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62-480: A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length . The perimeter of a circle or an ellipse is called its circumference . Calculating the perimeter has several practical applications. A calculated perimeter is the length of fence required to surround a yard or garden. The perimeter of a wheel/circle (its circumference) describes how far it will roll in one revolution . Similarly,

124-400: 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

186-613: 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 : [

248-425: 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

310-426: 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,

372-444: 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

434-411: 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,

496-432: 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

558-427: A circle by surrounding it with regular polygons . The perimeter of a polygon equals the sum of the lengths of its sides (edges) . In particular, the perimeter of a rectangle of width w {\displaystyle w} and length ℓ {\displaystyle \ell } equals 2 w + 2 ℓ . {\displaystyle 2w+2\ell .} An equilateral polygon

620-457: 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

682-468: 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

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744-541: 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

806-439: 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

868-407: A piece from a figure, its area decreases but its perimeter may not. The convex hull of a figure may be visualized as the shape formed by a rubber band stretched around it. In the animated picture on the left, all the figures have the same convex hull; the big, first hexagon . The isoperimetric problem is to determine a figure with the largest area, amongst those having a given perimeter. The solution

930-399: A reduction) of a shape make its area grow (or decrease) as well as its perimeter. For example, if a field is drawn on a 1/10,000 scale map, the actual field perimeter can be calculated multiplying the drawing perimeter by 10,000. The real area is 10,000 times the area of the shape on the map. Nevertheless, there is no relation between the area and the perimeter of an ordinary shape. For example,

992-450: 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

1054-488: 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}}

1116-508: A triangle is a segment from the midpoint of a side of a triangle to the opposite side such that the perimeter is divided into two equal lengths. The three cleavers of a triangle all intersect each other at the triangle's Spieker center . The perimeter of a circle , often called the circumference, is proportional to its diameter and its radius . That is to say, there exists a constant number pi , π (the Greek p for perimeter), such that if P

1178-559: 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}

1240-475: Is a polygon which has all sides of the same length. Except in the triangle case, an equilateral polygon does not need to also be equiangular (have all angles equal), but if it does then it is a regular polygon . If the number of sides is at least four, an equilateral polygon does not need to be a convex polygon : it could be concave or even self-intersecting . All regular polygons and edge-transitive polygons are equilateral. When an equilateral polygon

1302-418: 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

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1364-415: 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

1426-494: Is a polygon which has all sides of the same length (for example, a rhombus is a 4-sided equilateral polygon). To calculate the perimeter of an equilateral polygon, one must multiply the common length of the sides by the number of sides. A regular polygon may be characterized by the number of its sides and by its circumradius , that is to say, the constant distance between its centre and each of its vertices . The length of its sides can be calculated using trigonometry . If R

1488-465: Is a regular polygon's radius and n is the number of its sides, then its perimeter is A splitter of a triangle is a cevian (a segment from a vertex to the opposite side) that divides the perimeter into two equal lengths, this common length being called the semiperimeter of the triangle. The three splitters of a triangle all intersect each other at the Nagel point of the triangle. A cleaver of

1550-480: 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,

1612-420: 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

1674-487: 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}}

1736-506: 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

1798-483: 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

1860-408: Is described by the theory of Caccioppoli sets . Polygons are fundamental to determining perimeters, not only because they are the simplest shapes but also because the perimeters of many shapes are calculated by approximating them with sequences of polygons tending to these shapes. The first mathematician known to have used this kind of reasoning is Archimedes , who approximated the perimeter of

1922-424: Is important in the calculation. The computation of the digits of π is relevant to many fields, such as mathematical analysis , algorithmics and computer science . The perimeter and the area are two main measures of geometric figures. Confusing them is a common error, as well as believing that the greater one of them is, the greater the other must be. Indeed, a commonplace observation is that an enlargement (or

