^2) \left({1\over a^2} - {1\over b^2}\right) + x y \left({1\over a^2}+{1\over b^2}\right) </math>
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| and if <math>\ a=b</math> then this simplifies to
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| :<math> z = {2\over a^2} x y </math>.
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| Finally, letting <math> a=\sqrt{2} </math>, we see that the hyperbolic paraboloid
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| :<math> z = {x^2 - y^2 \over 2}. </math>
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| is congruent to the surface | |
| :<math>\ z = x y </math>
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| which can be thought of as the geometric representation (a three-dimensional [[nomogram|nomograph]], as it were) of a [[multiplication table]].
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| The two paraboloidal <math>\mathbb{R}^2 \rarr \mathbb{R}</math> functions
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| :<math> z_1 (x,y) = {x^2 - y^2 \over 2} </math>
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| and | |
| :<math>\ z_2 (x,y) = x y </math>
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| are [[harmonic conjugate]]s, and together form the [[analytic function]]
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| :<math> f(z) = {1\over 2} z^2 = f(x + i y) = z_1 (x,y) + i z_2 (x,y) </math>
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| which is the [[analytic continuation]] of the <math>\mathbb{R}\rarr \mathbb{R}</math> parabolic function <math>\ f(x) = {1\over 2} x^2. </math>
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| == Dimensions of a paraboloidal dish ==
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| The dimensions of a symmetrical paraboloidal dish are related by the equation: <math> \scriptstyle 4FD = R^2,</math> where <math> \scriptstyle F</math> is the focal length, <math> \scriptstyle D</math> is the depth of the dish (measured along the axis of symmetry from the vertex to the plane of the rim), and <math> \scriptstyle R</math> is the radius of the rim. Of course, they must all be in the same units. If two of these three quantities are known, this equation can be used to calculate the third.
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| A more complex calculation is needed to find the diameter of the dish ''measured along its surface''. This is sometimes called the "linear diameter", and equals the diameter of a flat, circular sheet of material, usually metal, which is the right size to be cut and bent to make the dish. Two intermediate results are useful in the calculation: <math>\scriptstyle P=2F</math> (or the equivalent: <math>\scriptstyle P=\frac{R^2}{2D})</math> and <math>\scriptstyle Q=\sqrt {P^2+R^2},</math> where <math> \scriptstyle F,</math> <math> \scriptstyle D,</math> and <math> \scriptstyle R</math> are defined as above. The diameter of the dish, measured along the surface, is then given by: <math>\scriptstyle \frac {RQ} {P} + P \ln \left ( \frac {R+Q} {P} \right ),</math> where <math>\scriptstyle \ln(x)</math> means the [[natural logarithm]] of <math> \scriptstyle x </math>, i.e. its logarithm to base "[[e (mathematical constant)|e]]".
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| The volume of the dish, the amount of liquid it could hold if the rim were horizontal and the vertex at the bottom (e.g. the capacity of a paraboloidal [[wok]]), is given by <math>\scriptstyle \frac {1} {2} \pi R^2 D ,</math> where the symbols are defined as above. This can be compared with the formulae for the volumes of a [[Cylinder (geometry)|cylinder]] <math>\scriptstyle (\pi R^2 D),</math> a [[sphere|hemisphere]] <math>\scriptstyle (\frac {2}{3} \pi R^2 D,</math> where <math>\scriptstyle D=R),</math> and a [[Cone (geometry)|cone]] <math>\scriptstyle ( \frac {1} {3} \pi R^2 D ).</math> Of course, <math>\scriptstyle \pi R^2 </math> is the aperture area of the dish, the area enclosed by the rim, which is proportional to the amount of sunlight a reflector dish can intercept.
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| == Applications ==
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| [[File:Pringles chips.JPG|thumb|right|200px|Pringles. An example of a hyperbolic paraboloid.]]
