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 .
103-516: 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 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
206-400: A 2 , . . . a n ) ∣ a 1 c 1 + a 2 c 2 + . . . a n c n = d } , {\displaystyle L=\lbrace (a_{1},a_{2},...a_{n})\mid a_{1}c_{1}+a_{2}c_{2}+...a_{n}c_{n}=d\rbrace ,} where c 1 through c n and d are constants and n
309-400: 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
412-613: 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 : [
515-425: 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
618-453: A 1 , a 2 , … , a n ) where n is the dimension of the space in which the point is located. Many constructs within Euclidean geometry consist of an infinite collection of points that conform to certain axioms. This is usually represented by a set of points; As an example, a line is an infinite set of points of the form L = { ( a 1 ,
721-426: 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,
824-444: 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
927-400: A compass , scriber , or pen, whose pointed tip can mark a small dot or prick a small hole representing a point, or can be drawn across a surface to represent a curve. Since the advent of analytic geometry , points are often defined or represented in terms of numerical coordinates . In modern mathematics, a space of points is typically treated as a set , a point set . An isolated point
1030-410: 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,
1133-415: A parlour game , rather than a serious practical problem; but the purpose of the restriction is to ensure that constructions can be proved to be exactly correct. The ancient Greek mathematicians first attempted straightedge-and-compass constructions, and they discovered how to construct sums , differences , products , ratios , and square roots of given lengths. They could also construct half of
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#17327764067581236-431: 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
1339-425: A point is an abstract idealization of an exact position , without size, in physical space , or its generalization to other kinds of mathematical spaces . As zero- dimensional objects, points are usually taken to be the fundamental indivisible elements comprising the space, of which one-dimensional curves , two-dimensional surfaces , and higher-dimensional objects consist; conversely, a point can be determined by
1442-462: 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
1545-537: A constructible point (and therefore of any constructible length) is a power of 2. In particular, any constructible point (or length) is an algebraic number , though not every algebraic number is constructible; for example, √ 2 is algebraic but not constructible. There is a bijection between the angles that are constructible and the points that are constructible on any constructible circle. The angles that are constructible form an abelian group under addition modulo 2π (which corresponds to multiplication of
1648-472: 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
1751-773: 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 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,
1854-540: 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
1957-426: A formula in the original points using only the operations of addition , subtraction , multiplication , division , complex conjugate , and square root , which is easily seen to be a countable dense subset of the plane. Each of these six operations corresponding to a simple straightedge-and-compass construction. From such a formula it is straightforward to produce a construction of the corresponding point by combining
2060-440: A given angle , a square whose area is twice that of another square, a square having the same area as a given polygon, and regular polygons of 3, 4, or 5 sides (or one with twice the number of sides of a given polygon ). But they could not construct one third of a given angle except in particular cases, or a square with the same area as a given circle , or regular polygons with other numbers of sides. Nor could they construct
2163-510: A less powerful instrument. However, by the compass equivalence theorem in Proposition 2 of Book 1 of Euclid's Elements , no power is lost by using a collapsing compass. Although the proposition is correct, its proofs have a long and checkered history. In any case, the equivalence is why this feature is not stipulated in the definition of the ideal compass. Each construction must be mathematically exact . "Eyeballing" distances (looking at
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#17327764067582266-437: 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
2369-467: A line or either of two circles (in turn, using each point as centre and passing through the other point). If we draw both circles, two new points are created at their intersections. Drawing lines between the two original points and one of these new points completes the construction of an equilateral triangle. Therefore, in any geometric problem we have an initial set of symbols (points and lines), an algorithm, and some results. From this perspective, geometry
2472-424: A number of ancient problems in plane geometry impose this restriction. The ancient Greeks developed many constructions, but in some cases were unable to do so. Gauss showed that some polygons are constructible but that most are not. Some of the most famous straightedge-and-compass problems were proved impossible by Pierre Wantzel in 1837 using field theory , namely trisecting an arbitrary angle and doubling
2575-495: A regular n -gon is constructible, then so is a regular 2 n -gon and hence a regular 4 n -gon, 8 n -gon, etc.). However, there are only 5 known Fermat primes, giving only 31 known constructible regular n -gons with an odd number of sides. Sixteen key points of a triangle are its vertices , the midpoints of its sides , the feet of its altitudes , the feet of its internal angle bisectors , and its circumcenter , centroid , orthocenter , and incenter . These can be taken three at
