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| [[Image:Hyperbolic tiling omnitruncated 3-7.png|thumb|Poincaré disc model of the [[truncated triheptagonal tiling]].]] | | Most people think how the [http://www.google.de/search?q=electric+toothbrush electric toothbrush] cleans teeth better than any manual toothbrush, but this is not the protective case. Electric toothbrush, only helps to accelerate more than again of washing the teeth, although not cleaning great quality.<br><br>In the event you loved this short article and you would like to receive details with regards to [https://www.youtube.com/watch?v=4To0cBm6IsM In tampa] kindly visit the page. You in addition want to look at their offices. Their office should make you're warm and welcome. However, many don't offer this. This is important because ought to you don't feel warm and welcomed, it's simply as bad when you go to sit in the chair and indulge in a dentist work rrn your teeth. Content articles aren't sure as as their offices look like and a person already your market San Diego area, then ask think about a take an excursion to.<br><br>If teeth whitening techniques is not as effective a person have minor chips with your teeth, you cosmetic dentist may suggest veneers. Veneers are made or porcelain or artificial. Your dentist may use a veneer to disguise a rather difficult or stubborn stain or to cosmetically improve the appearance associated with an broken for ladies chipped dentist. Crooked teeth or even gaps within your teeth can be fixed with veneers.<br><br>One on the most memorable features of one's face is the smile obviously you smile your teeth are displayed for all to recognize. For this reason, a lot of people take superior care with their teeth preserve a white smile. Have no idea have attended great lengths to get whiter teeth, and a very high [https://www.youtube.com/watch?v=4To0cBm6IsM In tampa] visit to your dentist can be a small price for some to look their most useful.<br><br>If dentist we really believe he can which will help us, then we do our part, which is to ask and believe. We could at times go to God in prayer, upset and discouraged, pouring our heart out to Him, However , we prefer to let Him do His part, which would be to provide, however He sees fit to obtain.<br><br>The dentist will more than your records and ask you some questions concerning your dental appearing. The dentist might need some x-rays of your mouth in order to acquire a good in the inside of your mouth and how your teeth have formed. If they see any issues they will discuss tips on how to handle those. The dentist usually do good cleaning of the teeth and take away any plaque that they find. This will help provide you a new beginning toward healthy teeth. Children will also likely get x-rays among the dentist to produce they have a point commence from because they watch one's teeth grow. This will be put onto the child's chart.<br><br>If sense that you've teeth want dental crowns, schedule a trip with your dentist to see if he agrees with you. If so, your dentist believe that the process and what preparation you'd like before you'll crown. For severe decay, you could possibly want a root canal. Otherwise, the dentist will prepare the tooth and schedule its crowning day, and soon you will have use in the damaged tooth once after again. |
| [[File:Hyperb icosahedral hc.png|thumb|Poincaré '[[Ball (mathematics)|ball]]' model view of the hyperbolic regular [[Order-3 icosahedral honeycomb|icosahedral honeycomb]], {3,5,3}]]
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| In geometry, the '''Poincaré disk model''' or '''Poincaré ball model''', also called the '''conformal disk model''', is a model of ''n''-dimensional [[hyperbolic geometry]] in which the points of the geometry are in an ''n''-dimensional disk, or [[ball (mathematics)|unit ball]], and the straight lines consist of all segments of circles contained within the disk that are orthogonal to the boundary of the disk, plus all diameters of the disk. Along with the [[Klein model]] and the [[Poincaré half-space model]], it was proposed by [[Eugenio Beltrami]] who used these models to show hyperbolic geometry was [[equiconsistency|equiconsistent]] with Euclidean geometry.
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| ==Metric==
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| If ''u'' and ''v'' are two vectors in real ''n''-dimensional vector space '''R'''<sup>''n''</sup> with the usual Euclidean norm, both of which have norm less than 1, then we may define an [[isometry|isometric invariant]] by
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| :<math>\delta (u, v) = 2 \frac{\lVert u-v \rVert^2}{(1-\lVert u \rVert^2)(1-\lVert v \rVert^2)},\,</math> | |
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| where <math>\lVert \cdot \rVert</math> denotes the usual Euclidean norm. Then the distance function is
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| :<math>d(u, v) = \operatorname{arcosh} (1+\delta (u,v)).\,</math>
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| Such a distance function is defined for any two vectors of norm less than one, and makes the set of such vectors into a metric space which is a model of hyperbolic space of constant curvature −1. The model has the conformal property that the angle between two intersecting curves in hyperbolic space is the same as the angle in the model.
