curve

• n.

• 曲線

• v.

• 曲がる、曲げる、カーブする

関

bend、curvilinear、flex

　 　 　 　

WordNet

1. the trace of a point whose direction of motion changes (同)curved shape
2. a pitch of a baseball that is thrown with spin so that its path curves as it approaches the batter (同)curve ball, breaking ball, bender
3. a line on a graph representing data
4. a circular segment of a curve; "a bend in the road"; "a crook in the path" (同)crook, twist, turn
5. change direction; "The road bends"
6. diagonal line traversing a shield from the upper right corner to the lower left (同)bend dexter
7. curved segment (of a road or river or railroad track etc.) (同)curve
8. form a curve; "The stick does not bend" (同)flex
9. exhibit the strength of; "The victorious army flexes its invincibility"
10. the act of flexing; "he gave his biceps a flex to impress the ladies"
11. bend a joint; "flex your wrists"; "bend your knees" (同)bend
12. cause (a plastic object) to assume a crooked or angular form; "bend the rod"; "twist the dough into a braid"; "the strong man could turn an iron bar" (同)bend, deform, twist, turn
13. contract; "flex a muscle"
14. a closed curve that does not intersect itself (同)Jordan curve
15. the curve of y=sin x (同)sinusoid
16. curving upward
17. an inferior dog or one of mixed breed (同)mongrel, mutt
18. a cowardly and despicable person
19. having or marked by a curve or smoothly rounded bend; "the curved tusks of a walrus"; "his curved lips suggested a smile but his eyes were hard" (同)curving

PrepTutorEJDIC

1. 『曲線』 / 『曲がり』,曲がったもの,湾曲部 / (野球で投球の)『カーブ』 / (…の方へ)『曲がる』,湾曲する《+『to』+『名』》 / …'を'曲げる;'を'湾曲させる
2. 〈まっすぐな物〉'を'『曲げる』 / (…に)〈人〉'を'『従わせる』,屈服させる,〈意志〉'を'曲げさせる《+『名』+『to』+『名』》 / (…の方向に)〈視線・歩みなど〉'を'『向ける』,〈心など〉'を'傾ける《+『名』+『to』(『toward』,『on』)+『名/(do『ing』)》 / 〈物が〉曲がる / 身をかがめる;(…に)かがみこむ《+[『down』]『over』+『名』》 / (運命・意志,人などに)『従う』,屈服する《+『before』(『to』)+『名』》 / (…を方向に)『曲がる』,向かう《+『to』(『toward』)+『名』》 / (仕事などに)取り組む,身を入れる《+『to』+『名』》 / 曲がり,湾曲部,曲がった所 / (…を)曲げる(曲がる)こと,(身を)かがめること《+『of』+『名』》 / (綱の)結び目 / 《話》《the bends》ケーソン病,潜水病
3. (仕事などを始める準備運動として)〈手足など〉‘を'曲げる,〈筋肉〉‘を'縮める
4. 電線,電気のコード
5. 内側に曲がった
6. のら犬 / くだらない人間

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2017/11/19 15:09:11」(JST)

wiki ja

[Wiki ja表示]

カーヴ Curve
トニ・ハリディ（1995年、ロンドン）
基本情報
出身地	イングランド ロンドン
ジャンル	オルタナティヴ・ロック シューゲイザー ドリーム・ポップ エレクトロニカ
活動期間	1990年 - 1994年、1996年 - 2005年
レーベル	Anxious、カリスマ、FatLip、ユニバーサル
メンバー	トニ・ハリディ ディーン・ガルシア

カーヴ（Curve）は、イギリスのロックデュオ。1990年にロンドンで結成。インダストリアルを取り入れたシューゲイザーバンドの一つとして人気を博した。

略歴

ハワイ人とアイルランド人の両親を持つディーン・ガルシア（ギター、ベース、ドラムス、キーボード、プログラミング）と、英国人ヴォーカリストのトニ・ハリデイが共通の知人であるユーリズミックスのデイヴ・スチュワートを通じて出会い結成された。1991年に3枚のEP『Blindfold』『Frozen』『Cherry』をリリースし、翌1992年3月に1stアルバム『二重人格』（原題：Doppelgänger）を発表し、全英11位に付けた。またハリディはデペッシュ・モードのメンバー（当時）であるアラン・ワイルダーのソロプロジェクト、リコイルのアルバム『Bloodline』に参加し、「Edge to Life」「Bloodline」の2曲にボーカル提供した。11月にはそれまでにリリースした3枚のEPとシングル1曲で構成されたコンピレーション・アルバム『Pubic Fruit』を発表した。

