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90-432: The term oval when used to describe curves in geometry is not well-defined, except in the context of projective geometry . Many distinct curves are commonly called ovals or are said to have an "oval shape". Generally, to be called an oval, a plane curve should resemble the outline of an egg or an ellipse . In particular, these are common traits of ovals: Here are examples of ovals described elsewhere: An ovoid

180-454: A b ‖ γ ′ ( t ) ‖ d t . {\displaystyle l~{\stackrel {\text{def}}{=}}~\int _{a}^{b}\left\|\gamma '(t)\right\|\,\mathrm {d} {t}.} The length of a parametric curve is invariant under reparametrization and is therefore a differential-geometric property of the parametric curve. For each regular parametric C -curve γ : [

270-400: 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

360-613: 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 : [

450-635: A , b ] → R n {\displaystyle \gamma :[a,b]\to \mathbb {R} ^{n}} , where r ≥ 1 , the function is defined ∀ t ∈ [ a , b ] : s ( t )   = def   ∫ a t ‖ γ ′ ( x ) ‖ d x . {\displaystyle \forall t\in [a,b]:\quad s(t)~{\stackrel {\text{def}}{=}}~\int _{a}^{t}\left\|\gamma '(x)\right\|\,\mathrm {d} {x}.} Writing γ (s) = γ ( t ( s )) , where t ( s )

540-425: 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

630-426: 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,

720-444: 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

810-646: A bijective C -map φ  : I 1 → I 2 such that ∀ t ∈ I 1 : φ ′ ( t ) ≠ 0 {\displaystyle \forall t\in I_{1}:\quad \varphi '(t)\neq 0} and ∀ t ∈ I 1 : γ 2 ( φ ( t ) ) = γ 1 ( t ) . {\displaystyle \forall t\in I_{1}:\quad \gamma _{2}{\bigl (}\varphi (t){\bigr )}=\gamma _{1}(t).} γ 2

900-411: 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,

990-432: 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

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1080-463: 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

1170-472: 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

1260-538: A curved figure. Curve 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 . 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 […] the first species of quantity, which has only one dimension, namely length, without any width nor depth, and

1350-541: 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

1440-424: A figure that resembles two semicircles joined by a rectangle, like a cricket infield , speed skating rink or an athletics track . However, this is most correctly called a stadium . The term "ellipse" is often used interchangeably with oval, but it has a more specific mathematical meaning. The term "oblong" is also used to mean oval, though in geometry an oblong refers to rectangle with unequal adjacent sides, not

1530-411: A lack of reflection symmetry across the equatorial plane, it may also refer to true prolate ellipsoids. It can also be used to describe the 2-dimensional figure that, if revolved around its major axis , produces the 3-dimensional surface. In technical drawing , an oval is a figure that is constructed from two pairs of arcs, with two different radii (see image on the right). The arcs are joined at

1620-439: 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

1710-426: A parametrized C curve γ = γ ( t ) , for every value t = t 0 of the parameter, the vector γ ′ ( t 0 ) = d d t γ ( t ) | t = t 0 {\displaystyle \gamma '(t_{0})=\left.{\frac {\mathrm {d} }{\mathrm {d} t}}{\boldsymbol {\gamma }}(t)\right|_{t=t_{0}}}

1800-399: A point in which lines tangential to both joining arcs lie on the same line, thus making the joint smooth. Any point of an oval belongs to an arc with a constant radius (shorter or longer), but in an ellipse , the radius is continuously changing. In common speech, "oval" means a shape rather like an egg or an ellipse, which may be two-dimensional or three-dimensional. It also often refers to

1890-404: A regular curve in a Euclidean space has no intrinsic geometry. Any regular curve may be parametrized by the arc length (the natural parametrization ). From the point of view of a theoretical point particle on the curve that does not know anything about the ambient space, all curves would appear the same. Different space curves are only distinguished by how they bend and twist. Quantitatively, this

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1980-450: 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

