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In [[geometry]], the '''exsphere''' of a face of a regular polyhedron is the sphere outside the polyhedron which touches the face and the planes defined by extending the adjacent faces outwards. It is tangent to the face externally and tangent to the adjacent faces internally.


It is the 3-dimensional equivalent of the [[excircle]].
Shares in LVMH (LVMH.PA), the world's biggest luxury goods group, fell sharply on Wednesday after an unexpected slowdown in sales growth at its fashion and leather business, which includes the Louis Vuitton, Celine and Dior brands.<br><br>The share price was down 6.4 percent at 135.50 euros by 1140 GMT, a six-week low and wiping  [http://www.pcs-systems.co.uk/Images/celinebag.aspx http://www.pcs-systems.co.uk/Images/celinebag.aspx] around 4.8 billion euros ($6.5 billion)off the market value of France's fourth-biggest listed company.<br><br>The group, which also owns Ruinart champagne and Hennessy cognac, saw sales growth at the fashion and leather division slide to 3 percent in the third quarter, against expectations of 7 to 8 percent.<br>In a [http://Www.ehow.com/search.html?s=conference conference] call, Chief Financial Officer Jean-Jacques Guiony blamed price increases in Japan for the slowdown, as well as softer demand for some brands.<br>However, one London-based analyst noted that Japan accounted for only around 15 percent of LVMH's fashion and leather sales, "so we did not get a full explanation".<br><br>LVMH has been trying to stem a decline in Louis Vuitton's sales growth by introducing new and pricier leather bags, which analysts expected would lead to short-term losses in sales.<br>"I understand that the repositioning of Louis Vuitton takes time and may be a bumpy ride," said Exane BNP Paribas analyst Luca Solca.<br>Before the results were announced, LVMH shares were up 4.4 percent since January 1, underperforming the overall luxury sector, which was up more than 20 percent.<br><br>Analysts said concerns about the future growth of Louis Vuitton had been exacerbated by the announcement this month that it was parting company with its star designer, Marc Jacobs.<br>SUPPLY CONSTRAINTS<br>Guiony said the recent launch of new leather collections had not come in time to have a meaningful impact on sales, and acknowledged that production was constrained by a lack of high-quality leather.<br><br>"Without these supply constraints, we would produce more than what we do," Guiony said.<br>Louis Vuitton shop assistants polled by Reuters last month said they had been provided with only a small number of new handbags, such as the Capucines model, priced at 3,500 euros, which had flown off the shelves.<br>LVMH has been buying tanneries to secure supplies but experts say the market is under pressure partly because the number of calves raised and slaughtered is driven more by demand for meat -- which has been in decline -- than by demand for quality hides.<br><br>In addition, China, the luxury industry's main driver since the late 2000s, has started to run out of steam in the last year due to an economic slowdown and a government crackdown on gift-giving.<br>Guiony said Vuitton's sales in mainland China were "flattish" but, thanks to sales to Chinese tourists, overall sales growth to Chinese customers was in the "mid-single digits plus".<br>Guiony said trends in watches and jewelry had slightly improved in China, but not in fashion and leather.<br><br>He said trading remained difficult in Europe, particularly for perfume and cosmetics, where sales were "flattish".<br>LVMH's overall sales grew 2 percent in the third quarter. Organic growth was 8 percent, of which 6 percentage points were accounted for by the relative weakness of the U.S. dollar and the Japanese yen against the euro.<br>($1=0.7406 euros)<br>(Additional reporting by James Regan; Editing by James Jukwey and Kevin Liffey)
 
The sphere is more generally well-defined for any face which is a regular
polygon and delimited by faces with the same dihedral angles
at the shared edges. Faces of semi-regular polyhedra often
have different types of faces, which define exspheres of different size with each type of face.
 
 
== Parameters ==
The exsphere touches the face of the regular polyedron at the center
of the incircle of that face. If the exsphere radius is denoted {{math|r<sub>ex</sub>}}, the radius of this incircle {{math|r<sub>in</sub>}}
and the dihedral angle between the face and the extension of the
adjacent face {{math|&delta;}}, the center of the exsphere
is located from the viewpoint at the middle of one edge of the
face by bisecting the dihedral angle. Therefore
 
:<math>\tan\frac{\delta}{2} = \frac{r_{ex}}{r_{in}}.</math>
 
{{math|&delta;}} is the 180-degree complement of the
internal face-to-face angle.
 
=== Tetrahedron ===
Applied to the geometry of the [[Tetrahedron]] of edge length {{math|a}},
we have an [[Incircle|incircle radius]] {{math|r<sub>in</sub>{{=}}a/(2&radic;3)}} (derived by
dividing twice the face area {{math|(a<sup>2</sup>&radic; 3)/4}} through the
perimeter {{math|3a}}), a dihedral angle {{math|&delta;{{=}}&pi;-arccos(1/3)}}, and in consequence {{math|r<sub>ex</sub>{{=}}a/&radic; 6}}.
 
=== Cube ===
The radius of the exspheres of the 6 faces of the [[Cube]]
is the same as the radius of the inscribed
sphere, since {{math|&delta;}} and its complement are the same, 90 degrees.
 