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| The associated [[metric tensor]] of the Poincaré disk model is given by
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| :<math>ds^2 = 4 \frac{\sum_i dx_i^2}{(1-\sum_i x_i^2)^2}</math>
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| where the ''x''<sub>''i''</sub> are the Cartesian coordinates of the ambient Euclidean space. The [[geodesic]]s of the disk model are circles perpendicular to the boundary sphere ''S''<sup>''n''−1</sup>.
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| ==Relation to the hyperboloid model==
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| [[File:HyperboloidProjection.png|320px|thumb|The [[hyperboloid model]] can be seen as the equation of t<sup>2</sup>=x<sup>2</sup>+y<sup>2</sup>+1 can can be used to construct a Poincaré disk model as a [[perspective projection]] viewed from (t=-1,x=0,y=0), projecting the upper half hyperboloid onto an (x,y) [[unit disk]] at t=0. Planes passing through the origin represents [[geodesic]]s on the hyperbolic plane. The red circular arc is geodesic in Poincaré disk model; it projects to the brown geodesic on the green hyperboloid.]]
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| The Poincaré disk model, as well as the [[Klein model]], are related to the [[hyperboloid model]] projectively. If we have a point [''t'', ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>] on the upper sheet of the hyperboloid of the hyperboloid model, thereby defining a point in the hyperboloid model, we may project it onto the hypersurface ''t'' = 0 by intersecting it with a line drawn through [−1, 0, ..., 0]. The result is the corresponding point of the Poincaré disk model.
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| For [[Cartesian coordinates]] (''t'', ''x<sub>i</sub>'') on the hyperboloid and (''y<sub>i</sub>'') on the plane, the conversion formulae are:
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| :<math>y_i = \frac{x_i}{1 + t}</math>
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| :<math>(t, x_i) = \frac {\left( 1+\sum{y_i^2},\, 2 y_i \right)} {1-\sum{y_i^2}}</math>
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| Compare the formulae for [[stereographic projection]] between a sphere and a plane.
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| ==Analytic geometry constructions in the hyperbolic plane==
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| A basic construction of [[analytic geometry]] is to find a line through two given points. In the Poincaré disk model, lines in the plane are defined by portions of circles having equations of the form
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| :<math>x^2 + y^2 + a x + b y + 1 = 0,\,</math>
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| which is the general form of a circle orthogonal to the unit circle, or else by diameters. Given two points ''u'' and ''v'' in the disk which do not lie on a diameter, we can solve for the circle of this form passing through both points, and obtain
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| :<math> | |
| \begin{align}
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| & {} \quad x^2 + y^2 + \frac{u_2(v_1^2+v_2^2)-v_2(u_1^2+u_2^2)+u_2-v_2}{u_1v_2-u_2v_1}x \\[8pt]
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| & {} + \frac{v_1(u_1^2+u_2^2)-u_1(v_1^2+v_2^2)+v_1-u_1}{u_1v_2-u_2v_1}y + 1 = 0.
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| \end{align}
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| </math>
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| If the points ''u'' and ''v'' are points on the boundary of the disk not lying at the endpoints of a diameter, the above simplifies to
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| :<math>x^2+y^2+\frac{2(u_2-v_2)}{u_1v_2-u_2v_1}x - \frac{2(u_1-v_1)}{u_1v_2-u_2v_1}y + 1 = 0.</math>
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| ==Angles==
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| We may compute the angle between the [[circular arc]] whose endpoints (''ideal points'') are given by unit vectors ''u'' and ''v'', and the arc whose endpoints are ''s'' and ''t'', by means of a formula. Since the ideal points are the same in the Klein model and the Poincaré disk model, the formulas are identical for each model.