2678-535: A regular 17-sided polygon can be constructed, and five years later showed that a regular n -sided polygon can be constructed with straightedge and compass if the odd prime factors of n are distinct Fermat primes . Gauss conjectured that this condition was also necessary ; the conjecture was proven by Pierre Wantzel in 1837. The first few constructible regular polygons have the following numbers of sides: There are known to be an infinitude of constructible regular polygons with an even number of sides (because if
2781-453: A regular polygon of n sides to be constructible. In 1837 Pierre Wantzel published a proof of the impossibility of trisecting an arbitrary angle or of doubling the volume of a cube, based on the impossibility of constructing cube roots of lengths. He also showed that Gauss's sufficient constructibility condition for regular polygons is also necessary. Then in 1882 Lindemann showed that π {\displaystyle \pi }
2884-410: A ruler and compass, provided that the given data and the data to be found consist of discrete points (not lines or circles). The truth of this theorem depends on the truth of Archimedes' axiom , which is not first-order in nature. Examples of compass-only constructions include Napoleon's problem . It is impossible to take a square root with just a ruler, so some things that cannot be constructed with
2987-486: A ruler can be constructed with a compass; but (by the Poncelet–Steiner theorem ) given a single circle and its center, they can be constructed. The ancient Greeks classified constructions into three major categories, depending on the complexity of the tools required for their solution. If a construction used only a straightedge and compass, it was called planar; if it also required one or more conic sections (other than
3090-424: A set of ratios is precisely the smallest field containing the original ratios and closed under taking complex conjugates and square roots. For example, the real part, imaginary part and modulus of a point or ratio z (taking one of the two viewpoints above) are constructible as these may be expressed as Doubling the cube and trisection of an angle (except for special angles such as any φ such that φ /(2 π ))
3193-404: A solid construction if it can be constructed using a straightedge, compass, and a (possibly hypothetical) conic drawing tool that can draw any conic with already constructed focus, directrix, and eccentricity. The same set of points can often be constructed using a smaller set of tools. For example, using a compass, straightedge, and a piece of paper on which we have the parabola y=x together with
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3296-450: A solid construction. In the language of fields, a complex number that is planar has degree a power of two, and lies in a field extension that can be broken down into a tower of fields where each extension has degree two. A complex number that has a solid construction has degree with prime factors of only two and three, and lies in a field extension that is at the top of a tower of fields where each extension has degree 2 or 3. A point has
3399-450: 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
3502-488: 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}}
3605-406: A time to yield 139 distinct nontrivial problems of constructing a triangle from three points. Of these problems, three involve a point that can be uniquely constructed from the other two points; 23 can be non-uniquely constructed (in fact for infinitely many solutions) but only if the locations of the points obey certain constraints; in 74 the problem is constructible in the general case; and in 39
3708-504: A topological space X {\displaystyle X} is defined to be the minimum value of n , such that every finite open cover A {\displaystyle {\mathcal {A}}} of X {\displaystyle X} admits a finite open cover B {\displaystyle {\mathcal {B}}} of X {\displaystyle X} which refines A {\displaystyle {\mathcal {A}}} in which no point
3811-524: A way that the operation "take a value at this point" may not be defined. A further tradition starts from some books of A. N. Whitehead in which the notion of region is assumed as a primitive together with the one of inclusion or connection . Often in physics and mathematics, it is useful to think of a point as having non-zero mass or charge (this is especially common in classical electromagnetism , where electrons are idealized as points with non-zero charge). The Dirac delta function , or δ function ,
3914-430: Is (informally) a generalized function on the real number line that is zero everywhere except at zero, with an integral of one over the entire real line. The delta function is sometimes thought of as an infinitely high, infinitely thin spike at the origin, with total area one under the spike, and physically represents an idealized point mass or point charge . It was introduced by theoretical physicist Paul Dirac . In
4017-559: 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}
4120-597: Is a rational number with denominator not divisible by 3) require ratios which are the solution to cubic equations , while squaring the circle requires a transcendental ratio. None of these are in the fields described, hence no straightedge-and-compass construction for these exists. The ancient Greeks thought that the construction problems they could not solve were simply obstinate, not unsolvable. With modern methods, however, these straightedge-and-compass constructions have been shown to be logically impossible to perform. (The problems themselves, however, are solvable, and
4223-407: Is a transcendental number , and thus that it is impossible by straightedge and compass to construct a square with the same area as a given circle. All straightedge-and-compass constructions consist of repeated application of five basic constructions using the points, lines and circles that have already been constructed. These are: For example, starting with just two distinct points, we can create
