Curve.In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. A simple example is the circle. In everyday use of the term "curve", a straight line is not curved, but in mathematical parlance curves include straight lines and line segments. A large number of other curves have been studied in geometry.This article is about the general theory. The term curve is also used in ways making it almost synonymous with mathematical function (as in learning curve), or graph of a function (Phillips curve).

An example of a (simple, closed) curve: a hypotrochoid. Definitions

Boundaries of hyperbolic components of Mandelbrot set as a closed curvesIn mathematics, a (topological) curve is defined as follows: let I be an interval of real numbers (i.e. a non-empty connected subset of ); then a curve is a continuous mapping , where X is a topological space. The curve is said to be simple if it is injective, i.e. if for all x, y in I, we have . If I is a closed bounded interval , we also allow the possibility (this convention makes it possible to talk about closed simple curve). If x y) for some (other than the extremities of I), then x) is called a double (or multiple) point of the curve.A curve is said to be closed or a loop if and if . A closed curve is thus a continuous mapping of the circle S1; a simple closed curve is also called a Jordan curve or a Jordan arc.A plane curve is a curve for which X is the Euclidean plane — these are the examples first encountered — or in some cases the projective plane. A space curve is a curve for which X is of three dimensions, usually Euclidean space; a skew curve is a space curve which lies in no plane. These definitions also apply to algebraic curves (see below). However, in the case of algebraic curves it is very common not to restrict the curve to having points only defined over the real numbers.This definition of curve captures our intuitive notion of a curve as a connected, continuous geometric figure that is "like" a line, without thickness and drawn without interruption, although it also includes figures that can hardly be called curves in common usage. For example, the image of a curve can cover a square in the plane (space-filling curve). The image of simple plane curve can have Hausdorff dimension bigger than one (see Koch snowflake) and even positive Lebesgue measure[1] (the last example can be obtained by small variation of the Peano curve construction). The dragon curve is another unusual example.

Conventions and terminologyThe distinction between a curve and its image is important. Two distinct curves may have the same image. For example, a line segment can be traced out at different speeds, or a circle can be traversed a different number of times. Many times, however, we are just interested in the image of the curve. It is important to pay attention to context and convention in reading.Terminology is also not uniform. Often, topologists use the term "path" for what we are calling a curve, and "curve" for what we are calling the image of a curve. The term "curve" is more common in vector calculus and differential geometry.

Lengths of curvesMain article: Arc lengthIf X is a metric space with metric d, then we can define the length of a curve byA rectifiable curve is a curve with finite length. A parametrization of is called natural (or unit speed or parametrised by arc length) if for any t1, t2 in [a,b], we haveIf is a Lipschitz-continuous function, then it is automatically rectifiable. Moreover, in this case, one can define speed of at t0 asand thenIn particular, if is Euclidean space and is differentiable thenDifferential geometryMain article: Differential geometry of curvesWhile the first examples of curves that are met are mostly plane curves (that is, in everyday words, curved lines in two-dimensional space), there are obvious examples such as the helix which exist naturally in three dimensions. The needs

of geometry, and also for example classical mechanics are to have a notion of curve in space of any number of dimensions. In general relativity, a world line is a curve in spacetime.If X is a differentiable manifold, then we can define the notion of differentiable curve in X. This general idea is enough to cover many of the applications of curves in mathematics. From a local point of view one can take X to be Euclidean space.On the other hand it is useful to be more general, in that (for example) it is possible to define the tangent vectors to X by means of this notion of curve.If X is a smooth manifold, a smooth curve in X is a smooth mapThis is a basic notion. There are less and more restricted ideas, too. If X is a Ck manifold (i.e., a manifold whose charts are k times continuously differentiable), then a Ck curve in X is such a curve which is only assumed to be Ck (i.e. k times continuously differentiable). If X is an analytic manifold (i.e. infinitely differentiable and charts are expressible as power series), and is an analytic map, then is said to be an analytic curve.A differentiable curve is said to be regular if its derivative never vanishes. (In words, a regular curve never slows to a stop or backtracks on itself.) Two Ck differentiable curvesand are said to be equivalent if there is a bijective Ck mapsuch that the inverse mapis also Ck, andfor all t. The map is called a reparametrisation of ; and this makes an equivalence relation on the set of all Ck differentiable curves in X. A Ck arc is an equivalence class of Ck curves under the relation of reparametrisation.

