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| | Sing to the four winds of bowed slightly after, and the emperors way: "soldiers fight fierce neville, lucky, but miss victory although hurt, maria knight, very ashamed." Song said ashamed, expression is insignificant, all can't see any guilty of idea.<br><br> |
| [[File:Parabola.svg|right|thumb|A [[parabola]], a simple example of a curve]]
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| In [[mathematics]], a '''curve''' (also called a '''curved line''' in older texts) is, generally speaking, an object similar to a [[line (geometry)|line]] but which is not required to be [[Curvature|straight]]. This entails that a line is a special case of curve, namely a curve with null [[curvature]].<ref>
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| In current language, a line is typically required to be straight. Historically, however, lines could be "curved" or "straight".
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| </ref> | |
| Often curves in two-dimensional ([[plane curves]]) or three-dimensional (space curves) [[Euclidean space]] are of interest.
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| Various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However many of these meanings are special instances of the definition which follows. A curve is a [[topological space]] which is locally [[homeomorphic]] to a line. In everyday language, this means that a curve is a set of points which, near each of its points, looks like a line, up to a deformation. A simple example of a curve is the [[parabola]], shown to the right. A [[list of curves|large number of other curves]] have been studied in multiple mathematical fields.
| | The win would way: "for the empire, and achieve active no. Enjoy too clear spirit Dan three of god." Too clear spirit god Dan can wash practice out, ghd australia is the best of the JinDan used to baby. This also includes Dan royal, the world had only the royal family.<br><br>Reward is rich. The fault of the all present JinDan, is where a.<br>Such a reward, also represents the win for the appreciation of sing and expectations. Although foreign things not sing so seriously, this is all the more the better.<br>Packed in exult, and the light of teaching more than ten delegates dark complexion, his eyes always follow sang. At the head of the cardinals candy pastor astringent voice asked: "how?"<br><br>A head pastor at white wavy hair, wore a law champions league, bishop started in a red robe, is extremely thin the dry. Look no bishops majesty, but the church archbishop cardinal, also is the Pope's most important assistant. This if it weren't for the holy cross found traces of the emperor sword, he also won't personally rate in the ten thousand group empire pageant.<br><br>"Strong, I'm not opponent. But this is not the adult opponent. Austin no chance. No cross the emperor of the sword breath." Back to the camp of the region, is to be sitting right beside him is a female godsworn. A bit cracky low voice, has the unspeakable strong glamour, let a person not independent wanted to believe her.<br><br>She wears a white robe, tall and slender, bright and clean and forehead is very broad, facial features a unique deep westerners, fine eyebrow inclined to choose, in a beautiful eyebrow peak place next fold radian, let her brow looks special beautiful, black eyes, deep and peaceful, high and straight nose manifestation her persistence and perseverance.<br><br>Big mouth, and his lips are the average woman want to be thick, does not accord with the aesthetic view of the east, but and her facial features together, coordination and natural, to a more let a person soul-stirring sexy.<br>The female is now the world the first young generation, [http://Lightknight.org/ light knight] Julie. Julie white robe barefoot, whole body and not any decorations, while sitting in that, simplicity, no authority the wind, nor any of ace your spirit, some just quiet and peaceful. As the light teach ascetic godsworn, jolie is a saint is a model of all christians.<br><br>Even if it is seen maria, death, Julie has any anger shocked color, just read it in low light's prayer, Nebuchadnezzar's praying out. To Julie, it was born, means the beginning of death. Death is not terrible, terrible is lose faith.<br>Julie is not eternal yearning not destroy, she believes is right faith itself. For her, the holy light huang, or hell a devil king, no matter how to call, as long as their ways is correct, it is worth beliefs. As for because of the faith, as in god domain life, Julie thought not.<br><br>Such a pure pure faith and looking into the light teach eight hundred million followers, is also no one can and. Julie can easily the suggestion of the emperor of the holy light reflects the body, almost unlimited play the ability of holy light emperor. In one sense, Julie is living the holy huang sword.<br><br>Some special moments, Julie can even say the first strong polish themselves.<br>She said of the defeat of the song, but said the real fix for the level. If it was let go for now, the ZhuLiYou qualification despising the yuan baby realm of the strong.