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[[Image:Small circle.svg|right|thumb|200px|Small circle of a sphere.]]
It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br><br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>If you have any questions relating to where and ways to utilize [http://www.youtube.com/watch?v=90z1mmiwNS8 Dentists in DC], you can call us at our own site.
[[Image:Esfera-raio-circulomenor.png|right|thumb|180px|<math>BC^2=AB^2+AC^2</math>, where ''C'' is the center of the sphere, ''A'' is the center of the small circle, and ''B'' is a point in the boundary of the small circle. Therefore, knowing the radius of the sphere, and the distance from the plane of the small circle to C, the radius of the small circle can be determined using the Pythagorean theorem.]]
A '''circle of a sphere''' is a circle that lies on a [[sphere]]. Such a circle can be formed as the intersection of a [[sphere]] and a [[Plane (geometry)|plane]], or of two spheres. A circle on a sphere whose plane passes through the center of the sphere then the circle is called a [[great circle]]; otherwise it is a '''small circle'''. Circles of a sphere have radius less than or equal to the sphere radius, with equality when the circle is a great circle.  
 
==On the earth==
In the [[geographic coordinate system]] on a globe, the parallels of [[latitude]] are such circles, with the [[Equator]] the only great circle. By contrast, all meridians of [[longitude]], paired with their opposite meridian in the other [[hemisphere of the Earth|hemisphere]], form great circles.
 
==Related terminology==
The diameter of the sphere which passes though the center of the circle is called its '''axis''' and the endpoints of this diameter are called its '''poles'''. A '''circle of a sphere''' can also be defined as the set of points at a given [[angular distance]] from a given pole.
 
==Sphere-plane intersection==
When the intersection of a sphere and a plane is not empty or a single point, it is a circle.
 
That the intersection of a sphere and a plane is, in fact, a circle can be seen as follows. Let the ''S'' be a sphere with center ''O'', ''P'' a plane which intersects ''S''. Draw {{overline|''OE''}} perpendicular to ''P'' and meeting ''P'' at ''E''., Let ''A'' and ''B'' be any two points in the intersection. Then ''AOE'' and ''BOE'' are right triangles with a common side, ''OE'', and hypotenuses ''AO'' and ''BO'', equal. Therefore the remaining sides ''AE'' and ''BE'' are equal. This proves that all points in the intersection are the same distance from the point ''E'' in the plane ''P'', in other words all points in the intersection lie on a circle with center ''E''.<ref>Proof follows Hobbs, Prop. 304</ref> Note that ''OE'' is the axis of the circle.
 
As a corollary, on a sphere there is exactly one circle that can be drawn though three given points.<ref>Hobbs, Prop. 308</ref>
 
The proof can be extended to show that the points on a circle are all a common angular distance from one of its poles.<ref>Hobbs, Prop. 310</ref>
 
==Sphere-sphere intersection==
To show that a non-trivial intersection of two spheres is a circle, assume (without loss of generality) that one sphere (with radius <math>R</math>) is centered at the origin. Points on this sphere satisfy
:<math>x^2 + y^2 + z^2 = R^2.</math>
Also without loss of generality, assume that the second sphere, with radius <math>r</math>, is centered at a point on the positive x-axis, at distance <math>a</math> from the origin. Its points satisfy
:<math>(x-a)^2 + y^2 + z^2 = r^2.</math>
The intersection of the circles is the set of points satisfying both equations. Subtracting the equations gives
:<math>\begin{align}
(x-a)^2 - x^2 & = r^2 - R^2 \\
    a^2 - 2ax & = r^2 - R^2 \\
            x & = \frac{a^2 + R^2 - r^2}{2a}.
\end{align}</math>
In the singular case <math>a = 0</math>, the spheres are concentric. There are two possibilities: if <math>R = r</math>, the spheres coincide, and the intersection is the entire sphere; if <math>R \not= r</math>, the spheres are disjoint and the intersection is empty.
When ''a'' is nonzero, the intersection lies in a vertical plane with this x-coordinate, which may intersect both of the spheres, be tangent to both spheres, or external to both spheres.
The result follows from the previous proof for sphere-plane intersections.
 
==See also==
*[[Line-plane intersection]]
*[[Line–sphere intersection]]
 
==References==
{{reflist}}
*{{cite book |title=Solid Geometry|first=C.A.|last=Hobbs|publisher=G.H. Kent|year=1921|pages=397 ff.}}
*{{cite book |title=Solid Geometry
|first1=M.|last1=Sykes|first2=C.E.|last2=Comstock|publisher=Rand McNally
|year=1922|pages=81 ff.}}
 
[[Category:Spherical geometry]]
[[Category:Curves]]
[[Category:Rotational symmetry]]

Latest revision as of 12:36, 9 December 2014

It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.



Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.

At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.

Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.

Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.

Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.

Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.

Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.

In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.

If you have any questions relating to where and ways to utilize Dentists in DC, you can call us at our own site.

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