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{{redirect|Kissing circles|Descartes' theorem on mutually tangent (kissing) circles|Descartes' theorem}}
== not for others ==


[[File:Osculating circle.svg|thumb|right|An osculating circle]]
Somewhat similar, but from a certain extent, this is a part of the General Assembly refining 'medicine' mainland real division between the top level.<br><br>by this Assembly, that is to let people get to know, who is ranked first on the continent today refining 'medicine' division!<br><br>able to participate in 'medicine' typical person, in addition to the 'medicine' family of the tribe, as well as some had [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html カシオ 腕時計 バンド] or now has a great reputation and high refining 'medicine' technique of refining 'medicine' guru on the Plains, also can be invited, but this number is generally invited to very few people, after all, can be 'medicine' family fancy eye refining 'medicine' division, on the mainland, but also [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html カシオ 電波ソーラー時計] the Phoenix 'gross' Linjiao like existence.<br><br>But anyway, it does not prevent the 'medicine' typical grand, not for others, just because that continent first refining 'medicine' division [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-11.html カシオ腕時計 g-shock] title!<br><br>stars fell Court from [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html casio 腕時計 説明書] the 'medicine' Shennong Mountain where the family very far, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html 時計 メンズ カシオ] but Xiao Yan duo
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In [[differential geometry of curves]], the '''osculating circle''' of a sufficiently smooth plane [[curve]] at a given point ''p'' on the curve has been traditionally defined as the circle passing through ''p'' and a pair of additional points on the curve [[infinitesimal]]ly close to ''p''.  Its center lies on the inner [[Normal (geometry)|normal line]], and its [[curvature]] is the same as that of the given curve at that point. This circle, which is the one among all '''[[tangent circles]]''' at the given point that approaches the curve most tightly, was named ''circulus osculans'' (Latin for "kissing circle") by [[Gottfried Wilhelm Leibniz|Leibniz]].
== looking at the broken chest open to the robes ==


The center and radius of the osculating circle at a given point are called '''center of curvature''' and '''[[radius of curvature (mathematics)|radius of curvature]]''' of the curve at that point. A geometric construction was described by [[Isaac Newton]] in his ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'':
, Yunshan face 'color' is gradually cold, chest a shrink, soon suddenly a drum, an impulsive majestic, overwhelming storm surge out, and bear the brunt, and that [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html 電波腕時計 カシオ] is the nearest Xiao Yan!<br><br>series of marching footsteps void, Xiao Yan also pick up behind the wings shake, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-11.html カシオ腕時計 g-shock] after a good moment, just the kind of momentum built shed, chest Department, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html カシオソーラー時計] also came the refreshing feeling of tightness.<br><br>cold eyes [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-3.html casio 腕時計 レディース] looked back rapidly Xiao Yan, Yunshan slowly down, looking at the broken chest open to the robes, robes cracked, 'dew' out of its chest, at the moment, a little bit above being printed red boxing India, obviously, this is a previously Xiao Yan stay.<br><br>'Oh, good, very good [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-2.html casio 腕時計 デジタル] ah, to the fight since I Jin cases, you are still left in me first boxing Indian man' looked at the road fist India, Yunshan face is completely cold down , it is the heart of an intention to kill the chill gradually spread out from its body.
{{Quotation|There being given, in any places, the velocity with which a body describes a given figure, by means of forces directed to some common centre: to find that centre.|Isaac Newton, ''Principia''; PROPOSITION V. PROBLEM I.}}
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==Description in lay terms==
== 'back Gama empire ==
Imagine a car moving along a curved road on a vast flat plane.  Suddenly, at one point along the road, the steering wheel locks in its present position.  Thereafter, the car moves in a circle that "kisses" the road at the point of locking.  The [[curvature]] of the circle is equal to that of the road at that point.  That circle is the osculating circle of the road curve at that point.