Another way to think about a curve is to look at the tangents at each point. A curve is defined by the condition that the X and Y intercepts tangents(eg. slopes) added up equals two. This can be explained using the differential equation:

Y y = (dy / dx)X x

Also known as point-slope form of a line, but this case we will use it to find the X and Y intercepts which are when x and y equal to 0:

X = x y(dy / dx)

Y = y x(dy / dx)

Representing the condition as a differential:

x y(dy / dx) + y x(dy / dx) = 2

Taken from text Differential Equations by Frank Ayers, Jr

Algebraic curveMain article: Algebraic curveAlgebraic curves are the curves considered in algebraic geometry. A plane algebraic curve is the locus of points f(x, y) = 0, where f(x, y) is a polynomial in two variables defined over some field F. Algebraic geometry normally looks at such curves in the context of algebraically closed fields. If K is the algebraic closure of F, and C is a curve defined by a polynomial f(x, y) defined over F, the points of the curve defined over F, consisting of pairs (a, b) with a and b in F, can be denoted C(F); the full curve itself being C(K).Algebraic curves can also be space curves, or curves in even higher dimensions, obtained as the intersection (common solution set) of more than one polynomial equation in more than two variables. By eliminating variables by means of the resultant, these can be reduced to plane algebraic curves, which however may introduce singularities such as cusps or double points. We may also consider these curves to have points defined in the projective plane; if f(x, y) = 0 then if x = u/w and y = v/w, and n is the total degree of f, then by expanding out wnf(u/w, v/w) = 0 we obtain g(u, v, w) = 0, where g is homogeneous of degree n. An example is the Fermat curve un + vn = wn, which has an affine form xn + yn = 1.Important examples of algebraic curves are the conics, which are nonsingular curves of degree two and genus zero, and ellipticcurves, which are nonsingular curves of genus one studied in number theory and which have important applications to cryptography. Because algebraic curves in fields of characteristic zero are most often studied over the complex numbers,

algbebraic curves in algebraic geometry look like real surfaces. Looking at them projectively, if we have a nonsingular curve in n dimensions, we obtain a picture in the complex projective space of dimension n, which corresponds to a real manifold of dimension 2n, in which the curve is an embedded smooth and compact surface with a certain number of holes in it, the genus. In fact, non-singular complex projective algebraic curves are compact Riemann surfaces.

HistoryA curve may be a locus, or a path. That is, it may be a graphical representation of some property of points; or it may be traced out, for example by a stick in the sand on a beach. Of course if one says curved in ordinary language, it means bent (not straight), so refers to a locus. This leads to the general idea of curvature. As we now understand, after Newtonian dynamics, to follow a curved path a body must experience acceleration. Before that, the application of current ideas to (for example) the physics of Aristotle is probably anachronistic. This is important because major examples of curves are the orbits of the planets. One reason for the use of the Ptolemaic system of epicycle and deferent was the special status accorded to the circle as curve.The conic sections had been deeply studied by Apollonius of Perga. They were applied in astronomy by Kepler. The Greek geometers had studied many other kinds of curves. One reason was their interest in geometric constructions, going beyond compass and straightedge. In that way, the intersection of curves could be used to solve some polynomial equations, such as that involved in trisecting an angle.Newton also worked on an early example in the calculus of variations. Solutions to variational problems, such as the brachistochrone and tautochrone questions, introduced properties of curves in new ways (in this case, the cycloid). The catenary gets its name as the solution to the problem of a hanging chain, the sort of question that became routinely accessible by means of differential calculus.In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the cubic curves, in the general description of the real points into 'ovals'. The statement of Bézout's theorem showed a number of aspects which were not directly accessible to the geometry of the time, to do with singular points and complex solutions.

