In mathematics , a curve (also called a curved line in older texts) is an object similar to a line , but that does not have to be straight .
51-409: Diocles may refer to: People [ edit ] Diocles (mathematician) (c. 240 BC–c. 180 BC), Greek mathematician and geometer Diocles (mythology) , one of the first priests of Demeter Diocles of Carystus (4th century BC), also known as Diocles Medicus , Greek physician Diocles of Cnidus (3rd or 2nd century BC), Greek philosopher who wrote
102-400: A ) = γ ( b ) {\displaystyle \gamma (a)=\gamma (b)} . A closed curve is thus the image of a continuous mapping of a circle . A non-closed curve may also be called an open curve . If the domain of a topological curve is a closed and bounded interval I = [ a , b ] {\displaystyle I=[a,b]} , the curve is called
153-613: A , b ] {\displaystyle [a,b]} . A rectifiable curve is a curve with finite length. A curve γ : [ a , b ] → X {\displaystyle \gamma :[a,b]\to X} is called natural (or unit-speed or parametrized by arc length) if for any t 1 , t 2 ∈ [ a , b ] {\displaystyle t_{1},t_{2}\in [a,b]} such that t 1 ≤ t 2 {\displaystyle t_{1}\leq t_{2}} , we have If γ : [
204-425: A , b ] → X {\displaystyle \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 γ {\displaystyle \gamma } at t ∈ [ a , b ] {\displaystyle t\in [a,b]} as and then show that While
255-426: A differentiable curve is a curve that is defined as being locally the image of an injective differentiable function γ : I → X {\displaystyle \gamma \colon I\rightarrow X} from an interval I of the real numbers into a differentiable manifold X , often R n . {\displaystyle \mathbb {R} ^{n}.} More precisely,
306-444: A path , also known as topological arc (or just arc ). A curve is simple if it is the image of an interval or a circle by an injective continuous function. In other words, if a curve is defined by a continuous function γ {\displaystyle \gamma } with an interval as a domain, the curve is simple if and only if any two different points of the interval have different images, except, possibly, if
357-411: A finite field are widely used in modern cryptography . 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 with a stick on the sand on a beach. Historically,
408-432: A plane algebraic curve , which however may introduce new singularities such as cusps or double points . A plane curve may also be completed to a curve in the projective plane : if a curve is defined by a polynomial f of total degree d , then w 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
459-463: A closed interval [ a , b ] {\displaystyle [a,b]} is which can be thought of intuitively as using the Pythagorean theorem at the infinitesimal scale continuously over the full length of the curve. More generally, if X {\displaystyle X} is a metric space with metric d {\displaystyle d} , then we can define
510-472: A 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
561-541: A differentiable curve is a subset C of X where every point of C has a neighborhood U such that C ∩ U {\displaystyle C\cap U} is diffeomorphic to an interval of the real numbers. In other words, a differentiable curve is a differentiable manifold of dimension one. In Euclidean geometry , an arc (symbol: ⌒ ) is a connected subset of a differentiable curve. Arcs of lines are called segments , rays , or lines , depending on how they are bounded. A common curved example
SECTION 10
#1732773105657612-427: A large influence on Arabic mathematicians, particularly on al-Haytham , the 11th-century polymath of Cairo whom Europeans knew as "Alhazen". The treatise contains sixteen propositions that are proved by conic sections . One of the fragments contains propositions seven and eight, which is a solution to the problem of dividing a sphere by a plane so that the resulting two volumes are in a given ratio. Proposition ten gives
663-439: 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: 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: A fundamental advance in
714-436: A solution to the problem of doubling the cube. This is equivalent to solving a certain cubic equation . Another fragment contains propositions eleven and twelve, which use the cissoid to solve the problem of finding two mean proportionals in between two magnitudes. Since this treatise covers more topics than just burning mirrors , it may be the case that On burning mirrors is the aggregate of three shorter works by Diocles. In
765-450: 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
816-488: A stop or backtracks on itself.) Two C k {\displaystyle C^{k}} differentiable curves are said to be equivalent if there is a bijective C k {\displaystyle C^{k}} map such that the inverse map is also C k {\displaystyle C^{k}} , and for all t {\displaystyle t} . The map γ 2 {\displaystyle \gamma _{2}}
867-570: A work quoted by Eusebius Diocles of Corinth , winner of the stadion race of the 13th Olympic Games in 728 BC Diocles of Magnesia (2nd or 1st century BC), Greek writer on ancient philosophers quoted many times by Diogenes Laertius Diocles of Megara , ancient Greek warrior from Athens Diocles of Messenia , winner of the stadion race of the 7th Olympic Games in 752 BC Diocles of Peparethus (3rd century BC), Greek historian Diocles of Phlius (fl. c. 400 BC ), comic poet Diocles of Syracuse (fl. 413–408 BC), Greek lawgiver in