2070-488: 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}}

2160-862: A unique (up to transformations using the Euclidean group ) C -curve γ which is regular of order n and has the following properties: ‖ γ ′ ( t ) ‖ = 1 t ∈ [ a , b ] χ i ( t ) = ⟨ e i ′ ( t ) , e i + 1 ( t ) ⟩ ‖ γ ′ ( t ) ‖ {\displaystyle {\begin{aligned}\|\gamma '(t)\|&=1&t\in [a,b]\\\chi _{i}(t)&={\frac {\langle \mathbf {e} _{i}'(t),\mathbf {e} _{i+1}(t)\rangle }{\|{\boldsymbol {\gamma }}'(t)\|}}\end{aligned}}} where

2250-720: Is simple if γ | ( a , b ) : ( a , b ) → R n {\displaystyle \gamma |_{(a,b)}:(a,b)\to \mathbb {R} ^{n}} is injective . It is analytic if each component function of γ is an analytic function , that is, it is of class C . The curve γ is regular of order m (where m ≤ r ) if, for every t ∈ I , { γ ′ ( t ) , γ ″ ( t ) , … , γ ( m ) ( t ) } {\displaystyle \left\{\gamma '(t),\gamma ''(t),\ldots ,{\gamma ^{(m)}}(t)\right\}}

2340-413: Is r -times continuously differentiable (that is, the component functions of γ are continuously differentiable), where n ∈ N {\displaystyle n\in \mathbb {N} } , r ∈ N ∪ { ∞ } {\displaystyle r\in \mathbb {N} \cup \{\infty \}} , and I is a non-empty interval of real numbers. The image of

2430-559: 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}

2520-508: Is a linearly independent subset of R n {\displaystyle \mathbb {R} ^{n}} . In particular, a parametric C -curve γ is regular if and only if γ ′ ( t ) ≠ 0 for any t ∈ I . Given the image of a parametric curve, there are several different parametrizations of the parametric curve. Differential geometry aims to describe the properties of parametric curves that are invariant under certain reparametrizations. A suitable equivalence relation on

2610-432: Is a circular helix. The first three Frenet vectors and generalized curvatures can be visualized in three-dimensional space. They have additional names and more semantic information attached to them. If a curve γ represents the path of a particle, then the instantaneous velocity of the particle at a given point P is expressed by a vector , called the tangent vector to the curve at P . Mathematically, given

2700-418: 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

2790-415: 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

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2880-480: 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,

2970-1227: Is always orthogonal to the unit tangent and normal vectors at t . It is defined as e 3 ( t ) = e 3 ¯ ( t ) ‖ e 3 ¯ ( t ) ‖ , e 3 ¯ ( t ) = γ ‴ ( t ) − ⟨ γ ‴ ( t ) , e 1 ( t ) ⟩ e 1 ( t ) − ⟨ γ ‴ ( t ) , e 2 ( t ) ⟩ e 2 ( t ) {\displaystyle \mathbf {e} _{3}(t)={\frac {{\overline {\mathbf {e} _{3}}}(t)}{\left\|{\overline {\mathbf {e} _{3}}}(t)\right\|}},\quad {\overline {\mathbf {e} _{3}}}(t)={\boldsymbol {\gamma }}'''(t)-{\bigr \langle }{\boldsymbol {\gamma }}'''(t),\mathbf {e} _{1}(t){\bigr \rangle }\,\mathbf {e} _{1}(t)-{\bigl \langle }{\boldsymbol {\gamma }}'''(t),\mathbf {e} _{2}(t){\bigr \rangle }\,\mathbf {e} _{2}(t)} In 3-dimensional space,

3060-420: 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

3150-487: 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}}

3240-506: 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

3330-483: 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

3420-627: Is called curvature and measures the deviance of γ from being a straight line relative to the osculating plane. It is defined as κ ( t ) = χ 1 ( t ) = ⟨ e 1 ′ ( t ) , e 2 ( t ) ⟩ ‖ γ ′ ( t ) ‖ {\displaystyle \kappa (t)=\chi _{1}(t)={\frac {{\bigl \langle }\mathbf {e} _{1}'(t),\mathbf {e} _{2}(t){\bigr \rangle }}{\left\|{\boldsymbol {\gamma }}'(t)\right\|}}} and