1993年9月に2ndアルバム『Cuckoo』を発表。全英23位。アルバムを引っ提げたツアーによる疲れから、1994年に2人はカーヴを一旦解散させることを決意する。その後2人は新バンドを結成したりソロで活動を開始。

1996年9月にシングル「Pink Girl with the Blues」をリリースし、カーヴは再び活動を開始した。1997年11月リリースのシングル「Chinese Burn」を挟み、1998年2月に3rdアルバム『Come Clean』を発表。よりエレクトロニカに接近したサウンドを追求し、その後ヨーロッパで小規模のライヴツアーを展開した。カーヴの次のリリースは3年後である2001年5月発表のコンピレーション・アルバム『Open Day at the Hate Fest』だった。9月に4枚目のアルバム『Gift』をリリースした。アルバムにはマイ・ブラッディ・ヴァレンタインのケヴィン・シールズが2曲でギタリストとして参加した。

2002年に5枚目で最後のスタジオ・アルバムである『The New Adventures of Curve』を発表。同作はインターネットのみのリリースとなった。2003年にハリディはL'Arc〜en〜Cielのドラマーであるyukihiroのプロジェクト、acid androidの楽曲「Faults」でボーカル提供を行った。2004年5月に2枚組ベスト・アルバム『The Way of Curve』をリリース。翌2005年始めにハリディはカーヴからの脱退を発表、これ以上カーヴでの活動を行わないことを宣言しカーヴは解散した。2010年10月にレア曲や未発表曲を収録したダウンロードオンリーアルバム『Rare and Unreleased』を発表した。

メンバー

• トニ・ハリディ（Toni Halliday、1964年7月5日 - ）：ボーカル、ギター
• ディーン・ガルシア（Dean Garcia、1958年5月3日 - ）：ギター、ベース、ドラムス、キーボード、プログラミング

ディスコグラフィー

アムステルダムにて（1992年）

スタジオ・アルバム

• 二重人格 - Doppelgänger （1992年）
• Cuckoo （1993年）
• Come Clean （1998年）
• Gift （2002年）
• The New Adventures of Curve （2002年）

コンピレーション・アルバム

• Pubic Fruit （1992年）
• Radio Sessions （1993年）
• Open Day at the Hate Fest （2001年）
• The Way of Curve （2004年）
• Rare and Unreleased （2010年）

EP、シングル

• Blindfold EP （1991年）
• Frozen EP （1991年）
• Cherry EP （1991年）
• "Faît Accompli" （1992年）
• "Horror Head" （1992年）
• "Faît Accompli" （1992年）
• "Blackerthreetracker" （1993年）
• "Superblaster" （1993年）
• "Pink Girl With the Blues" （1996年）
• "Chinese Burn" （1997年）
• "Coming Up Roses" （1998年）
• "Perish" （2002年）
• "Want More Need Less" （2003年）

関連項目

• ユーリズミックス
• acid android
• リコイル

wiki en

[Wiki en表示]

For other uses, see Curve (disambiguation).

A parabola, a simple example of a curve

In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that curvature is not necessarily zero.

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 large number of other curves have been studied in multiple mathematical fields.

A closed curve is a curve that forms a path whose starting point is also its ending point—that is, a path from any of its points to the same point.

Closely related meanings include the graph of a function (as in Phillips curve) and a two-dimensional graph.

History

Megalithic art from Newgrange showing an early interest in curves

Interest in 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 times. Curves, or at least their graphical representations, are simple to create, for example by a stick in the sand on a beach.