==Mathematical description==
'Clouds, haze, were it seems that old [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html カシオ 時計 電波 ソーラー] bastard Yunshan, really want ruthless ah!' Black eye [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html カシオ 時計 プロトレック] pupil almost been bloodshot, her face 'color' terribly grim, Xiao Yan body trembling rapid surge strong intention to kill filled the room, after a moment, he suddenly stood up, but it was hastily pulled smoked children: '? Xiao Yan brother, you have to go.'<br><br>'back Gama empire! I want cloud-lan everyone buried!' sepulchral voice, with a strong intention to kill and ferocious.<br><br>'You can now go back and die in vain in [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html 電波時計 casio] addition to what is the use? xiaojia now suffering from the disaster, but you still have who can save? if you die, how Xiao Shushu accountable?' askew hideous to see Xiao [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html カシオ 腕時計 チタン] Yan [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-15.html カシオ 腕時計 激安] look, Kaoru children know, because he is somewhat out of control rage up, the moment hastily loudly.<br><br>'Xiao Yan, you have nothing to say! Brother let me vapid Mania Wanlitiaotiao come find you, not let you
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Let ''γ''(''s'') be a [[regular parametric curve|regular parametric plane curve]], where ''s'' is the [[arc length]], or natural parameter. This determines the unit tangent vector ''T'', the unit normal vector ''N'', the [[Curvature#Precise definition|signed curvature]] ''k''(''s'') and the radius of curvature at each point:
== solidification in this space ==


: <math> T(s)=\gamma'(s),\quad T'(s)=k(s)N(s),\quad R(s)=\frac{1}{\left|k(s)\right|}.</math>
Three Baotui, Cao Ying [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html カシオ 電波ソーラー時計] space in front of a sudden surge of violent fluctuations, immediately crystal layer gripper forcibly [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html カシオ 腕時計 バンド] ripped space, a fiercely arrested at [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-10.html カシオ腕時計 価格] her, look at this look, that black statue is actually like to fight for Cao Ying start.<br>Under<br>energy gripper grasping, Cao Ying is somewhat shocked to find that the surrounding space, direct solidification is life and life is down!<br><br>solidification in this space, is no longer her stature could not move! Therefore, it can only watch as a crystal [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html 時計 メンズ カシオ] layer behemoth severely captured.<br><br>suddenly in danger of Cao Ying, Xiao Yan and Song Qing also be aware, they face 'color' simply shot the lights changed, and that Song Qing eyes shimmering, after an instant, but it is a vicious bite, little feet trunk, stature is lightning coming into the forest, the trail between the blink of [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-3.html カシオ gps 時計] an eye is gone.<br><br>Qiaode Song Qing Cao Ying actually throwing alone escape, Xiao Yan
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Suppose that ''P'' is a point on ''γ'' where ''k'' ≠ 0. The corresponding center of curvature is the point ''Q'' at distance ''R'' along ''N'', in the same direction if ''k'' is positive and in the opposite direction if ''k'' is negative. The circle with center at ''Q'' and with radius ''R'' is called the '''osculating circle''' to the curve ''γ'' at the point ''P''.
== this indifference glanced at Xiao Yan ==