<br>Another cardinal Gary also agree in a way: "the sword skill magical, god knows a power, huge for aircraft grasp, however, fine. Just to build for could neutralize, can be in each other beyond levels of sword skill, maria, almost losing. But it is the man's limits, and never again higher levels of the strong challenge. I don't have any breath of ShengJian induction to."<br><br>At this time, there is already a da, the body of the back up. At the last moment, maria, eyes still yuanzheng, cannot believe their own destruction.<br>Pastor at silence for a long time, to light [http://www.mtcaravans.co.nz/ghdnz.html GHD Straighteners NZ] a sigh, [http://www.twitpic.com/tag/gently+close gently close]. The spread of dull, eyes, low word praise "may the emperor's out, bless, forgive all her, lead her into the temple of god and never calm domain..."<br><br>Others heard together with low voice praise, solemn and solemn atmosphere, lead [http://www.mtcaravans.co.nz/ghdnz.html GHD Straighteners NZ] of main halls in many people's attention.<br>Her win's face of the works is way: "that can really sing cruel, clearly had the upper hand, but with bent to a sword to kill each other. To let those whose native language ManYi thought that lack of tolerance. But it is also the empire, heard that he had soured cruel, before school is just killed a many people, a narrow escape the law sanctions. Ah, father emperor was also some people veiled, that such people knighted." |
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| The term ''curve'' has several meanings in non-mathematical language as well. For example, it can be almost synonymous with [[mathematical function]] (as in ''[[learning curve]]''), or [[graph of a function]] (as in ''[[Phillips curve]]'').
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| An [[Arc (geometry)|arc]] or segment of a curve is a part of a curve that is bounded by two distinct end points and contains every point on the curve between its end points. Depending on how the arc is defined, either of the two end points may or may not be part of it. When the arc is straight, it is typically called a [[line segment]].
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| ==History==
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| [[File:Newgrange Entrance Stone.jpg|thumb|225px|[[Megalithic art]] from Newgrange showing an early interest in curves]]
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| Fascination with curves began long before they were the subject of mathematical study. This can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric
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| times.<ref name="Lockwood">Lockwood p. ix</ref> Curves, or at least their graphical representations, are simple to create, for example by a stick in the sand on a beach.
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| Historically, the term "line" was used in place of the more modern term "curve". Hence the phrases "straight line" and "right line" were used to distinguish what are today called lines from "curved lines". For example, in Book I of [[Euclid's Elements]], a line is defined as a "breadthless length" (Def. 2), while a ''straight'' line is defined as "a line that lies evenly with the points on itself" (Def. 4). Euclid's idea of a line is perhaps clarified by the statement "The extremities of a line are points," (Def. 3).<ref>Heath p. 153</ref> Later commentators further classified lines according to various schemes. For example:<ref>Heath p. 160</ref>
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| *Composite lines (lines forming an angle)
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| *Incomposite lines
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| **Determinate (lines that do not extend indefinitely, such as the circle)
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| **Indeterminate (lines that extend indefinitely, such as the straight line and the parabola)
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| [[File:Conic sections with plane.svg|thumb|225px|The curves created by slicing a cone ([[conic section]]s) were among the curves studied in ancient Greece.]] | |
| The Greek [[geometers]] had studied many other kinds of curves. One reason was their interest in solving geometrical problems that could not be solved using standard [[compass and straightedge]] construction.
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| These curves include:
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| *The [[conic section]]s, deeply studied by [[Apollonius of Perga]]
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| *The [[cissoid of Diocles]], studied by [[Diocles (mathematician)|Diocles]] and use a method to [[doubling the cube|double the cube]].<ref>Lockwood p. 132</ref>
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| *The [[conchoid of Nicomedes]], studied by [[Nicomedes (mathematician)|Nicomedes]] as a method to both double the cube and to [[angle trisection|trisect an angle]].<ref>Lockwood p. 129</ref>
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| *The [[Archimedean spiral]], studied by [[Archimedes]] as a method to trisect an angle and [[Squaring the circle|square the circle]].<ref>{{MacTutor|class=Curves|id=Spiral|title=Spiral of Archimedes}}</ref>
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| *The [[spiric section]]s, sections of [[torus|tori]] studied by [[Perseus (geometer)|Perseus]] as sections of cones had been studied by Apollonius.