3510-440: Is called the curvature of γ at point t . It can be shown that κ ( t ) = ‖ e 1 ′ ( t ) ‖ ‖ γ ′ ( t ) ‖ . {\displaystyle \kappa (t)={\frac {\left\|\mathbf {e} _{1}'(t)\right\|}{\left\|{\boldsymbol {\gamma }}'(t)\right\|}}.} The reciprocal of

3600-652: Is called the torsion of γ at point t . The third derivative may be used to define aberrancy , a metric of non-circularity of a curve. Given n − 1 functions: χ i ∈ C n − i ( [ a , b ] , R n ) , χ i ( t ) > 0 , 1 ≤ i ≤ n − 1 {\displaystyle \chi _{i}\in C^{n-i}([a,b],\mathbb {R} ^{n}),\quad \chi _{i}(t)>0,\quad 1\leq i\leq n-1} then there exists

3690-673: Is far easier and more natural to describe local properties (e.g. curvature, torsion) in terms of a local reference system than using a global one such as Euclidean coordinates. Given a C -curve γ in R n {\displaystyle \mathbb {R} ^{n}} which is regular of order n the Frenet frame for the curve is the set of orthonormal vectors e 1 ( t ) , … , e n ( t ) {\displaystyle \mathbf {e} _{1}(t),\ldots ,\mathbf {e} _{n}(t)} called Frenet vectors . They are constructed from

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3780-880: Is illustrated by the examples of a right-handed helix and a left-handed helix. The second generalized curvature χ 2 ( t ) is called torsion and measures the deviance of γ from being a plane curve . In other words, if the torsion is zero, the curve lies completely in the same osculating plane (there is only one osculating plane for every point t ). It is defined as τ ( t ) = χ 2 ( t ) = ⟨ e 2 ′ ( t ) , e 3 ( t ) ⟩ ‖ γ ′ ( t ) ‖ {\displaystyle \tau (t)=\chi _{2}(t)={\frac {{\bigl \langle }\mathbf {e} _{2}'(t),\mathbf {e} _{3}(t){\bigr \rangle }}{\left\|{\boldsymbol {\gamma }}'(t)\right\|}}} and

3870-449: Is measured by the differential-geometric invariants called the curvature and the torsion of a curve. The fundamental theorem of curves asserts that the knowledge of these invariants completely determines the curve. A parametric C - curve or a C - parametrization is a vector-valued function γ : I → R n {\displaystyle \gamma :I\to \mathbb {R} ^{n}} that

3960-402: 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, the function that defines the curve is called a parametrization , and

4050-641: Is often very difficult to calculate the natural parametrization of a parametric curve, but it is useful for theoretical arguments. For a given parametric curve γ , the natural parametrization is unique up to a shift of parameter. The quantity E ( γ )   = def   1 2 ∫ a b ‖ γ ′ ( t ) ‖ 2   d t {\displaystyle E(\gamma )~{\stackrel {\text{def}}{=}}~{\frac {1}{2}}\int _{a}^{b}\left\|\gamma '(t)\right\|^{2}~\mathrm {d} {t}}

4140-480: Is sometimes called the energy or action of the curve; this name is justified because the geodesic equations are the Euler–Lagrange equations of motion for this action. A Frenet frame is a moving reference frame of n orthonormal vectors e i ( t ) which are used to describe a curve locally at each point γ ( t ) . It is the main tool in the differential geometric treatment of curves because it

4230-400: 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

4320-424: Is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus . Many specific curves have been thoroughly investigated using the synthetic approach . Differential geometry takes another path: curves are represented in a parametrized form , and their geometric properties and various quantities associated with them, such as