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4326-471: Is a constructible number. A number is constructible if and only if it can be written using the four basic arithmetic operations and the extraction of square roots but of no higher-order roots. The "straightedge" and "compass" of straightedge-and-compass constructions are idealized versions of real-world rulers and compasses . Actual compasses do not collapse and modern geometric constructions often use this feature. A 'collapsing compass' would appear to be
4429-418: 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
4532-415: 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
4635-479: 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,
4738-409: Is also easily solved when a straightedge with two marks on it is allowed (a neusis construction). The line segment from any point in the plane to the nearest point on a circle can be constructed, but the segment from any point in the plane to the nearest point on an ellipse of positive eccentricity cannot in general be constructed. See Note that results proven here are mostly a consequence of
4841-420: 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
4944-487: 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}}
5047-401: Is an element of some subset of points which has some neighborhood containing no other points of the subset. Points, considered within the framework of Euclidean geometry , are one of the most fundamental objects. Euclid originally defined the point as "that which has no part". In the two-dimensional Euclidean plane , a point is represented by an ordered pair ( x , y ) of numbers, where
5150-506: 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
5253-418: Is an unimportant restriction since, using a multi-step procedure, a distance can be transferred even with a collapsing compass; see compass equivalence theorem . Note however that whilst a non-collapsing compass held against a straightedge might seem to be equivalent to marking it, the neusis construction is still impermissible and this is what unmarked really means: see Markable rulers below.) More formally,
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#17327764067585356-483: 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
5459-404: Is defined by dim H ( X ) := inf { d ≥ 0 : C H d ( X ) = 0 } . {\displaystyle \operatorname {dim} _{\operatorname {H} }(X):=\inf\{d\geq 0:C_{H}^{d}(X)=0\}.} A point has Hausdorff dimension 0 because it can be covered by a single ball of arbitrarily small radius. Although
5562-530: Is equivalent to an axiomatic algebra , replacing its elements by symbols. Probably Gauss first realized this, and used it to prove the impossibility of some constructions; only much later did Hilbert find a complete set of axioms for geometry . The most-used straightedge-and-compass constructions include: One can associate an algebra to our geometry using a Cartesian coordinate system made of two lines, and represent points of our plane by vectors . Finally we can write these vectors as complex numbers. Using
5665-398: Is included in more than n +1 elements. If no such minimal n exists, the space is said to be of infinite covering dimension. A point is zero-dimensional with respect to the covering dimension because every open cover of the space has a refinement consisting of a single open set. Let X be a metric space . If S ⊂ X and d ∈ [0, ∞) , the d -dimensional Hausdorff content of S
5768-652: Is the infimum of the set of numbers δ ≥ 0 such that there is some (indexed) collection of balls { B ( x i , r i ) : i ∈ I } {\displaystyle \{B(x_{i},r_{i}):i\in I\}} covering S with r i > 0 for each i ∈ I that satisfies ∑ i ∈ I r i d < δ . {\displaystyle \sum _{i\in I}r_{i}^{d}<;\delta .} The Hausdorff dimension of X
5871-400: 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
5974-437: Is the construction of lengths, angles , and other geometric figures using only an idealized ruler and a pair of compasses . The idealized ruler, known as a straightedge , is assumed to be infinite in length, have only one edge, and no markings on it. The compass is assumed to have no maximum or minimum radius, and is assumed to "collapse" when lifted from the page, so it may not be directly used to transfer distances. (This
6077-521: Is the dimension of the space. Similar constructions exist that define the plane , line segment , and other related concepts. A line segment consisting of only a single point is called a degenerate line segment. In addition to defining points and constructs related to points, Euclid also postulated a key idea about points, that any two points can be connected by a straight line. This is easily confirmed under modern extensions of Euclidean geometry, and had lasting consequences at its introduction, allowing
6180-437: 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
6283-550: The Oxford mathematician Peter M. Neumann proved the theorem that there is no ruler-and-compass construction for the general solution of the ancient Alhazen's problem (billiard problem or reflection from a spherical mirror). Some regular polygons (e.g. a pentagon ) are easy to construct with straightedge and compass; others are not. This led to the question: Is it possible to construct all regular polygons with straightedge and compass? Carl Friedrich Gauss in 1796 showed that
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#17327764067586386-402: 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
6489-515: The intersection of two curves or three surfaces, called a vertex or corner . In classical Euclidean geometry , a point is a primitive notion , defined as "that which has no part". Points and other primitive notions are not defined in terms of other concepts, but only by certain formal properties, called axioms , that they must satisfy; for example, "there is exactly one straight line that passes through two distinct points" . As physical diagrams, geometric figures are made with tools such as