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| [[File:Folium Of Descartes.svg|thumb|225px|left|Analytic geometry allowed curves, such as the [[Folium of Descartes]], to be defined using equations instead of geometrical construction.]]
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| A fundamental advance in theory of curves was the advent of [[analytic geometry]] in the seventeenth century. This enabled a curve to be described using an equation rather than an elaborate geometrical construction. This not only allowed new curves to be defined and studied, but it enabled a formal distinction to be made between curves that can be defined using [[algebraic equation]]s, [[algebraic curve]]s, and those that cannot, [[transcendental curve]]s. Previously, curves had been described as "geometrical" or "mechanical" according to how they were, or supposedly could be, generated.<ref name="Lockwood" />
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| Conic sections were applied in [[astronomy]] by [[Johannes Kepler|Kepler]].
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| Newton also worked on an early example in the [[calculus of variations]]. Solutions to variational problems, such as the [[brachistochrone]] and [[tautochrone]] questions, introduced properties of curves in new ways (in this case, the [[cycloid]]). The [[catenary]] gets its name as the solution to the problem of a hanging chain, the sort of question that became routinely accessible by means of [[differential calculus]].
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| In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the [[cubic curve]]s, in the general description of the real points into 'ovals'. The statement of [[Bézout's theorem]] showed a number of aspects which were not directly accessible to the geometry of the time, to do with singular points and complex solutions.
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| From the nineteenth century there is not a separate curve theory, but rather the appearance of curves as the one-dimensional aspect of [[projective geometry]], and [[differential geometry]]; and later [[topology]], when for example the [[Jordan curve theorem]] was understood to lie quite deep, as well as being required in [[complex analysis]]. The era of the [[space-filling curve]]s finally provoked the modern definitions of curve.
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| =={{anchor|Definitions}} Topology==
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| [[File:Mandelbrot Components.svg|250px|right|thumb|Boundaries of hyperbolic components of [[Mandelbrot set]] as closed curves]]
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| In [[topology]], a '''curve''' is defined as follows. Let <math>I</math> be an [[Interval (mathematics)|interval]] of [[real number]]s (i.e. a [[non-empty set|non-empty]] [[connected space|connected]] [[subset]] of <math>\mathbb{R}</math>). Then a curve <math>\!\,\gamma</math> is a [[continuous function (topology)|continuous]] [[Map (mathematics)|mapping]] <math>\,\!\gamma : I \rightarrow X</math>, where <math>X</math> is a [[topological space]].
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| *The curve <math>\!\,\gamma</math> is said to be '''simple''', or a '''Jordan arc''', if it is [[injective]], i.e. if for all <math>x</math>, <math>y</math> in <math>I</math>, we have <math>\,\!\gamma(x) = \gamma(y)</math> implies <math>x = y</math>. If <math>I</math> is a closed bounded interval <math>\,\![a, b]</math>, we also allow the possibility <math>\,\!\gamma(a) = \gamma(b)</math> (this convention makes it possible to talk about "closed" simple curves, see below).
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| In other words this curve "does not cross itself and has no missing points".<ref>{{cite web|url=http://dictionary.reference.com/browse/jordan%20arc |title=Jordan arc definition at Dictionary.com. Dictionary.com Unabridged. Random House, Inc |publisher=Dictionary.reference.com |date= |accessdate=2012-03-14}}</ref>
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| <!-- I think the use of all \!\, above is against WP guideline "avoid at all cost inline PNGs", I can't see justification for it. -MFH -->
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| *If <math>\gamma(x)=\gamma(y)</math> for some <math>x\ne y</math> (other than the extremities of <math>I</math>), then <math>\gamma(x)</math> is called a '''double''' (or '''multiple''') '''point''' of the curve.