If ''C'' is a regular space curve then the osculating circle is defined in a similar way, using the [[principal normal vector]] ''N''. It lies in the ''[[osculating plane]]'', the plane spanned by the tangent and principal normal vectors ''T'' and ''N'' at the point ''P''.
When, as you so, but overdone, go to step down, just this once. '<br>deep<br>mountains, a faint voice, but also sailed from the depths of [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html カシオ ソーラー電波腕時計] the mountains spread, the last piece of the sky in this sounded.<br><br>'yes!'<br><br>heard, which is directed at modest depths of ancient mountains respectful arch of the hand, and then facing Kaoru children after a ceremony, this indifference glanced at [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-2.html 腕時計 casio] Xiao Yan, turned that is swept into the depths of the mountains.<br><br>'Xiao Yan brother, okay?'<br><br>looking at the left of the ancient Qian, Kaoru children turned over, softly.<br><br>'all right.' Xiao Yan smiled and shook his head.<br><br>'Xiao Yan brother, you should now know why [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-15.html カシオ 腕時計 gps] Kaoru children always delays the time you come to the ancient tribe that?' Kaoru children [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html casio 電波時計] sigh, she'd never expect, these pedantic old guy could not even be so [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html カシオ レディース 電波ソーラー腕時計] , just arrived in Xiaoyan Gang ancient tribe
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The plane curve can also be given in a different regular parametrization
== but added feel quite satisfied ==
<math> \gamma(t)\,= \, \begin{pmatrix} x_1(t) \\
x_2(t) \end{pmatrix}\, </math>
where regular means that <math>\gamma'(t)\ne 0</math> for all <math>t</math>. Then the formulas for the signed curvature ''k''(''t''), the normal unit vector ''N''(''t''), the radius of curvature ''R''(''t''), and the center ''Q''(''t'') of the osculating cicle are
: <math>k(t) = \frac{x_1'(t) \cdot x_2''(t) - x_1''(t) \cdot x_2'(t)}{\Big( x_1'(t)^2+x_2'(t)^2 \Big)^{\frac{3}{2}}} \qquad\qquad \qquad\qquad\qquad N(t)\,=\,\frac{1}{|| \gamma'(t)||}\cdot\begin{pmatrix} -x_2'(t)  \\ x_1'(t) \end{pmatrix}</math>,


: <math>R(t) = \left| \frac{\Big( x_1'(t)^2+x_2'(t)^2 \Big)^{\frac{3}{2}}}{x_1'(t) \cdot x_2''(t) - x_1''(t) \cdot x_2'(t)} \right|\qquad\qquad \mathrm{and} \qquad\qquad Q(t)\,=\,\gamma(t)\,+ \, \frac{1}{k(t)\cdot|| \gamma'(t)||}\cdot\begin{pmatrix} -x_2'(t)  \\ x_1'(t) \end{pmatrix}\,. </math>
Spit out immediately condensed into a 'liquid' body, floating down, penetrated among some boulders, suddenly, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html casio 腕時計 ゴールド] those stones, is gradually becoming crystal clear, like jade.<br><br>'Oh, congratulations brother Xiao Yan Jin stars into a holy late breakthrough to two stars St., is just around the corner [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html カシオ 腕時計 チタン] ah.' saw Xiao Yan received power, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html カシオ 掛け時計] sky demon Dusk is rapidly falling down on, Baoquan laughed.<br><br>for demon Dusk congratulations, Xiao Yan is polite smile, shook his fist, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-13.html 時計 カシオ] but added feel quite satisfied, a drop of demon holy blood, is to get him into the two stars is almost holy, imagine, if He alone those demon blood Espírito Santo completely engulfed it, perhaps to catch the demon Dusk are not impossible, he has a [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html casio 腕時計 説明書] different fire, the ability to maximize energy absorption BES demon blood.<br>Broken body<br>Xiao Yan robes, when it appeared from the clouds early, body energy is automatically formed in the surface of clothes,
 
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==Properties==
<ul>
 
 
For a curve ''C'' given by a sufficiently smooth parametric equations (twice continuously differentiable), the osculating circle may be obtained by a limiting procedure: it is the limit of the circles passing through three distinct points on ''C'' as these points approach ''P''.<ref name=Lamb>Actually, point ''P'' plus two additional points, one on either side of ''P'' will do. See Lamb (on line): {{cite book |title=An Elementary Course of Infinitesimal Calculus |author=Horace Lamb |page=406 |url=http://books.google.com/books?id=eDM6AAAAMAAJ&pg=PA406&dq=%22osculating+circle%22&lr=&as_brr=0 |publisher=University Press |year=1897}}</ref> This is entirely analogous to the construction of the [[tangent]] to a curve as a limit of the secant lines through pairs of distinct points on ''C'' approaching ''P''.
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The osculating circle ''S'' to a plane curve ''C'' at a regular point ''P'' can be characterized by the following properties:
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* The circle ''S'' passes through ''P''.
 