6592-476: 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,
6695-455: 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
6798-457: The "quadrature of the circle" can be achieved using a Kepler triangle . Doubling the cube is the construction, using only a straightedge and compass, of the edge of a cube that has twice the volume of a cube with a given edge. This is impossible because the cube root of 2, though algebraic, cannot be computed from integers by addition, subtraction, multiplication, division, and taking square roots. This follows because its minimal polynomial over
6901-611: The Greeks knew how to solve them without the constraint of working only with straightedge and compass.) The most famous of these problems, squaring the circle , otherwise known as the quadrature of the circle, involves constructing a square with the same area as a given circle using only straightedge and compass. Squaring the circle has been proved impossible, as it involves generating a transcendental number , that is, √ π . Only certain algebraic numbers can be constructed with ruler and compass alone, namely those constructed from
7004-460: The allowable tools for constructions under various rules, in order to determine what is still constructible and how it may be constructed, as well as determining the minimum criteria necessary to still be able to construct everything that compass and straightedge can. It is possible (according to the Mohr–Mascheroni theorem ) to construct anything with just a compass if it can be constructed with
7107-401: 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
7210-429: The circle), then it was called solid; the third category included all constructions that did not fall into either of the other two categories. This categorization meshes nicely with the modern algebraic point of view. A complex number that can be expressed using only the field operations and square roots (as described above ) has a planar construction. A complex number that includes also the extraction of cube roots has
7313-406: The circle, and Nicomedes in the second century BCE showed how to use a conchoid to trisect an arbitrary angle; but these methods also cannot be followed with just straightedge and compass. No progress on the unsolved problems was made for two millennia, until in 1796 Gauss showed that a regular polygon with 17 sides could be constructed; five years later he showed the sufficient criterion for
7416-401: 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
7519-527: The common definitions, a point is 0-dimensional. The dimension of a vector space is the maximum size of a linearly independent subset. In a vector space consisting of a single point (which must be the zero vector 0 ), there is no linearly independent subset. The zero vector is not itself linearly independent, because there is a non-trivial linear combination making it zero: 1 ⋅ 0 = 0 {\displaystyle 1\cdot \mathbf {0} =\mathbf {0} } . The topological dimension of
7622-548: The complex numbers). The only angles of finite order that may be constructed starting with two points are those whose order is either a power of two, or a product of a power of two and a set of distinct Fermat primes . In addition there is a dense set of constructible angles of infinite order. Given a set of points in the Euclidean plane , selecting any one of them to be called 0 and another to be called 1 , together with an arbitrary choice of orientation allows us to consider
7725-469: The construction and guessing at its accuracy) or using markings on a ruler, are not permitted. Each construction must also terminate . That is, it must have a finite number of steps, and not be the limit of ever closer approximations. (If an unlimited number of steps is permitted, some otherwise-impossible constructions become possible by means of infinite sequences converging to a limit .) Stated this way, straightedge-and-compass constructions appear to be
7828-494: The construction of almost all the geometric concepts known at the time. However, Euclid's postulation of points was neither complete nor definitive, and he occasionally assumed facts about points that did not follow directly from his axioms, such as the ordering of points on the line or the existence of specific points. In spite of this, modern expansions of the system serve to remove these assumptions. There are several inequivalent definitions of dimension in mathematics. In all of
7931-483: The constructions for each of the arithmetic operations. More efficient constructions of a particular set of points correspond to shortcuts in such calculations. Equivalently (and with no need to arbitrarily choose two points) we can say that, given an arbitrary choice of orientation, a set of points determines a set of complex ratios given by the ratios of the differences between any two pairs of points. The set of ratios constructible using straightedge and compass from such
8034-480: The context of signal processing it is often referred to as the unit impulse symbol (or function). Its discrete analog is the Kronecker delta function which is usually defined on a finite domain and takes values 0 and 1. Compass and straightedge In geometry , straightedge-and-compass construction – also known as ruler-and-compass construction , Euclidean construction , or classical construction –
8137-400: The equations for lines and circles, one can show that the points at which they intersect lie in a quadratic extension of the smallest field F containing two points on the line, the center of the circle, and the radius of the circle. That is, they are of the form x + y = k {\displaystyle x+y={\sqrt {k}}} , where x , y , and k are in F . Since