4410-437: 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

4500-763: Is the inverse function of s ( t ) . This is a re-parametrization γ of γ that is called an arc-length parametrization , natural parametrization , unit-speed parametrization . The parameter s ( t ) is called the natural parameter of γ . This parametrization is preferred because the natural parameter s ( t ) traverses the image of γ at unit speed, so that ∀ t ∈ I : ‖ γ ¯ ′ ( s ( t ) ) ‖ = 1. {\displaystyle \forall t\in I:\quad \left\|{\overline {\gamma }}'{\bigl (}s(t){\bigr )}\right\|=1.} In practice, it

4590-405: Is the natural parameter, then the tangent vector has unit length. The formula simplifies: e 1 ( s ) = γ ′ ( s ) . {\displaystyle \mathbf {e} _{1}(s)={\boldsymbol {\gamma }}'(s).} The unit tangent vector determines the orientation of the curve, or the forward direction, corresponding to the increasing values of

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4680-425: Is the surface in 3-dimensional space generated by rotating an oval curve about one of its axes of symmetry. The adjectives ovoidal and ovate mean having the characteristic of being an ovoid, and are often used as synonyms for "egg-shaped". For finite planes (i.e. the set of points is finite) there is a more convenient characterization: An ovoid in a projective space is a set Ω of points such that: In

4770-398: Is the tangent vector at the point P = γ ( t 0 ) . Generally speaking, the tangent vector may be zero . The tangent vector's magnitude ‖ γ ′ ( t 0 ) ‖ {\displaystyle \left\|{\boldsymbol {\gamma }}'(t_{0})\right\|} is the speed at the time t 0 . The first Frenet vector e 1 ( t )

4860-553: Is the torsion. A Bertrand curve is a regular curve in R 3 {\displaystyle \mathbb {R} ^{3}} with the additional property that there is a second curve in R 3 {\displaystyle \mathbb {R} ^{3}} such that the principal normal vectors to these two curves are identical at each corresponding point. In other words, if γ 1 ( t ) and γ 2 ( t ) are two curves in R 3 {\displaystyle \mathbb {R} ^{3}} such that for any t ,

4950-430: Is the unit tangent vector in the same direction, defined at each regular point of γ : e 1 ( t ) = γ ′ ( t ) ‖ γ ′ ( t ) ‖ . {\displaystyle \mathbf {e} _{1}(t)={\frac {{\boldsymbol {\gamma }}'(t)}{\left\|{\boldsymbol {\gamma }}'(t)\right\|}}.} If t = s

5040-417: Is then said to be a re-parametrization of γ 1 . Re-parametrization defines an equivalence relation on the set of all parametric C -curves of class C . The equivalence class of this relation simply a C -curve. An even finer equivalence relation of oriented parametric C -curves can be defined by requiring φ to satisfy φ ′ ( t ) > 0 . Equivalent parametric C -curves have

5130-402: 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

5220-512: The curvature and the arc length , are expressed via derivatives and integrals using vector calculus . One of the most important tools used to analyze a curve is the Frenet frame , a moving frame that provides a coordinate system at each point of the curve that is "best adapted" to the curve near that point. The theory of curves is much simpler and narrower in scope than the theory of surfaces and its higher-dimensional generalizations because

5310-432: The finite case only for dimension 3 there exist ovoids. A convenient characterization is: The shape of an egg is approximated by the "long" half of a prolate spheroid , joined to a "short" half of a roughly spherical ellipsoid , or even a slightly oblate spheroid . These are joined at the equator and share a principal axis of rotational symmetry , as illustrated above. Although the term egg-shaped usually implies

5400-476: 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,

5490-455: 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

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5580-450: The trajectory of a moving point in space. When I is a closed interval [ a , b ] , γ ( a ) is called the starting point and γ ( b ) is the endpoint of γ . If the starting and the end points coincide (that is, γ ( a ) = γ ( b ) ), then γ is a closed curve or a loop . To be a C -loop, the function γ must be r -times continuously differentiable and satisfy γ ( a ) = γ ( b ) for 0 ≤ k ≤ r . The parametric curve