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| *A curve <math>\!\,\gamma</math> is said to be '''closed''' or '''a loop''' if <math>\,\!I = [a,
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| b]</math> and if <math>\!\,\gamma(a) = \gamma(b)</math>. A closed curve is thus a continuous mapping of the circle <math>S^1</math>; a '''simple closed curve''' is also called a '''Jordan curve'''. The [[Jordan curve theorem]] states that such curves divide the plane into an "interior" and an "exterior".
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| A '''[[plane curve]]''' is a curve for which ''X'' is the [[Euclidean plane]]—these are the examples first encountered—or in some cases the [[projective plane]]. A '''space curve''' is a curve for which ''X'' is of three dimensions, usually [[Euclidean space]]; a '''skew curve''' {{anchor|skew curve}} is a space curve which lies in no plane. These definitions also apply to [[algebraic curve]]s (see below). However, in the case of algebraic curves it is very common to consider number systems more general than the reals.
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| This definition of curve captures our intuitive notion of a curve as a connected, continuous geometric figure that is "like" a line, without thickness and drawn without interruption, although it also includes figures that can hardly be called curves in common usage. For example, the image of a curve can cover a [[Square (geometry)|square]] in the plane ([[space-filling curve]]). The image of simple plane curve can have [[Hausdorff dimension]] bigger than one (see [[Koch snowflake]]) and even [[positive number|positive]] [[Lebesgue measure]]<ref>{{cite journal|last=Osgood|first=William F.| authorlink1=William Fogg Osgood |year=1903|month=January|title=A Jordan Curve of Positive Area|journal=Transactions of the American Mathematical Society|publisher=American Mathematical Society|volume=4|issue=1|pages=107–112|accessdate=2008-06-04|doi=10.2307/1986455|issn=0002-9947|jstor=1986455}}</ref> (the last example can be obtained by small variation of the [[Peano curve]] construction). The [[dragon curve]] is another unusual example.
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| ==Conventions and terminology==
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| The distinction between a curve and its [[image (mathematics)|image]] is important. Two distinct curves may have the same image. For example, a [[line segment]] can be traced out at different speeds, or a circle can be traversed a different number of times. Many times, however, we are just interested in the image of the curve. It is important to pay attention to context and convention in reading.
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| Terminology is also not uniform. Often, topologists use the term "[[path (topology)|path]]" for what we are calling a curve, and "curve" for what we are calling the image of a curve. The term "curve" is more common in [[vector calculus]] and [[differential geometry]].
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| ==Lengths of curves==
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| {{main|Arc length}}
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| If <math>X</math> is a [[metric space]] with metric <math>d</math>, then we can define the ''length'' of a curve <math>\!\,\gamma : [a, b] \rightarrow X</math> by
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| :<math>\text{length} (\gamma)=\sup \left\{ \sum_{i=1}^n d(\gamma(t_i),\gamma(t_{i-1})) : n \in \mathbb{N} \text{ and } a = t_0 < t_1 < \cdots < t_n = b \right\}. </math>
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| where the sup is over all <math>n</math> and all partitions <math>t_0 < t_1 < \cdots < t_n</math> of <math>[a, b]</math>.
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| A '''{{visible anchor|rectifiable curve}}''' is a curve with [[wiktionary:finite|finite]] length.