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| [[File:Pengrowth_Saddledome.jpg|thumb|right|200px|The Calgary [[Saddledome]]]]
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| Paraboloidal mirrors are frequently used to bring parallel light to a point focus, e.g. in [[astronomical telescope]]s, or to [[collimate]] light that has originated from a source at the focus into a parallel beam, e.g. in a [[searchlight]].
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| The top surface of a fluid in an open-topped rotating container will form a paraboloid. This property can be used to make a [[liquid mirror telescope]] with a rotating pool of a reflective liquid, such as mercury, for the primary mirror. The same technique is used to make solid paraboloids, in [[rotating furnace]]s.
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| The widely-sold fried snack food [[Pringles]] potato crisps resemble a truncated hyperbolic paraboloid. According to Pringles [[marketing]], the shape allows the snack to be securely stacked in a canister to prevent breakage during packaging and transport.<ref>http://www.pringles.com.au/faq</ref>
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| == Examples in Architecture ==
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| * [[St. Mary's Cathedral, Tokyo]]
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| * [[Cathedral of Saint Mary of the Assumption]]
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| * [[Saddledome]] in Calgary, Alberta, Canada
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| * [[London Velopark]]
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| == See also ==
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| * [[Quadratic form]]
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| * [[Ellipsoid]]
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| * [[Hyperboloid]]
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| * [[Hyperboloid structure]]
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| * [[Saddle roof]]
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| * [[Holophones]]
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| * [[Parabola]]
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| * [[Parabolic reflector]]
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| ==References==
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| {{Reflist}}
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| [[Category:Geometric shapes]]
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| [[Category:Surfaces]]
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| [[Category:Quadrics]]
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| [[Category:Parabolas]]
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This article will now explain stretching exercises that will strengthen and thicken your cartilage, which results in an increase of your stature by at least 2-3 inches. Any time you install or adjust an antenna, perform a channel scan with your digital converter to see if you have improved reception and perhaps more channels available to you. Giraffes can go several days without drinking because the leaves they eat contain a lot of water. Although some studies have shown that calcium and vitamins A and D do not help in increasing the height of an adult, they all agree that they are extremely important when it comes to preserving height. You can increase your height with stimulation of the pituitary gland that enables the release of HGH to facilitate natural growth.
As you perform height gain exercises, maintain an effective breathing technique. You must be confident enough in your looks and always maintain your straight posture, which will help you to become confident. The height of a person plays an essential role in several areas. There are a number of factors that will help determine how tall someone can grow to become. ' 'How much height do I realistically expect to gain from using a growing pill.
It is becoming increasingly hard to prepare nutritiously balanced food because of the daily demands of living. Do not concentrate only on stretching your legs but make sure that you also stretch your spinal column, arms and neck with the right type of stretching exercises that focus on the particular part of your body. These hormones get sent to our bloodstream after production, and they target certain organs and tissues that aid growth and development. They should practice exercises, or carry out yoga that would help them grow taller. At any given time during the day or night, you can find thousands of people already playing this game online.
t have an adequate diet or you live in a polluted atmosphere you simply would probably not be in a position to succeed in your tallest height prospective. They're typically created of nutritional supplements and they supply you with bound benefits. Now I can walk with her without having feeling individuals searching at us and saying I'm smaller than her. Are you those types of people who look good enough but could look much better if were a few inches taller. There is some truth on this - height is genetic, however that doesn't explain why so many tall folks, have a lot shorter parents and grandparents.
We'll start with the simplest method, which unfortunately may also be the most impractical as it involves measuring the shadow of a tall building on the ground. Backward bend- the backward bend is another great exercise to increase height. This exercise works since it makes your oftenly weak muscle in the body that is the muscle in the center of the spine, to work. As a abridge person, I acquire the annual to abound taller so it leads me to amateur the achievement the program. But then all this cannot be permanently overcome because of the short time and effects of every day reversal.
If you loved this report and you would like to acquire extra info concerning growing taller after 20 - http://www.estampas.info - kindly visit our page.