* The circle ''S'' and the curve ''C'' have the [[tangent lines to circles|common tangent]] line at ''P'', and therefore the common normal line.
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* Close to ''P'', the distance between the points of the curve ''C'' and the circle ''S'' in the normal direction decays as the cube or a higher power of the distance to ''P'' in the tangential direction.
 
This is usually expressed as "the curve and its osculating circle have the third or higher order contact" at ''P''. Loosely speaking, the vector functions representing ''C'' and ''S'' agree together with their first and second derivatives at ''P''.
</ul>
 
If the derivative of the curvature with respect to ''s'' is nonzero at ''P'' then the osculating circle crosses the curve ''C'' at ''P''. Points ''P'' at which the derivative of the curvature is zero are called [[vertex (curve)|vertices]]. If ''P'' is a vertex then ''C'' and its osculating circle have contact of order at least four. If, moreover, the curvature has a non-zero [[local maximum]] or minimum at ''P'' then the osculating circle touches the curve ''C'' at ''P'' but does not cross it.
 
The curve ''C'' may be obtained as the [[envelope (mathematics)|envelope]] of the one-parameter family of its osculating circles. Their centers, i.e. the centers of curvature,  form another curve, called the ''[[evolute]]'' of ''C''. Vertices of ''C'' correspond to singular points on its evolute.
 
==Examples==
 
===Parabola===
[[File:Krümmungskreis-Parabel.png|thumb|250px|The osculating circle of the parabola at its vertex has radius 0.5 and fourth order contact.]]
For the parabola
:<math>\gamma(t) = \begin{pmatrix} t\\t^2 \end{pmatrix}</math>
the radius of curvature is
:<math>R(t)= \left| \frac{ \left(1+4\cdot t^2 \right)^{\frac{3}{2}}}{2} \right| </math>
At the vertex <math>\gamma(0) = \begin{pmatrix} 0\\0 \end{pmatrix}</math>  the radius of curvature equals ''R(0)=0.5'' (see figure). The parabola has fourth order contact with its osculating circle there. For large ''t'' the radius of curvature increases ~ ''t<sup>3</sup>'', that is, the curve straightens more and more.
 
===Lissajous curve===
[[File:Lissajous-Curve+OsculatingCircle+3vectors animated.gif|thumb|250px|Animation of the osculating circle to a Lissajous curve]]
A [[Lissajous curve]] with ratio of frequencies (3:2) can be parametrized as follows
:<math> \gamma(t)\,= \, \begin{pmatrix} \cos(3t) \\
\sin(2t) \end{pmatrix}\,. </math>
 
It has signed curvature ''k''(''t''), normal unit vector ''N''(''t'') and radius of curvature ''R''(''t'') given by
: <math>k(t) = \frac{6\cos(t)(8\cos(t)^4-10\cos(t)^2+5)}{(232\cos(t)^4-97\cos(t)^2+13-144\cos(t)^6)^{3/2}}\,, </math>
: <math>N(t)\,=\,\frac{1}{|| \gamma'(t)||}\cdot\begin{pmatrix} -2\cos(2t) \\ -3\sin(3t)
\end{pmatrix}</math>
and
: <math>R(t) = \left| \frac{(232\cos(t)^4-97\cos(t)^2+13-144\cos(t)^6)^{3/2}}{6\cos(t)(8\cos(t)^4-10\cos(t)^2+5)} \right|\,.</math>
 
See the figure for an animation. There the "acceleration vector" is the second derivative <math>\frac{\mathrm{d}^2\gamma(s)}{\mathrm{d}s^2}</math> with respect to the [[arc length]] <math>s</math>.
 