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). Later commentators further classified lines according to various schemes. For example:

• Composite lines (lines forming an angle)
• Incomposite lines
  • Determinate (lines that do not extend indefinitely, such as the circle)
  • Indeterminate (lines that extend indefinitely, such as the straight line and the parabola)

The curves created by slicing a cone (conic sections) 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. These curves include:

• The conic sections, deeply studied by Apollonius of Perga
• The cissoid of Diocles, studied by Diocles and used as a method to double the cube.
• The conchoid of Nicomedes, studied by Nicomedes as a method to both double the cube and to trisect an angle.
• The Archimedean spiral, studied by Archimedes as a method to trisect an angle and square the circle.
• The spiric sections, sections of tori studied by Perseus as sections of cones had been studied by Apollonius.

Analytic geometry allowed curves, such as the Folium of Descartes, to be defined using equations instead of geometrical construction.

A fundamental advance in the 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 equations, algebraic curves, and those that cannot, transcendental curves. Previously, curves had been described as "geometrical" or "mechanical" according to how they were, or supposedly could be, generated.

Conic sections were applied in astronomy by Kepler. 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.

In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the cubic curves, 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.

Since the nineteenth century there has not been a separate theory of curves, 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 curves finally provoked the modern definitions of curve.

Definition

Boundaries of hyperbolic components of Mandelbrot set as closed curves

In general, a curve is defined through a continuous function $\gamma \colon I\rightarrow X$ from an interval I of the real numbers into a topological space X. Depending on the context, it is either $\gamma$ or its image $\gamma (I)$ which is called a curve.

In general topology, when non-differentiable functions are considered, it is the map $\gamma$, which is called a curve, because its image may look very differently from what is commonly called a curve. For example, the image of the Peano curve completely fills the square. On the other hand, when one considers curves defined by a differentiable function (or, at least, a piecewise differentiable function), this is commonly the image of the function which is called a curve.

• The curve is said to be simple, or a Jordan arc, if $\gamma$ is injective, i.e. if for all $x$, $y$ in $I$, we have $\gamma (x)=\gamma (y)$ implies $x=y$. If $I$ is a closed bounded interval $[a,b]$, we also allow the possibility $\gamma (a)=\gamma (b)$ (this convention makes it possible to talk about "closed" simple curves, see below). In other words, this curve "does not cross itself and has no missing points".
• If $\gamma (x)=\gamma (y)$ for some $x\neq y$ (other than the extremities of $I$), then $\gamma (x)$ is called a double (or multiple) point of the curve. This is a special case of a singular point of a curve.
• A curve $\gamma$ is said to be closed or a loop if $I=[a,b]$ and if $\gamma (a)=\gamma (b)$. A closed curve is thus the image of a continuous mapping of the circle $S^{1}$; 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".

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 is a space curve which lies in no plane. These definitions of plane, space and skew curves apply also to real algebraic curves, although the above definition of a curve does not apply (a real algebraic curve may be disconnected).

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 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 Lebesgue measure (the last example can be obtained by small variation of the Peano curve construction). The dragon curve is another unusual example.

Differentiable curve

Roughly speaking a differentiable curve is a curve that is defined as being locally the image of an injective differentiable function $\gamma \colon I\rightarrow X$ from an interval I of the real numbers into a differentiable manifold X, often $\mathbb {R} ^{n}.$

More precisely, a differentiable curve is a subset C of X where every point of C has a neighborhood U such that $C\cap U$ is diffeomorphic to an interval of the real numbers. In other words, a differentiable curve differentiable manifold is of dimension one.

Length of a curve

Main article: Arc length

If $X=\mathbb {R} ^{n}$ is the $n$-dimensional Euclidean space, and if $\gamma :[a,b]\to \mathbb {R} ^{n}$ is an injective and continuously differentiable function, then the length of $\gamma$ is defined as the quantity

$\operatorname {Length} (\gamma )~{\stackrel {\text{df}}{=}}~\int _{a}^{b}|\gamma \,'(t)|~\mathrm {d} {t}.$

The length of a curve is independent of the parametrization $\gamma$.

In particular, the length $s$ of the graph of a continuously differentiable function $y=f(x)$ defined on a closed interval $[a,b]$ is

$s=\int _{a}^{b}{\sqrt {1+[f'(x)]^{2}}}~\mathrm {d} {x}.$

More generally, if $X$ is a metric space with metric $d$, then we can define the length of a curve $\gamma :[a,b]\to X$ by

$\operatorname {Length} (\gamma )~{\stackrel {\text{df}}{=}}~\sup \!\left(\left\{\sum _{i=1}^{n}d(\gamma (t_{i}),\gamma (t_{i-1}))~{\Bigg |}~n\in \mathbb {N} ~{\text{and}}~a=t_{0}<t_{1}<\ldots <t_{n}=b\right\}\right),$

where the supremum is taken over all $n\in \mathbb {N}$ and all partitions $t_{0}<t_{1}<\ldots <t_{n}$ of $[a,b]$.