5670-401: 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

5760-401: 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

5850-441: The curvature 1 κ ( t ) {\displaystyle {\frac {1}{\kappa (t)}}} is called the radius of curvature . A circle with radius r has a constant curvature of κ ( t ) = 1 r {\displaystyle \kappa (t)={\frac {1}{r}}} whereas a line has a curvature of 0. The unit binormal vector is the third Frenet vector e 3 ( t ) . It

5940-535: 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,

6030-2101: The derivatives of γ ( t ) using the Gram–Schmidt orthogonalization algorithm with e 1 ( t ) = γ ′ ( t ) ‖ γ ′ ( t ) ‖ e j ( t ) = e j ¯ ( t ) ‖ e j ¯ ( t ) ‖ , e j ¯ ( t ) = γ ( j ) ( t ) − ∑ i = 1 j − 1 ⟨ γ ( j ) ( t ) , e i ( t ) ⟩ e i ( t ) ⟨ {\displaystyle {\begin{aligned}\mathbf {e} _{1}(t)&={\frac {{\boldsymbol {\gamma }}'(t)}{\left\|{\boldsymbol {\gamma }}'(t)\right\|}}\\[1ex]\mathbf {e} _{j}(t)&={\frac {{\overline {\mathbf {e} _{j}}}(t)}{\left\|{\overline {\mathbf {e} _{j}}}(t)\right\|}},&{\overline {\mathbf {e} _{j}}}(t)&={\boldsymbol {\gamma }}^{(j)}(t)-\sum _{i=1}^{j-1}\left\langle {\boldsymbol {\gamma }}^{(j)}(t),\,\mathbf {e} _{i}(t)\right\rangle \,\mathbf {e} _{i}(t){\vphantom {\Bigg \langle }}\end{aligned}}} The real-valued functions χ i ( t ) are called generalized curvatures and are defined as χ i ( t ) = ⟨ e i ′ ( t ) , e i + 1 ( t ) ⟩ ‖ γ ′ ( t ) ‖ {\displaystyle \chi _{i}(t)={\frac {{\bigl \langle }\mathbf {e} _{i}'(t),\mathbf {e} _{i+1}(t){\bigr \rangle }}{\left\|{\boldsymbol {\gamma }}^{'}(t)\right\|}}} The Frenet frame and

6120-546: The equation simplifies to e 3 ( t ) = e 1 ( t ) × e 2 ( t ) {\displaystyle \mathbf {e} _{3}(t)=\mathbf {e} _{1}(t)\times \mathbf {e} _{2}(t)} or to e 3 ( t ) = − e 1 ( t ) × e 2 ( t ) , {\displaystyle \mathbf {e} _{3}(t)=-\mathbf {e} _{1}(t)\times \mathbf {e} _{2}(t),} That either sign may occur

6210-401: 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

6300-421: The generalized curvatures are invariant under reparametrization and are therefore differential geometric properties of the curve. For curves in R 3 {\displaystyle \mathbb {R} ^{3}} χ 1 ( t ) {\displaystyle \chi _{1}(t)} is the curvature and χ 2 ( t ) {\displaystyle \chi _{2}(t)}

6390-478: 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 that w is not zero. An example is the Fermat curve u + v = w , which has an affine form x + y = 1 . A similar process of homogenization may be defined for curves in higher dimensional spaces. Differentiable curve Differential geometry of curves

6480-423: 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

6570-490: 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 (

6660-466: 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 [

6750-513: The normal vector at point t define the osculating plane at point t . It can be shown that ē 2 ( t ) ∝ e ′ 1 ( t ) . Therefore, e 2 ( t ) = e 1 ′ ( t ) ‖ e 1 ′ ( t ) ‖ . {\displaystyle \mathbf {e} _{2}(t)={\frac {\mathbf {e} _{1}'(t)}{\left\|\mathbf {e} _{1}'(t)\right\|}}.} The first generalized curvature χ 1 ( t )