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| A [[Parametric equation|parametrization]] of <math>\!\,\gamma</math> is called '''natural''' (or '''unit speed''' or '''parametrised by arc length''') if for any <math>t_1</math>, <math>t_2</math> in <math>[a, b]</math>, we have
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| :<math> \text{length} (\gamma|_{[t_1,t_2]})=|t_2-t_1|. </math>
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| If <math>\!\,\gamma</math> is a [[Lipschitz continuity|Lipschitz-continuous]] function, then it is automatically rectifiable. Moreover, in this case, one can define the '''speed''' (or [[metric derivative]]) of <math>\!\,\gamma</math> at <math>t_0</math> as
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| :<math>\text{speed}(t_0)=\limsup_{t\to t_0} {d(\gamma(t),\gamma(t_0))\over |t-t_0|} </math>
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| and then
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| :<math>\text{length}(\gamma)=\int_a^b \text{speed}(t) \, dt.</math>
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| In particular, if <math>X = \mathbb{R}^n</math> is an [[Euclidean space]] and <math>\gamma : [a, b] \rightarrow \mathbb{R}^n</math> is [[differentiable]] then
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| :<math>\text{length}(\gamma)=\int_a^b | \gamma '(t) | \, dt. </math>
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| ==Differential geometry==
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| {{main|Differential geometry of curves}}
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| While the first examples of curves that are met are mostly plane curves (that is, in everyday words, ''curved lines'' in ''two-dimensional space''), there are obvious examples such as the [[helix]] which exist naturally in three dimensions. The needs of geometry, and also for example [[classical mechanics]] are to have a notion of curve in space of any number of dimensions. In [[general relativity]], a [[world line]] is a curve in [[spacetime]].
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| If <math>X</math> is a [[differentiable manifold]], then we can define the notion of ''differentiable curve'' in <math>X</math>. This general idea is enough to cover many of the applications of curves in mathematics. From a local point of view one can take <math>X</math> to be [[Euclidean space]]. On the other hand it is useful to be more general, in that (for example) it is possible to define the [[Differential geometry of curves|tangent vector]]s to <math>X</math> by means of this notion of curve.
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| If <math>X</math> is a [[smooth manifold]], a ''smooth curve'' in <math>X</math> is a [[smooth map]]
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| :<math>\!\,\gamma : I \rightarrow X.</math>
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| This is a basic notion. There are less and more restricted ideas, too. If <math>X</math> is a <math>C^k</math> manifold (i.e., a manifold whose [[chart (topology)|charts]] are <math>k</math> times [[continuously differentiable]]), then a <math>C^k</math> curve in <math>X</math> is such a curve which is only assumed to be <math>C^k</math> (i.e. <math>k</math> times continuously differentiable). If <math>X</math> is an [[manifold|analytic manifold]] (i.e. infinitely differentiable and charts are expressible as [[power series]]), and <math>\gamma</math> is an analytic map, then <math>\gamma</math> is said to be an ''analytic curve''.
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| A differentiable curve is said to be ''regular'' if its [[derivative]] never vanishes. (In words, a regular curve never slows to a stop or backtracks on itself.) Two <math>C^k</math> differentiable curves
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| :<math>\!\,\gamma_1 :I \rightarrow X</math> and
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| :<math>\!\,\gamma_2 : J \rightarrow X</math>
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| are said to be ''equivalent'' if there is a [[bijection|bijective]] <math>C^k</math> map
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| :<math>\!\,p : J \rightarrow I</math>
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| such that the [[inverse map]]
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| :<math>\!\,p^{-1} : I \rightarrow J</math>
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| is also <math>C^k</math>, and
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| :<math>\!\,\gamma_{2}(t) = \gamma_{1}(p(t))</math>
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| for all <math>t</math>. The map <math>\gamma_2</math> is called a ''reparametrisation'' of <math>\gamma_1</math>; and this makes an [[equivalence relation]] on the set of all <math>C^k</math> differentiable curves in <math>X</math>. A <math>C^k</math> ''arc'' is an [[equivalence class]] of <math>C^k</math> curves under the relation of reparametrisation.
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| ==Algebraic curve==
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| {{main|Algebraic curve}}
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| Algebraic curves are the curves considered in [[algebraic geometry]]. A plane algebraic curve is the locus of the points of coordinates ''x'', ''y'' such that ''f''(''x'', ''y'') = 0, where ''f'' is a polynomial in two variables defined over some field ''F''. Algebraic geometry normally looks not only on points with coordinates in ''F'' but on all the points with coordinates in an [[algebraically closed field]] ''K''. If ''C'' is a curve defined by a polynomial ''f'' with coefficients in ''F'', the curve is said '''defined''' over ''F''. The points of the curve ''C'' with coordinates in a field ''G'' are said '''rational''' over ''G'' and can be denoted ''C''(''G'')); thus the full curve ''C'' = ''C''(''K'').