A rectifiable curve is a curve with finite length. A curve $\gamma :[a,b]\to X$ is called natural (or unit-speed or parametrized by arc length) if for any $t_{1},t_{2}\in [a,b]$ such that $t_{1}\leq t_{2}$, we have

$\operatorname {Length} \!\left(\gamma |_{[t_{1},t_{2}]}\right)=t_{2}-t_{1}.$

If $\gamma :[a,b]\to X$ is a Lipschitz-continuous function, then it is automatically rectifiable. Moreover, in this case, one can define the speed (or metric derivative) of $\gamma$ at $t\in [a,b]$ as

${\operatorname {Speed} _{\gamma }}(t)~{\stackrel {\text{df}}{=}}~\limsup _{[a,b]\ni s\to t}{\frac {d(\gamma (s),\gamma (t))}{|s-t|}}$

and then show that

$\operatorname {Length} (\gamma )=\int _{a}^{b}{\operatorname {Speed} _{\gamma }}(t)~\mathrm {d} {t}.$

Differential geometry

Main article: Differential geometry of curves

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.

If $X$ is a differentiable manifold, then we can define the notion of differentiable curve in $X$. This general idea is enough to cover many of the applications of curves in mathematics. From a local point of view one can take $X$ 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 tangent vectors to $X$ by means of this notion of curve.

If $X$ is a smooth manifold, a smooth curve in $X$ is a smooth map

$\gamma \colon I\rightarrow X$.

This is a basic notion. There are less and more restricted ideas, too. If $X$ is a $C^{k}$ manifold (i.e., a manifold whose charts are $k$ times continuously differentiable), then a $C^{k}$ curve in $X$ is such a curve which is only assumed to be $C^{k}$ (i.e. $k$ times continuously differentiable). If $X$ is an analytic manifold (i.e. infinitely differentiable and charts are expressible as power series), and $\gamma$ is an analytic map, then $\gamma$ is said to be an analytic curve.

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 $C^{k}$ differentiable curves

$\gamma _{1}\colon I\rightarrow X$ and

$\gamma _{2}\colon J\rightarrow X$

are said to be equivalent if there is a bijective $C^{k}$ map

$p\colon J\rightarrow I$

such that the inverse map

$p^{-1}\colon I\rightarrow J$

is also $C^{k}$, and

$\gamma _{2}(t)=\gamma _{1}(p(t))$

for all $t$. The map $\gamma _{2}$ is called a reparametrisation of $\gamma _{1}$; and this makes an equivalence relation on the set of all $C^{k}$ differentiable curves in $X$. A $C^{k}$ arc is an equivalence class of $C^{k}$ curves under the relation of reparametrisation.

Algebraic curve

Main article: Algebraic curve

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 to be defined over F. The points of the curve C with coordinates in a field G are said to be rational over G and can be denoted C(G)). When G is the field of the rational numbers, one simply talks of rational points. For example, Fermat's Last Theorem may be restated as: For n > 2, every rational point of the Fermat curve of degree n has a zero coordinate.

Algebraic curves can also be space curves, or curves in a space of higher dimension, say n. They are defined as algebraic varieties of dimension one. They may be obtained as the common solutions of at least n–1 polynomial equations in n variables. If n–1 polynomials are sufficient to define a curve in a space of dimension n, the curve is said to be a complete intersection. By eliminating variables (by any tool of elimination theory), an algebraic curve may be projected onto a plane algebraic curve, which however may introduce new singularities such as cusps or double points.

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^df(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 uⁿ + vⁿ = wⁿ, which has an affine form xⁿ + yⁿ = 1. A similar process of homogenization may be defined for curves in higher dimensional spaces

Important examples of algebraic curves are the conics, which are nonsingular curves of degree two and genus zero, and elliptic curves, 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 zero are most often studied over the complex numbers, algebraic curves in algebraic geometry may be considered as real surfaces. In particular, the nonsingular complex projective algebraic curves are called Riemann surfaces.