6840-473: 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}

6930-783: The parameter. The unit tangent vector taken as a curve traces the spherical image of the original curve. A curve normal vector , sometimes called the curvature vector , indicates the deviance of the curve from being a straight line. It is defined as e 2 ¯ ( t ) = γ ″ ( t ) − ⟨ γ ″ ( t ) , e 1 ( t ) ⟩ e 1 ( t ) . {\displaystyle {\overline {\mathbf {e} _{2}}}(t)={\boldsymbol {\gamma }}''(t)-{\bigl \langle }{\boldsymbol {\gamma }}''(t),\mathbf {e} _{1}(t){\bigr \rangle }\,\mathbf {e} _{1}(t).} Its normalized form,

7020-471: The parametric curve is γ [ I ] ⊆ R n {\displaystyle \gamma [I]\subseteq \mathbb {R} ^{n}} . The parametric curve γ and its image γ [ I ] must be distinguished because a given subset of R n {\displaystyle \mathbb {R} ^{n}} can be the image of many distinct parametric curves. The parameter t in γ ( t ) can be thought of as representing time, and γ

7110-409: 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

7200-400: 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

7290-425: 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

7380-400: The same image, and equivalent oriented parametric C -curves even traverse the image in the same direction. The length l of a parametric C -curve γ : [ a , b ] → R n {\displaystyle \gamma :[a,b]\to \mathbb {R} ^{n}} is defined as l   = def   ∫

7470-442: The same two-dimensional plane are characterized by the existence of a linear relation a κ ( t ) + b τ ( t ) = 1 where κ ( t ) and τ ( t ) are the curvature and torsion of γ 1 ( t ) and a and b are real constants with a ≠ 0 . Furthermore, the product of torsions of a Bertrand pair of curves is constant. If γ 1 has more than one Bertrand mate then it has infinitely many. This only occurs when γ 1

7560-916: The set e 1 ( t ) , … , e n ( t ) {\displaystyle \mathbf {e} _{1}(t),\ldots ,\mathbf {e} _{n}(t)} is the Frenet frame for the curve. By additionally providing a start t 0 in I , a starting point p 0 in R n {\displaystyle \mathbb {R} ^{n}} and an initial positive orthonormal Frenet frame { e 1 , ..., e n − 1 } with γ ( t 0 ) = p 0 e i ( t 0 ) = e i , 1 ≤ i ≤ n − 1 {\displaystyle {\begin{aligned}{\boldsymbol {\gamma }}(t_{0})&=\mathbf {p} _{0}\\\mathbf {e} _{i}(t_{0})&=\mathbf {e} _{i},\quad 1\leq i\leq n-1\end{aligned}}}

7650-817: The set of all parametric curves must be defined. The differential-geometric properties of a parametric curve (such as its length, its Frenet frame , and its generalized curvature) are invariant under reparametrization and therefore properties of the equivalence class itself. The equivalence classes are called C -curves and are central objects studied in the differential geometry of curves. Two parametric C -curves, γ 1 : I 1 → R n {\displaystyle \gamma _{1}:I_{1}\to \mathbb {R} ^{n}} and γ 2 : I 2 → R n {\displaystyle \gamma _{2}:I_{2}\to \mathbb {R} ^{n}} , are said to be equivalent if and only if there exists

7740-446: 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

7830-466: 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

7920-678: 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

8010-529: The two principal normals N 1 ( t ), N 2 (t) are equal, then γ 1 and γ 2 are Bertrand curves, and γ 2 is called the Bertrand mate of γ 1 . We can write γ 2 ( t ) = γ 1 ( t ) + r N 1 ( t ) for some constant r . According to problem 25 in Kühnel's "Differential Geometry Curves – Surfaces – Manifolds", it is also true that two Bertrand curves that do not lie in

8100-463: The unit normal vector, is the second Frenet vector e 2 ( t ) and is defined as e 2 ( t ) = e 2 ¯ ( t ) ‖ e 2 ¯ ( t ) ‖ . {\displaystyle \mathbf {e} _{2}(t)={\frac {{\overline {\mathbf {e} _{2}}}(t)}{\left\|{\overline {\mathbf {e} _{2}}}(t)\right\|}}.} The tangent and

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