==See also==
*[[Circle packing theorem]]
*[[Contact (mathematics)]]
*[[Osculating curve]]
 
==Notes==
<references/>
 
==Further reading==
For some historical notes on the study of curvature, see
*{{cite book |title=From the Calculus to Set Theory 1630-1910: An Introductory History |author=Grattan-Guinness & H. J. M. Bos |url=http://books.google.com/books?id=OLNeNIbD3jUC&pg=PA72&vq=curvature&dq=Leibniz+Calculus+Bos&lr=&source=gbs_search_s&sig=ACfU3U1ksyW8pE2DkRUdMQ6-Dw5jo4WdJA
|page=72 |isbn=0-691-07082-2 |year=2000 |publisher=Princeton University Press}}
*{{cite book |url=http://books.google.com/books?id=KDSqLsOHc9UC&pg=PA313&dq=motion+%22center+of+curvature%22&lr=&as_brr=0&sig=ACfU3U1xXmwMgWfk0Is_CToKzTbY591kXQ#PPA313,M1
|page=313 |author=Roy Porter, editor  |isbn=0-521-57243-6 |year=2003 |publisher=Cambridge University Press   |title=The Cambridge History of Science: v4 - Eighteenth Century Science}}
For application to maneuvering vehicles see
*JC Alexander and JH Maddocks: [https://drum.umd.edu/dspace/bitstream/1903/4630/1/TR_87-122.pdf  ''On the maneuvering of vehicles'']
*{{cite book |title= Problems in Applied Mathematics: selections from SIAM review |author=Murray S. Klamkin |page=1
|url=http://books.google.com/books?id=-b2hQ7_ARocC&pg=PA1&dq=motion+%22center+of+curvature%22&lr=&as_brr=0&sig=ACfU3U0e0jN1k65IAWtXyKbYwYxtdeGlRw#PPA1,M1
|isbn=0-89871-259-9 |publisher=Society for Industrial and Applied Mathematics |year=1990}}
 
==External links==
{{Commons category|Illustrations for curvature and torsion of curves|Graphical illustrations of curvature and osculating circles}}
*[http://www.math.uni-muenster.de/u/urs.hartl/gifs/CurvatureAndTorsionOfCurves.mw Create your own animated illustrations of osculating circles] ([[Maple (software)|Maple]]-Worksheet)
* {{MathWorld | urlname= OsculatingCircle | title= Osculating Circle }}
*[http://math.fullerton.edu/mathews/n2003/CurvatureMod.html Module for Curvature]
 
[[Category:Circles]]
[[Category:Differential geometry]]
[[Category:Curves]]

Latest revision as of 14:54, 25 September 2014

not for others

Somewhat similar, but from a certain extent, this is a part of the General Assembly refining 'medicine' mainland real division between the top level.

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stars fell Court from casio 腕時計 説明書 the 'medicine' Shennong Mountain where the family very far, 時計 メンズ カシオ but Xiao Yan duo 相关的主题文章：

looking at the broken chest open to the robes

, Yunshan face 'color' is gradually cold, chest a shrink, soon suddenly a drum, an impulsive majestic, overwhelming storm surge out, and bear the brunt, and that 電波腕時計 カシオ is the nearest Xiao Yan!

series of marching footsteps void, Xiao Yan also pick up behind the wings shake, カシオ腕時計 g-shock after a good moment, just the kind of momentum built shed, chest Department, カシオソーラー時計 also came the refreshing feeling of tightness.

cold eyes casio 腕時計 レディース looked back rapidly Xiao Yan, Yunshan slowly down, looking at the broken chest open to the robes, robes cracked, 'dew' out of its chest, at the moment, a little bit above being printed red boxing India, obviously, this is a previously Xiao Yan stay.

'Oh, good, very good casio 腕時計 デジタル ah, to the fight since I Jin cases, you are still left in me first boxing Indian man' looked at the road fist India, Yunshan face is completely cold down , it is the heart of an intention to kill the chill gradually spread out from its body. 相关的主题文章：

'back Gama empire

'Clouds, haze, were it seems that old カシオ 時計 電波 ソーラー bastard Yunshan, really want ruthless ah!' Black eye カシオ 時計 プロトレック pupil almost been bloodshot, her face 'color' terribly grim, Xiao Yan body trembling rapid surge strong intention to kill filled the room, after a moment, he suddenly stood up, but it was hastily pulled smoked children: '? Xiao Yan brother, you have to go.'