From the nineteenth century there is not a separate curve theory, but rather the appearance of curves as the one-dimensional aspect of projective geometry, and differential geometry; and later topology, when for example the Jordan curve theorem was understood to lie quite deep, as well as being required in complex analysis. The era of the space-filling curves finally provoked the modern definitions of curve.

Notes

1. ̂ Osgood, William F. (January 1903). "A Jordan Curve of Positive Area". Transactions of the American Mathematical Society (American Mathematical Society) 4 (1): 107–112. doi:10.2307/1986455, http://www.jstor.org/sici?sici=0002-9947(190301)4%3A1%3C107%3AAJCOPA%3E2.0.CO%3B2-T. Retrieved on 4 June 2008. 

[edit References

B.I. Golubov (2001), "Rectifiable curve", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 

Famous Curves Index, School of Mathematics and Statistics, University of St Andrews, Scotland

Arc lengthFrom Wikipedia, the free encyclopedia  (Redirected from Length of arc)Jump to: navigation, searchDetermining the length of an irregular arc segment — also called rectification of a curve — was historically difficult. Although many methods were used for specific curves, the advent of calculus led to a general formula that provides closed

form solutions in some cases.

Contents[hide]1 General approach 2 Definition 3 Modern methods 3.1 Derivation 3.2 Another way to obtain the integral formula 4 Historical methods 4.1 Ancient 4.2 1600s 4.3 Integral form 5 Curves with infinite length 6 Generalization to (pseudo-)Riemannian manifolds 7 See also 8 References

9 External links

[edit] General approach

A curve in, say, the plane can be approximated by connecting a finite number of points on the curve using line segments to create a polygonal path. Since it is straightforward to calculate the length of each linear segment (using the theorem of Pythagoras in Euclidean space, for example), the total length of the approximation can be found by summing the lengths of each linear segment.If the curve is not already a polygonal path, better approximations to the curve can be obtained by following the shape of the curve increasingly more closely. The approach is to use an increasingly larger number of segments of smaller lengths. The lengths of the successive approximations do not decrease and will eventually keep increasing – possibly indefinitely, but for smooth curves this will tend to a limit as the lengths of the segments get arbitrarily small.For some curves there is a smallest number L that is an upper bound on the length of any polygonal approximation. If such a number exists, then the curve is said to be rectifiable and the curve is defined to have arc length L.

[edit] DefinitionSee also: Curve#Lengths of curves Let C be a curve in Euclidean (or, generally, a metric) space X = Rn, so C is the image of a continuous function f : [a, b X of the interval [a, b] into X.From a partition a = t0 < t1 < … < tn < tn = b of the interval [a, b] we obtain a finite collection of points f(t0), f(t1), …, f(tn ), f(tn) on the curve C. Denote the distance from f(ti) to f(ti+1) by d(f(ti), f(ti+1)), which is the length of the line segment connecting the two points.The arc length L of C is then defined to bewhere the supremum is taken over all possible partitions of [a, b] and n is unbounded.The arc length L is either finite or infinite. If L C is rectifiable, and is non-rectifiable otherwise. This definition of arc length does not require that C is defined by a differentiable function.

[edit] Modern methodsConsider a real function f(x) such that f(x) and f x) (its derivative with respect to x) are continuous on [a, b] . The length s of the part of the graph of f between x = a and x = b is found by the formulawhich is derived from the distance formula approximating the arc length with many small lines. As the number of line segments increases (to infinity by use of the integral) this approximation becomes an exact value.If a curve is defined parametrically by x = X(t) and y = Y(t), then its arc length between t = a and t = b is

x y , we take the limit. A useful mnemonic isIf a function is defined in polar coordinates by r = fIn most cases, including even simple curves, there are no closed-form solutions of arc length and numerical integration is necessary.