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| If both models' lines are diameters, so that ''v'' = −''u'' and ''t'' = −''s'', then we are merely finding the angle between two unit vectors, and the formula for the angle θ is | |
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| :<math>\cos(\theta) = u \cdot s.\,</math>
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| If ''v'' = −''u'' but not ''t'' = −''s'', the formula becomes, in terms of the [[wedge product]] (<math>\wedge</math>),
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| :<math>\cos^2(\theta) = \frac{P^2}{QR},</math>
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| where
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| :<math>P = u \cdot (s-t),\,</math>
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| :<math>Q = u \cdot u,\,</math>
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| :<math>R = (s-t) \cdot (s-t) - (s \wedge t) \cdot (s \wedge t).\,</math>
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| If both chords are not diameters, the general formula obtains | |
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| :<math>\cos^2(\theta) = \frac{P^2}{QR},</math>
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| where
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| :<math>P = (u-v) \cdot (s-t) - (u \wedge v) \cdot (s \wedge t),\,</math>
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| :<math>Q = (u-v) \cdot (u-v) - (u \wedge v) \cdot (u \wedge v),\,</math>
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| :<math>R = (s-t) \cdot (s-t) - (s \wedge t) \cdot (s \wedge t).\,</math>
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| Using the [[Binet–Cauchy identity]] and the fact that these are unit vectors we may rewrite the above expressions purely in terms of the [[dot product]], as
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| :<math>P = (u-v) \cdot (s-t) + (u \cdot t)(v \cdot s) - (u \cdot s)(v \cdot t),\,</math>
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| :<math>Q = (1 - u \cdot v)^2,\,</math>
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| :<math>R = (1 - s \cdot t)^2.\,</math>
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| ==Artistic realizations==
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| [[M. C. Escher]] explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician [[H.S.M. Coxeter]] around 1956 inspired Escher's interest in hyperbolic tessellations, which are regular tilings of the hyperbolic plane. Escher's wood engravings Circle Limit I–IV demonstrate this concept between 1958 and 1960, the final one being ''[http://www.mcescher.com/Gallery/recogn-bmp/LW436.jpg Circle Limit IV: Heaven and Hell]'' in 1960.<ref>[http://euler.slu.edu/escher/index.php/Escher%27s_Circle_Limit_Exploration Escher's Circle Limit Exploration]</ref> According to Bruno Ernst, the best of them is [[Circle Limit III]].
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| ==See also==
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| {{Commonscat|Poincaré Disc}}
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| * [[Hyperbolic geometry]]
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| * [[Klein model]]
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| * [[Poincaré half-plane model]]
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| * [[Poincaré metric]]
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| * [[Pseudosphere]]
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| * [[Hyperboloid model]]
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| * [[Inversive geometry]]
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| * [[Uniform tilings in hyperbolic plane]]
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| ==References==
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| * James W. Anderson, ''Hyperbolic Geometry'', second edition, Springer, 2005
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| * Eugenio Beltrami, ''Teoria fondamentale degli spazii di curvatura costante'', Annali. di Mat., ser II 2 (1868), 232-255
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| * Saul Stahl, ''The Poincaré Half-Plane'', Jones and Bartlett, 1993
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| {{reflist}}
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| {{DEFAULTSORT:Poincare disk model}}
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| [[Category:Multi-dimensional geometry]]
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| [[Category:Hyperbolic geometry]]
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Most people think how the electric toothbrush cleans teeth better than any manual toothbrush, but this is not the protective case. Electric toothbrush, only helps to accelerate more than again of washing the teeth, although not cleaning great quality.
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If teeth whitening techniques is not as effective a person have minor chips with your teeth, you cosmetic dentist may suggest veneers. Veneers are made or porcelain or artificial. Your dentist may use a veneer to disguise a rather difficult or stubborn stain or to cosmetically improve the appearance associated with an broken for ladies chipped dentist. Crooked teeth or even gaps within your teeth can be fixed with veneers.
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The dentist will more than your records and ask you some questions concerning your dental appearing. The dentist might need some x-rays of your mouth in order to acquire a good in the inside of your mouth and how your teeth have formed. If they see any issues they will discuss tips on how to handle those. The dentist usually do good cleaning of the teeth and take away any plaque that they find. This will help provide you a new beginning toward healthy teeth. Children will also likely get x-rays among the dentist to produce they have a point commence from because they watch one's teeth grow. This will be put onto the child's chart.
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