918-559: Is a C k {\displaystyle C^{k}} manifold (i.e., a manifold whose charts are k {\displaystyle k} times continuously differentiable ), then a C k {\displaystyle C^{k}} curve in X {\displaystyle X} is such a curve which is only assumed to be C k {\displaystyle C^{k}} (i.e. k {\displaystyle k} times continuously differentiable). If X {\displaystyle X}
969-418: Is a curve for which X {\displaystyle X} is at least three-dimensional; 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 ). A plane simple closed curve is also called a Jordan curve . It
1020-415: Is a curve in spacetime . If X {\displaystyle X} is a differentiable manifold , then we can define the notion of differentiable curve in X {\displaystyle 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 {\displaystyle X} to be Euclidean space. On
1071-480: Is also defined as a non-self-intersecting continuous loop in the plane. The Jordan curve theorem states that the set complement in a plane of a Jordan curve consists of two connected components (that is the curve divides the plane in two non-intersecting regions that are both connected). The bounded region inside a Jordan curve is known as Jordan domain . The definition of a curve includes figures that can hardly be called curves in common usage. For example,
SECTION 20
#17327731056571122-420: Is an analytic manifold (i.e. infinitely differentiable and charts are expressible as power series ), and γ {\displaystyle \gamma } is an analytic map, then γ {\displaystyle \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
1173-487: Is an arc of a circle , called a circular arc . In a sphere (or a spheroid ), an arc of a great circle (or a great ellipse ) is called a great arc . If X = R n {\displaystyle X=\mathbb {R} ^{n}} is the n {\displaystyle n} -dimensional Euclidean space, and if γ : [ a , b ] → R n {\displaystyle \gamma :[a,b]\to \mathbb {R} ^{n}}
1224-506: Is an injective and continuously differentiable function, then the length of γ {\displaystyle \gamma } is defined as the quantity The length of a curve is independent of the parametrization γ {\displaystyle \gamma } . In particular, the length s {\displaystyle s} of the graph of a continuously differentiable function y = f ( x ) {\displaystyle y=f(x)} defined on
1275-483: Is called a reparametrization of γ 1 {\displaystyle \gamma _{1}} ; and this makes an equivalence relation on the set of all C k {\displaystyle C^{k}} differentiable curves in X {\displaystyle X} . A C k {\displaystyle C^{k}} arc is an equivalence class of C k {\displaystyle C^{k}} curves under
1326-408: Is different from Wikidata All article disambiguation pages All disambiguation pages Diocles (mathematician) Diocles ( ‹See Tfd› Greek : Διοκλῆς ; c. 240 BC – c. 180 BC) was a Greek mathematician and geometer . Although little is known about the life of Diocles, it is known that he was a contemporary of Apollonius and that he flourished sometime around the end of
1377-400: Is the zero set of a polynomial in two indeterminates . More generally, an algebraic curve is the zero set of a finite set of polynomials, which satisfies the further condition of being an algebraic variety of dimension one. If the coefficients of the polynomials belong to a field k , the curve is said to be defined over k . In the common case of a real algebraic curve , where k
1428-437: Is the field of real numbers , an algebraic curve is a finite union of topological curves. When complex zeros are considered, one has a complex algebraic curve , which, from the topological point of view, is not a curve, but a surface , and is often called a Riemann surface . Although not being curves in the common sense, algebraic curves defined over other fields have been widely studied. In particular, algebraic curves over
1479-402: 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
1530-476: The real numbers into a topological space X . Properly speaking, the curve is the image of γ . {\displaystyle \gamma .} However, in some contexts, γ {\displaystyle \gamma } itself is called a curve, especially when the image does not look like what is generally called a curve and does not characterize sufficiently γ . {\displaystyle \gamma .} For example,
1581-455: The real part of the curve. It is therefore only the real part of an algebraic curve that can be a topological curve (this is not always the case, as the real part of an algebraic curve may be disconnected and contain isolated points). The whole curve, that is the set of its complex point is, from the topological point of view a surface. In particular, the nonsingular complex projective algebraic curves are called Riemann surfaces . The points of
Diocles - Misplaced Pages Continue
1632-478: The 3rd century BC and the beginning of the 2nd century BC. Diocles is thought to be the first person to prove the focal property of the parabola . His name is associated with the geometric curve called the Cissoid of Diocles , which was used by Diocles to solve the problem of doubling the cube . The curve was alluded to by Proclus in his commentary on Euclid and attributed to Diocles by Geminus as early as
1683-511: The beginning of the 1st century. Fragments of a work by Diocles entitled On burning mirrors were preserved by Eutocius in his commentary of Archimedes ' On the Sphere and the Cylinder and also survived in an Arabic translation of the lost Greek original titled Kitāb Dhiyūqlīs fī l-marāyā l-muḥriqa (lit. “The book of Diocles on burning mirrors”). Historically, On burning mirrors had
1734-401: 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, curve theory is viewed as