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| Algebraic curves can also be space curves, or curves in even higher dimension, obtained as the intersection (common solution set) of more than one polynomial equation in more than two variables. By eliminating variables (by any tool of [[elimination theory]]), an algebraic curve may be projected onto a [[plane algebraic curve]], which however may introduce singularities such as [[cusp (singularity)|cusp]]s or double points.
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| A plane curve may also be completed in a curve in the [[projective plane]]: if a curve is defined by a polynomial ''f'' of total degree ''d'', then ''w''<sup>''d''</sup>''f''(''u''/''w'', ''v''/''w'') simplifies to a [[homogeneous polynomial]] ''g''(''u'', ''v'', ''w'') of degree ''d''. The values of ''u'', ''v'', ''w'' such that ''g''(''u'', ''v'', ''w'') = 0 are the homogeneous coordinates of the points of the completion of the curve in the projective plane and the points of the initial curve are those such ''w'' is not zero. An example is the [[Fermat curve]] ''u''<sup>''n''</sup> + ''v''<sup>''n''</sup> = ''w''<sup>''n''</sup>, which has an affine form ''x''<sup>''n''</sup> + ''y''<sup>''n''</sup> = 1. A similar process of homogenization may be defined for curves in higher dimensional spaces
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| Important examples of algebraic curves are the [[conic]]s, which are nonsingular curves of degree two and [[genus (mathematics)|genus]] zero, and [[elliptic curve]]s, which are nonsingular curves of genus one studied in [[number theory]] and which have important applications to [[cryptography]]. Because algebraic curves in fields of [[characteristic (algebra)|characteristic]] zero are most often studied over the [[complex number]]s, algebraic curves in algebraic geometry may be considered as [[real number|real]] surfaces. In particular, the non-singular complex projective algebraic curves are called [[Riemann surface]]s.
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| ==See also==
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| {{multicol}}
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| *[[Curvature]]
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| *[[Curve orientation]]
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| *[[Curve sketching]]
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| *[[Differential geometry of curves]]
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| *[[Gallery of curves]]
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| *[[List of curve topics]]
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| {{multicol-break}}
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| *[[List of curves]]
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| *[[Osculating circle]]
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| *[[Parametric surface]]
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| *[[Path (topology)]]
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| *[[Position vector]]
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| *[[Vector-valued function]]
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| {{multicol-end}}
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| ==Notes==
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| {{Commonscat|Curves}}
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| {{reflist|2}}
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| ==References==
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| * {{springer|author=A.S. Parkhomenko|id=l/l059020|title=Line (curve)}}
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| * {{springer|author=B.I. Golubov|id=r/r080130|title=Rectifiable curve}}
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| * [[Euclid]], commentary and trans. by [[T. L. Heath]] ''Elements'' Vol. 1 (1908 Cambridge) [http://books.google.com/books?id=UhgPAAAAIAAJ Google Books]
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| * E. H. Lockwood ''A Book of Curves'' (1961 Cambridge)
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| ==External links==
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| *[http://www-gap.dcs.st-and.ac.uk/~history/Curves/Curves.html Famous Curves Index], School of Mathematics and Statistics, University of St Andrews, Scotland
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| *[http://www.2dcurves.com/ Mathematical curves] A collection of 874 two-dimensional mathematical curves
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| *[http://faculty.evansville.edu/ck6/Gallery/Introduction.html Gallery of Space Curves Made from Circles, includes animations by Peter Moses]
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| *[http://faculty.evansville.edu/ck6/GalleryTwo/Introduction2.html Gallery of Bishop Curves and Other Spherical Curves, includes animations by Peter Moses]
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| *YAN Kun. [http://www.nature.ac.cn/papers/paper-pdf/curveandequation-pdf.pdf Research on adaptive connection equation in discontinuous area of data curve]. {{doi|10.3969/j.issn.1004-2903.2011.01.018}}
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| * The Encyclopedia of Mathematics article on [http://www.encyclopediaofmath.org/index.php/Line_(curve) lines].