Curves with closed-form solution for arc length include the catenary, circle, cycloid, logarithmic spiral, parabola, semicubical parabola and (mathematically, a curve) straight line. The lack of closed form solution for the arc length of an elliptic arc led to the development of the elliptic integrals.[edit] Derivation

Pythagorean theorem

A representative linear element of the function In order to approximate the arc length of the curve, it is split into many linear segments. To make the value exact, and not an approximation, infinitely many linear elements are needed. This means that each element is infinitely small. This fact manifests itself later on when an integral is used.Begin by looking at a representative linear segment (see image) and observe that its length (element of the arc length) will be the differential ds. We will call the horizontal element of this distance dx, and the vertical element dy.The Pythagorean theorem tells us thatSince the function is defined in time, segments (ds) are added up across infintesimally small intervals of time (dt) yielding the integralIf y is a function of x, so that we could take t = x, then we have:which is the arc length from x = a to x = b of the graph of the function ƒ.For example, the curve in this figure is defined bySubsequently, the arc length integral for values of tUsing computational approximations, we can obtain a very accurate (but still approximate) arc length of 2.905. An expression in terms of the hypergeometric function can be obtained: it is[edit] Another way to obtain the integral formula

This section may require cleanup to meet Wikipedia's quality standards.Please improve this article

if you can. (July 2008)

Suppose that we have a rectifiable curve given by a function f(x), and that we want to approximate the arc length S along f between two points a and b in that curve. We can construct a series of rectangle triangles whose concatenated hypotenuses "cover" the arch of curve chosen as it's shown in the figure. To make this a "more functional" method we can also

x y cathetus will exist, depending on the type of curve and on the chosen arch, being then every hypotenuse equal to , as a result of the Pythagorean theorem. This way, an approximation of S would be given by the summation of all those n unfolded hypotenuses. Because of it we have that;To continue, let's algebraically operate on the form in which we calculate every hypotenuse to come to a new expression:Then, our previous result takes the following look:Now, the smaller these n segments are, the better our looked approximation is; they will be as small as we want doing that x tends to zero. This way, x develops in dx, and every incremental quotient yi xi becomes into a general dy / dx,that is by definition . Given these changes, our previous approximation turns into a thinner and at this point exact summation; an integration of infinite infinitesimal segments;

[edit] Historical methods[edit] AncientFor much of the history of mathematics, even the greatest thinkers considered it impossible to compute the length of an irregular arc. Although Archimedes had pioneered a rectangular approximation for finding the area beneath a curve with his method of exhaustion, few believed it was even possible for curves to have definite lengths, as do straight lines. The first ground was broken in this field, as it often has been in calculus, by approximation. People began to inscribe polygons within the curves and compute the length of the sides for a somewhat accurate measurement of the length. By using more segments, and by decreasing the length of each segment, they were able to obtain a more and more accurate approximation.[edit] 1600sIn the 1600s, the method of exhaustion led to the rectification by geometrical methods of several transcendental curves: the logarithm

ic spiral by Evangelista Torricelli in 1645 (some sources say John Wallis in the 1650s), the cycloid by Christopher Wren in 1658, and the catenary by Gottfried Leibniz in 1691.In 1659, Wallis credited William Neile's discovery of the first rectification of a nontrivial algebraic curve, the semicubical parabola.[edit] Integral formBefore the full formal development of the calculus, the basis for the modern integral form for arc length was independently discovered by Hendrik van Heuraet and Pierre Fermat.In 1659 van Heuraet published a construction showing that arc length could be interpreted as the area under a curve - this integral, in effect - and applied it to the parabola. In 1660, Fermat published a more general theory containing the same result in his De linearum curvarum cum lineis rectis comparatione dissertatio geometrica.

Fermat's method of determining arc lengthBuilding on his previous work with tangents, Fermat used the curvewhose tangent at x = a had a slope ofso the tangent line would have the equationNext, he increased a by a small amount to a + , making segment AC a relatively good approximation for the length of the curve from A to D. To find the length of the segment AC, he used the Pythagorean theorem:which, when solved, yieldsIn order to approximate the length, Fermat would sum up a sequence of short segments.

[edit] Curves with infinite length

The Koch curve.