1785-579: The city-state of Syracuse Diocletian (244–311), Roman emperor formerly named Diocles Diocles (1st century BC), or Tyrannion the Younger Gaius Appuleius Diocles (104–after 146 AD), Roman charioteer Other [ edit ] Diocles (bug) , a genus of bugs in the family Coreidae Diocles laser, a laser that uses Chirped pulse amplification at the University of Nebraska–Lincoln Topics referred to by
1836-401: The class of topological curves is very broad, and contains some curves that do not look as one may expect for a curve, or even cannot be drawn. This is the case of space-filling curves and fractal curves . For ensuring more regularity, the function that defines a curve is often supposed to be differentiable , and the curve is then said to be a differentiable curve . A plane algebraic curve
1887-401: 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
1938-440: The first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width." This definition of a curve has been formalized in modern mathematics as: A curve is the image of an interval to a topological space by a continuous function . In some contexts,
1989-606: The function that defines the curve is called a parametrization , and the curve is a parametric curve . In this article, these curves are sometimes called topological curves to distinguish them from more constrained curves such as differentiable curves . This definition encompasses most curves that are studied in mathematics; notable exceptions are level curves (which are unions of curves and isolated points), and algebraic curves (see below). Level curves and algebraic curves are sometimes called implicit curves , since they are generally defined by implicit equations . Nevertheless,
2040-423: The image of a curve can cover a square in the plane ( space-filling curve ), and a simple curve may have a positive area. Fractal curves can have properties that are strange for the common sense. For example, a fractal curve can have a Hausdorff dimension bigger than one (see Koch snowflake ) and even a positive area. An example is the dragon curve , which has many other unusual properties. Roughly speaking
2091-490: The image of the Peano curve or, more generally, a space-filling curve completely fills a square, and therefore does not give any information on how γ {\displaystyle \gamma } is defined. A curve γ {\displaystyle \gamma } is closed or is a loop if I = [ a , b ] {\displaystyle I=[a,b]} and γ (
Diocles - Misplaced Pages Continue
2142-466: The length of a curve γ : [ a , b ] → X {\displaystyle \gamma :[a,b]\to X} by where the supremum is taken over all n ∈ N {\displaystyle n\in \mathbb {N} } and all partitions t 0 < t 1 < … < t n {\displaystyle t_{0}<t_{1}<\ldots <t_{n}} of [
2193-473: The other hand, it is useful to be more general, in that (for example) it is possible to define the tangent vectors to X {\displaystyle X} by means of this notion of curve. If X {\displaystyle X} is a smooth manifold , a smooth curve in X {\displaystyle X} is a smooth map This is a basic notion. There are less and more restricted ideas, too. If X {\displaystyle X}
2244-465: The points are the endpoints of the interval. Intuitively, a simple curve is a curve that "does not cross itself and has no missing points" (a continuous non-self-intersecting curve). A plane curve is a curve for which X {\displaystyle X} is the Euclidean plane —these are the examples first encountered—or in some cases the projective plane . A space curve
2295-400: 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 . In the case of a curve defined over the real numbers , one normally considers points with complex coordinates. In this case, a point with real coordinates is a real point , and the set of all real points is
2346-425: The relation of reparametrization. Algebraic curves are the curves considered in algebraic geometry . A plane algebraic curve is the set 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 . One says that the curve is defined over F . Algebraic geometry normally considers not only points with coordinates in F but all
2397-450: The same term [REDACTED] This disambiguation page lists articles associated with the title Diocles . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Diocles&oldid=1241566543 " Categories : Disambiguation pages Human name disambiguation pages Hidden categories: Short description
2448-451: The same work, Diocles, just after demonstrating that the parabolic mirror could focus the rays in a single point, he mentioned that It is possible to obtain a lens with the same property. Curve Intuitively, a curve may be thought of as the trace left by a moving point . This is the definition that appeared more than 2000 years ago in Euclid's Elements : "The [curved] line is […]
2499-446: The special case of dimension one of the theory of manifolds and algebraic varieties . Nevertheless, many questions remain specific to curves, such as space-filling curves , Jordan curve theorem and Hilbert's sixteenth problem . A topological curve can be specified by a continuous function γ : I → X {\displaystyle \gamma \colon I\rightarrow X} from an interval I of
2550-466: The term line was used in place of the more modern term curve . Hence the terms 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
2601-678: The theory of curves was the introduction of analytic geometry by René Descartes 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 algebraic curves that can be defined using polynomial equations , and transcendental curves that cannot. 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
SECTION 50
#1732773105657#656343