=== Icosahedron ===
The dihedral angle applicable to the [[Icosahedron]] is derived by
considering the coordinates of two triangles with a common edge,
for [[Icosahedron#Cartesian coordinates|example]] one face with vertices
at
 
:<math>(0,-1,g), (g,0,1), (0,1,g),</math>
 
the other at
 
:<math>(1,-g,0), (g,0,1), (0,-1,g),</math>
 
where {{math|g}} is the [[golden ratio]]. Subtracting vertex coordinates
defines edge vectors,
 
:<math>(g,1,1-g), (-g,1,g-1)</math>
 
of the first face and
 
:<math>(g-1,g,1), (-g,-1,g-1)</math>
 
of the other. [[Cross product]]s of the edges of the first face and second
face yield (not normalized) face [[Normal (geometry)|normal vectors]]
 
:<math>(2g-2,0,2g) \sim (g-1,0,g)</math>
of the first and
:<math>(g^2-g+1,-g-(g-1)^2,1-g+g^2) = (2,-2,2)\sim (1,-1,1)</math>
of the second face, using {{math|g<sup>2</sub>{{=}}1+g}}.
The [[dot product]] between these two face normals yields the cosine
of the dihedral angle,
 
:<math>\cos\delta = \frac{(g-1)\cdot 1+g\cdot 1}{\sqrt{(g-1)^2+g^2} \sqrt{3}} =\frac{2g-1}{3} =\frac{\surd 5}{3}\approx 0.74535599.</math> {{OEIS2C|A208899}}
:<math>\therefore \delta \approx 0.72973 \,\mathrm{rad} \approx 41.81^\circ</math>
:<math>\therefore \tan\frac{\delta}{2} = \frac{\sin\delta}{1+\cos\delta}
=\frac{2}{3+\surd 5} \approx 0.3819660</math> {{OEIS2C|A132338}}
 
For an icosahedron of edge length {{math|a}}, the incircle radius
of the triangular faces is {{math|r<sub>in</sub>{{=}}a/(2&radic; 3)}}, and finally the
radius of the 20 exspheres
:<math>r_{ex} = \frac{a}{(3+\sqrt{5})\sqrt 3} \approx 0.1102641 a.</math>
 
== See also ==
[[Inscribed sphere|Insphere]]
 
== External links ==
*{{cite journal
|first1=Leon
|last1=Gerber
|title=Associated and skew-orthologic simplexes
|year=1977
|journal=Trans. Am. Math. Soc.
|volume=231
|number=1
|pages=47–63
|jstor=1997867
|mr=0445393
|doi=10.1090/S0002-9947-1977-0445393-6
}}
*{{cite journal
|first1=Mowaffaq
|last1=Hajja
|title=The Gergonne and Nagel centers of an n-dimensional simplex
|journal=J. Geom.
|doi=10.1007/s00022-005-0011-3
|volume=83
|number=1-2
|pages=46–56
|year=2005
}}
 
[[Category:Geometry]]

Latest revision as of 21:30, 4 October 2014


Shares in LVMH (LVMH.PA), the world's biggest luxury goods group, fell sharply on Wednesday after an unexpected slowdown in sales growth at its fashion and leather business, which includes the Louis Vuitton, Celine and Dior brands.

The share price was down 6.4 percent at 135.50 euros by 1140 GMT, a six-week low and wiping http://www.pcs-systems.co.uk/Images/celinebag.aspx around 4.8 billion euros ($6.5 billion)off the market value of France's fourth-biggest listed company.

The group, which also owns Ruinart champagne and Hennessy cognac, saw sales growth at the fashion and leather division slide to 3 percent in the third quarter, against expectations of 7 to 8 percent.
In a conference call, Chief Financial Officer Jean-Jacques Guiony blamed price increases in Japan for the slowdown, as well as softer demand for some brands.
However, one London-based analyst noted that Japan accounted for only around 15 percent of LVMH's fashion and leather sales, "so we did not get a full explanation".

LVMH has been trying to stem a decline in Louis Vuitton's sales growth by introducing new and pricier leather bags, which analysts expected would lead to short-term losses in sales.
"I understand that the repositioning of Louis Vuitton takes time and may be a bumpy ride," said Exane BNP Paribas analyst Luca Solca.
Before the results were announced, LVMH shares were up 4.4 percent since January 1, underperforming the overall luxury sector, which was up more than 20 percent.

Analysts said concerns about the future growth of Louis Vuitton had been exacerbated by the announcement this month that it was parting company with its star designer, Marc Jacobs.
SUPPLY CONSTRAINTS
Guiony said the recent launch of new leather collections had not come in time to have a meaningful impact on sales, and acknowledged that production was constrained by a lack of high-quality leather.

"Without these supply constraints, we would produce more than what we do," Guiony said.
Louis Vuitton shop assistants polled by Reuters last month said they had been provided with only a small number of new handbags, such as the Capucines model, priced at 3,500 euros, which had flown off the shelves.
LVMH has been buying tanneries to secure supplies but experts say the market is under pressure partly because the number of calves raised and slaughtered is driven more by demand for meat -- which has been in decline -- than by demand for quality hides.

In addition, China, the luxury industry's main driver since the late 2000s, has started to run out of steam in the last year due to an economic slowdown and a government crackdown on gift-giving.
Guiony said Vuitton's sales in mainland China were "flattish" but, thanks to sales to Chinese tourists, overall sales growth to Chinese customers was in the "mid-single digits plus".
Guiony said trends in watches and jewelry had slightly improved in China, but not in fashion and leather.

He said trading remained difficult in Europe, particularly for perfume and cosmetics, where sales were "flattish".
LVMH's overall sales grew 2 percent in the third quarter. Organic growth was 8 percent, of which 6 percentage points were accounted for by the relative weakness of the U.S. dollar and the Japanese yen against the euro.
($1=0.7406 euros)
(Additional reporting by James Regan; Editing by James Jukwey and Kevin Liffey)

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