'back Gama empire! I want cloud-lan everyone buried!' sepulchral voice, with a strong intention to kill and ferocious.

'You can now go back and die in vain in 電波時計 casio addition to what is the use? xiaojia now suffering from the disaster, but you still have who can save? if you die, how Xiao Shushu accountable?' askew hideous to see Xiao カシオ 腕時計 チタン Yan カシオ 腕時計 激安 look, Kaoru children know, because he is somewhat out of control rage up, the moment hastily loudly.

'Xiao Yan, you have nothing to say! Brother let me vapid Mania Wanlitiaotiao come find you, not let you 相关的主题文章：

solidification in this space

Three Baotui, Cao Ying カシオ 電波ソーラー時計 space in front of a sudden surge of violent fluctuations, immediately crystal layer gripper forcibly カシオ 腕時計 バンド ripped space, a fiercely arrested at カシオ腕時計 価格 her, look at this look, that black statue is actually like to fight for Cao Ying start.
Under
energy gripper grasping, Cao Ying is somewhat shocked to find that the surrounding space, direct solidification is life and life is down!

solidification in this space, is no longer her stature could not move! Therefore, it can only watch as a crystal 時計 メンズ カシオ layer behemoth severely captured.

suddenly in danger of Cao Ying, Xiao Yan and Song Qing also be aware, they face 'color' simply shot the lights changed, and that Song Qing eyes shimmering, after an instant, but it is a vicious bite, little feet trunk, stature is lightning coming into the forest, the trail between the blink of カシオ gps 時計 an eye is gone.

Qiaode Song Qing Cao Ying actually throwing alone escape, Xiao Yan 相关的主题文章：

this indifference glanced at Xiao Yan

When, as you so, but overdone, go to step down, just this once. '
deep
mountains, a faint voice, but also sailed from the depths of カシオ ソーラー電波腕時計 the mountains spread, the last piece of the sky in this sounded.

'yes!'

heard, which is directed at modest depths of ancient mountains respectful arch of the hand, and then facing Kaoru children after a ceremony, this indifference glanced at 腕時計 casio Xiao Yan, turned that is swept into the depths of the mountains.

'Xiao Yan brother, okay?'

looking at the left of the ancient Qian, Kaoru children turned over, softly.

'all right.' Xiao Yan smiled and shook his head.

'Xiao Yan brother, you should now know why カシオ 腕時計 gps Kaoru children always delays the time you come to the ancient tribe that?' Kaoru children casio 電波時計 sigh, she'd never expect, these pedantic old guy could not even be so カシオ レディース 電波ソーラー腕時計 , just arrived in Xiaoyan Gang ancient tribe 相关的主题文章：

but added feel quite satisfied

Spit out immediately condensed into a 'liquid' body, floating down, penetrated among some boulders, suddenly, casio 腕時計 ゴールド those stones, is gradually becoming crystal clear, like jade.

'Oh, congratulations brother Xiao Yan Jin stars into a holy late breakthrough to two stars St., is just around the corner カシオ 腕時計 チタン ah.' saw Xiao Yan received power, カシオ 掛け時計 sky demon Dusk is rapidly falling down on, Baoquan laughed.

for demon Dusk congratulations, Xiao Yan is polite smile, shook his fist, 時計 カシオ but added feel quite satisfied, a drop of demon holy blood, is to get him into the two stars is almost holy, imagine, if He alone those demon blood Espírito Santo completely engulfed it, perhaps to catch the demon Dusk are not impossible, he has a casio 腕時計 説明書 different fire, the ability to maximize energy absorption BES demon blood.
Broken body
Xiao Yan robes, when it appeared from the clouds early, body energy is automatically formed in the surface of clothes, 相关的主题文章：

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