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1984-418: Is intuitive; it is the circle . In particular, this can be used to explain why drops of fat on a broth surface are circular. This problem may seem simple, but its mathematical proof requires some sophisticated theorems. The isoperimetric problem is sometimes simplified by restricting the type of figures to be used. In particular, to find the quadrilateral , or the triangle, or another particular figure, with

2046-494: Is non-crossing and cyclic (its vertices are on a circle) it must be regular. An equilateral quadrilateral must be convex; this polygon is a rhombus (possibly a square ). A convex equilateral pentagon can be described by two consecutive angles, which together determine the other angles. However, equilateral pentagons, and equilateral polygons with more than five sides, can also be concave, and if concave pentagons are allowed then two angles are no longer sufficient to determine

2108-402: 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

2170-400: 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

2232-468: Is the circle's perimeter and D its diameter then, In terms of the radius r of the circle, this formula becomes, To calculate a circle's perimeter, knowledge of its radius or diameter and the number π suffices. The problem is that π is not rational (it cannot be expressed as the quotient of two integers ), nor is it algebraic (it is not a root of a polynomial equation with rational coefficients). So, obtaining an accurate approximation of π

2294-437: 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

2356-689: Is the length of the path and d s {\displaystyle ds} is an infinitesimal line element. Both of these must be replaced by algebraic forms in order to be practically calculated. If the perimeter is given as a closed piecewise smooth plane curve γ : [ a , b ] → R 2 {\displaystyle \gamma :[a,b]\to \mathbb {R} ^{2}} with then its length L {\displaystyle L} can be computed as follows: A generalized notion of perimeter, which includes hypersurfaces bounding volumes in n {\displaystyle n} - dimensional Euclidean spaces ,

2418-402: 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

2480-476: 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,

2542-455: 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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2604-426: The amount of string wound around a spool is related to the spool's perimeter; if the length of the string was exact, it would equal the perimeter. The perimeter is the distance around a shape. Perimeters for more general shapes can be calculated, as any path , with ∫ 0 L d s {\textstyle \int _{0}^{L}\mathrm {d} s} , where L {\displaystyle L}

2666-401: 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

2728-401: 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

2790-535: 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,

2852-401: 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

2914-479: 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. Equilateral polygon In geometry , an equilateral polygon

2976-423: 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

3038-486: 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 (

3100-403: The largest area amongst those with the same shape having a given perimeter. The solution to the quadrilateral isoperimetric problem is the square , and the solution to the triangle problem is the equilateral triangle . In general, the polygon with n sides having the largest area and a given perimeter is the regular polygon , which is closer to being a circle than is any irregular polygon with

3162-466: 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 [

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3224-473: 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}

3286-484: The perimeter of a rectangle of width 0.001 and length 1000 is slightly above 2000, while the perimeter of a rectangle of width 0.5 and length 2 is 5. Both areas are equal to 1. Proclus (5th century) reported that Greek peasants "fairly" parted fields relying on their perimeters. However, a field's production is proportional to its area, not to its perimeter, so many naive peasants may have gotten fields with long perimeters but small areas (thus, few crops). If one removes

3348-417: The perpendicular distances from an interior point to the sides of an equilateral polygon is independent of the location of the interior point. The principal diagonals of a hexagon each divide the hexagon into quadrilaterals. In any convex equilateral hexagon with common side a , there exists a principal diagonal d 1 such that and a principal diagonal d 2 such that When an equilateral polygon

3410-460: 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

3472-400: 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

3534-425: 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

3596-519: The same number of sides. The word comes from the Greek περίμετρος perimetros , from περί peri "around" and μέτρον metron "measure". Path (geometry) 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

3658-417: The shape of the pentagon. A tangential polygon (one that has an incircle tangent to all its sides) is equilateral if and only if the alternate angles are equal (that is, angles 1, 3, 5, ... are equal and angles 2, 4, ... are equal). Thus if the number of sides n is odd, a tangential polygon is equilateral if and only if it is regular. Viviani's theorem generalizes to equilateral polygons: The sum of

3720-446: 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

3782-462: 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

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3844-678: 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

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