See also

• Coordinate curve
• Curvature
• Curve orientation
• Curve sketching
• Differential geometry of curves
• Gallery of curves
• Implicit curve
• List of curves topics
• List of curves
• Osculating circle
• Parametric surface
• Path (topology)
• Position vector
• Vector-valued function
• Curve fitting
• Winding number

References

• A.S. Parkhomenko (2001) [1994], "Line (curve)", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 
• B.I. Golubov (2001) [1994], "Rectifiable curve", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 
• Euclid, commentary and trans. by T. L. Heath Elements Vol. 1 (1908 Cambridge) Google Books
• E. H. Lockwood A Book of Curves (1961 Cambridge)

External links

• Famous Curves Index, School of Mathematics and Statistics, University of St Andrews, Scotland
• Mathematical curves A collection of 874 two-dimensional mathematical curves
• Gallery of Space Curves Made from Circles, includes animations by Peter Moses
• Gallery of Bishop Curves and Other Spherical Curves, includes animations by Peter Moses
• The Encyclopedia of Mathematics article on lines.
• The Manifold Atlas page on 1-manifolds.

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• NAID 150000107831

• A Novel Growth Method of Freestanding GaN Using In situ Removal of Si Substrate in Hydride Vapor Phase Epitaxy

• Lee Moonsang,Mikulik Dmitry,Kim Joosung,Tak Youngjo,Kim Junyoun,Shim Munbo,Park Youngsoo,Chung Uin,Yoon Euijoon,Park Sungsoo
• Applied Physics Express 6(12), 125502-125502-4, 2013-12-25
• … The freestanding GaN exhibited high quality with FWHM of 65 arcsec in (0002) X-ray rocking curve and etch pit density of less than 1\times 10^{6}/cm². …
• NAID 150000107817

• Global dynamics of a stochastic neuronal oscillator

• Yamanobe Takanobu
• Physical Review E 88(5), 052709-1-052709-19, 2013-11
• … To do this, we use a Markov operator that both reflects the density evolution of the oscillator and is an extension of the phase transition curve, which describes the phase shift due to a single isolated impulse. …
• NAID 120005345327

• 非均一相似変換を用いた形状洗練化のためのグラフクラスタリングによる形状セグメンテーション

• 佐藤 信
• 情報処理学会研究報告. UBI, [ユビキタスコンピューティングシステム] 2013-UBI-40(7), 1-8, 2013-10-29
• 非均一相似変換を用いて平面上の形状を洗練化するために,グラフのクラスタリングを用いた形状セグメンテーション手法を提案する.提案手法では,視覚的な自然さを表現する形状特徴量をエッジの重みとするグラフをクラスタリングすることにより,構造化されていない曲線セグメントの集合から,形状セグメントを生成する.そして,その形状セグメントについて非均一相似変換をおこない,基準とする形状に類似な形状を生成する.
• NAID 110009616906

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Related Pictures

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★リンクテーブル★

リンク元	「bend」「flex」「曲がる」「曲げる」「曲線」
拡張検索	「Starling curve」「hyperbolic curve」

「bend」

　　[★]

• v.

• (過去-過去分詞bent)曲がる、曲げる、折り曲げる、ベンドする

• n.

• 屈曲

関

bending、caisson disease、curvature、curve、decompression sickness、flex、flexion、inflection

　 　 　 　 　 　

「flex」

　　[★]

• n.

• 屈曲

• v.

• 曲がる、曲げる

関

bend、bending、curvature、curve、flexion、inflection

　 　 　 　 　 　 　

「曲がる」

　　[★]

英

bend、curve、flex

関

曲線、屈曲、ベンド、折り曲げる、曲げる、カーブ

「曲げる」

　　[★]

英

bend、curve、flex

関

曲線、屈曲、ベンド、折り曲げる、曲がる、カーブ

「曲線」

　　[★]

英

• curve、curvilinear

関

曲げる、曲がる、カーブ

「Starling curve」

　　[★]

スターリング曲線、Starling曲線

「hyperbolic curve」

　　[★]

• 双曲線

関

hyperbola、hyperbolic