8240-410: The field of constructible points is closed under square roots , it contains all points that can be obtained by a finite sequence of quadratic extensions of the field of complex numbers with rational coefficients. By the above paragraph, one can show that any constructible point can be obtained by such a sequence of extensions. As a corollary of this, one finds that the degree of the minimal polynomial for
8343-401: 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
8446-479: The first number conventionally represents the horizontal and is often denoted by x , and the second number conventionally represents the vertical and is often denoted by y . This idea is easily generalized to three-dimensional Euclidean space , where a point is represented by an ordered triplet ( x , y , z ) with the additional third number representing depth and often denoted by z . Further generalizations are represented by an ordered tuplet of n terms, (
8549-452: 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. Point (geometry) In geometry ,
8652-422: 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
8755-489: 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 (
8858-471: The integers with a finite sequence of operations of addition, subtraction, multiplication, division, and taking square roots. The phrase "squaring the circle" is often used to mean "doing the impossible" for this reason. Without the constraint of requiring solution by ruler and compass alone, the problem is easily solvable by a wide variety of geometric and algebraic means, and was solved many times in antiquity. A method which comes very close to approximating
8961-466: 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 [
9064-405: The non-constructivity of conics. If the initial conic is considered as a given, then the proof must be reviewed to check if other distinct conic needs to be generated. As an example, constructions for normals of a parabola are known, but they need to use an intersection between circle and the parabola itself. So they are not constructible in the sense that the parabola is not constructible. In 1997,
9167-525: The notion of a point is generally considered fundamental in mainstream geometry and topology, there are some systems that forgo it, e.g. noncommutative geometry and pointless topology . A "pointless" or "pointfree" space is defined not as a set , but via some structure ( algebraic or logical respectively) which looks like a well-known function space on the set: an algebra of continuous functions or an algebra of sets respectively. More precisely, such structures generalize well-known spaces of functions in
9270-410: The only permissible constructions are those granted by the first three postulates of Euclid's Elements . It turns out to be the case that every point constructible using straightedge and compass may also be constructed using compass alone , or by straightedge alone if given a single circle and its center. Ancient Greek mathematicians first conceived straightedge-and-compass constructions, and
9373-473: 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}
9476-408: 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
9579-515: The points as a set of complex numbers . Given any such interpretation of a set of points as complex numbers, the points constructible using valid straightedge-and-compass constructions alone are precisely the elements of the smallest field containing the original set of points and closed under the complex conjugate and square root operations (to avoid ambiguity, we can specify the square root with complex argument less than π). The elements of this field are precisely those that may be expressed as
9682-451: The points on the unit circle viewed as complex numbers). The angles that are constructible are exactly those whose tangent (or equivalently, sine or cosine) is constructible as a number. For example, the regular heptadecagon (the seventeen-sided regular polygon ) is constructible because as discovered by Gauss . The group of constructible angles is closed under the operation that halves angles (which corresponds to taking square roots in
9785-400: 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
9888-476: The rationals has degree 3. This construction is possible using a straightedge with two marks on it and a compass. Angle trisection is the construction, using only a straightedge and a compass, of an angle that is one-third of a given arbitrary angle. This is impossible in the general case. For example, the angle 2 π /5 radians (72° = 360°/5) can be trisected, but the angle of π /3 radians (60 ° ) cannot be trisected. The general trisection problem
9991-425: 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
10094-417: The required triangle exists but is not constructible. Twelve key lengths of a triangle are the three side lengths, the three altitudes , the three medians , and the three angle bisectors . Together with the three angles, these give 95 distinct combinations, 63 of which give rise to a constructible triangle, 30 of which do not, and two of which are underdefined. Various attempts have been made to restrict
10197-418: The side of a cube whose volume is twice the volume of a cube with a given side. Hippocrates and Menaechmus showed that the volume of the cube could be doubled by finding the intersections of hyperbolas and parabolas , but these cannot be constructed by straightedge and compass. In the fifth century BCE, Hippias used a curve that he called a quadratrix to both trisect the general angle and square
10300-446: 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
10403-466: 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
10506-677: 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
10609-403: The volume of a cube (see § impossible constructions ). Many of these problems are easily solvable provided that other geometric transformations are allowed; for example, neusis construction can be used to solve the former two problems. In terms of algebra , a length is constructible if and only if it represents a constructible number , and an angle is constructible if and only if its cosine
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