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| * The Manifold Atlas page on [http://www.map.mpim-bonn.mpg.de/1-manifolds 1-manifolds].
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| [[Category:Curves| ]]
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| [[Category:Metric geometry]]
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| [[Category:Topology]]
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| [[Category:General topology]]
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Sing to the four winds of bowed slightly after, and the emperors way: "soldiers fight fierce neville, lucky, but miss victory although hurt, maria knight, very ashamed." Song said ashamed, expression is insignificant, all can't see any guilty of idea.
The win would way: "for the empire, and achieve active no. Enjoy too clear spirit Dan three of god." Too clear spirit god Dan can wash practice out, ghd australia is the best of the JinDan used to baby. This also includes Dan royal, the world had only the royal family.
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A head pastor at white wavy hair, wore a law champions league, bishop started in a red robe, is extremely thin the dry. Look no bishops majesty, but the church archbishop cardinal, also is the Pope's most important assistant. This if it weren't for the holy cross found traces of the emperor sword, he also won't personally rate in the ten thousand group empire pageant.
"Strong, I'm not opponent. But this is not the adult opponent. Austin no chance. No cross the emperor of the sword breath." Back to the camp of the region, is to be sitting right beside him is a female godsworn. A bit cracky low voice, has the unspeakable strong glamour, let a person not independent wanted to believe her.
She wears a white robe, tall and slender, bright and clean and forehead is very broad, facial features a unique deep westerners, fine eyebrow inclined to choose, in a beautiful eyebrow peak place next fold radian, let her brow looks special beautiful, black eyes, deep and peaceful, high and straight nose manifestation her persistence and perseverance.
Big mouth, and his lips are the average woman want to be thick, does not accord with the aesthetic view of the east, but and her facial features together, coordination and natural, to a more let a person soul-stirring sexy.
The female is now the world the first young generation, light knight Julie. Julie white robe barefoot, whole body and not any decorations, while sitting in that, simplicity, no authority the wind, nor any of ace your spirit, some just quiet and peaceful. As the light teach ascetic godsworn, jolie is a saint is a model of all christians.
Even if it is seen maria, death, Julie has any anger shocked color, just read it in low light's prayer, Nebuchadnezzar's praying out. To Julie, it was born, means the beginning of death. Death is not terrible, terrible is lose faith.
Julie is not eternal yearning not destroy, she believes is right faith itself. For her, the holy light huang, or hell a devil king, no matter how to call, as long as their ways is correct, it is worth beliefs. As for because of the faith, as in god domain life, Julie thought not.
Such a pure pure faith and looking into the light teach eight hundred million followers, is also no one can and. Julie can easily the suggestion of the emperor of the holy light reflects the body, almost unlimited play the ability of holy light emperor. In one sense, Julie is living the holy huang sword.
Some special moments, Julie can even say the first strong polish themselves.
She said of the defeat of the song, but said the real fix for the level. If it was let go for now, the ZhuLiYou qualification despising the yuan baby realm of the strong.
Another cardinal Gary also agree in a way: "the sword skill magical, god knows a power, huge for aircraft grasp, however, fine. Just to build for could neutralize, can be in each other beyond levels of sword skill, maria, almost losing. But it is the man's limits, and never again higher levels of the strong challenge. I don't have any breath of ShengJian induction to."
At this time, there is already a da, the body of the back up. At the last moment, maria, eyes still yuanzheng, cannot believe their own destruction.
Pastor at silence for a long time, to light GHD Straighteners NZ a sigh, gently close. The spread of dull, eyes, low word praise "may the emperor's out, bless, forgive all her, lead her into the temple of god and never calm domain..."
Others heard together with low voice praise, solemn and solemn atmosphere, lead GHD Straighteners NZ of main halls in many people's attention.
Her win's face of the works is way: "that can really sing cruel, clearly had the upper hand, but with bent to a sword to kill each other. To let those whose native language ManYi thought that lack of tolerance. But it is also the empire, heard that he had soured cruel, before school is just killed a many people, a narrow escape the law sanctions. Ah, father emperor was also some people veiled, that such people knighted."