The graph of x sin(1/x).As mentioned above, some curves are non-rectifiable, that is, they have infinite length. There are continuous curves for which any arc on the curve (containing more than a single point) has infinite length. An example of such a curve is the Koch curve. Another example of a curve with infinite length is the graph of the function defined by f(x) = x sin(1/x x f(0) = 0. Sometimes the Hausdorff dimension and Hausdorff measure are used to "measure" the size of infinite length curves.

[edit] Generalization to (pseudo-)Riemannian manifoldsLet M be a (pseudo-)Riemannian manifold, M a curve in M and g the (pseudo-) metric tensor.The length of is defined to bewhere d T (t)M is the tangent vector of at t. The sign in the square root is chosen once for a given curve, to ensure that the square root is a real number. The positive sign is chosen for spacelike curves; in a pseudo-Riemannian manifold, the negative sign may be chosen for timelike curves.In theory of relativity, arc-length of timelike curves (world lines) is the proper time elapsed along the world line.

[edit] See also

Arc (geometry) Integral approximations Geodesics

[edit] References

Farouki, Rida T. (1999). Curves from motion, motion from curves. In P-J. Laurent, P. Sablonniere, and L. L. Schumaker (Eds.), Curve and Surface Design: Saint-Malo 1999, pp.63-90, Vanderbilt Univ. Press. ISBN 0-8265-1356-5.

Multiple integralFrom Wikipedia, the free encyclopedia  (Redirected from Double integral)Jump to: navigation, search

It has been suggested that Volume integral be merged into this article or section. (Discuss)

Integral as area between two curves.The multiple integral is a type of definite integral extended to functions of more than one real variable, for example, f(x, y) or f(x, y, z).

Double integral as volume under a surface z = x2 + y2. The rectangular region at the bottom of the body is the domain of integration, while the surface is the graph of the two-variable function to be integrated.

Contents[hide]1 Introduction 2 Examples 3 Mathematical definition 3.1 Properties 3.2 Particular cases 4 Methods of integration 4.1 Direct examination 4.1.1 Constants 4.1.2 Use of the possible symmetries 4.2 Formulae of reduction 4.2.1 Normal domains on R2 4.2.1.1 x-axis 4.2.1.2 y-axis 4.2.1.3 Example 4.2.2 Normal domains on R3 4.3 Change of variables 4.3.1 Polar coordinates 4.3.2 Cylindrical coordinates 4.3.3 Spherical coordinates 5 Example of mathematical applications - Computing a volume 6 Multiple improper integral 7 Multiple integrals and iterated integrals 8 Some practical applications 9 See also 10 References

11 External links

[edit] IntroductionJust as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the x-axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three dimensional Cartesian plane where z = f(x,y)) and the plane which contains its domain. (Note that the same volume can be obtained via the triple integral — the integral of a function in three variables — of the constant function f(x, y, z) = 1 over the above-mentioned region between the surface and the plane.) If there are more variables, a multiple integral will yield hypervolumes of multi-dimensional

functions.Multiple integration of a function in n variables: f(x1, x2, …, xn) over a domain D is most commonly represented by nesting integral signs in the reverse order of execution (the leftmost integral sign is computed last) proceeded by the function and integrand arguments in proper order (the rightmost argument is computed last). The domain of integration is either represented symbolically for every integrand over each integral sign, or is often abbreviated by a variable at the rightmost integral sign:

Since it is impossible to calculate the antiderivative of a function of more than one variable, indefinite multiple integrals do not exist. Therefore all multiple integrals are definite integrals.

[edit] ExamplesFor example, the volume of the parallelepiped of sides 4 × 6 × 5 may be obtained in two ways:

By the double integral

of the function f(x, y) = 5 calculated in the region D in the xy-plane which is the base of the parallelepiped.

By the triple integral

of the constant function 1 calculated on the parallelepiped itself.

[edit] Mathematical definitionLet n be an integer greater than 1. Consider a so-called half-open n-dimensional rectangle (from here on simply called rectangle). For a plane, n = 2, and the multiple integral is just a double integral.

Divide each interval (ai, bi) into a finite number of non-overlapping subintervals, with each subinterval closed at the left end, and open at the right end. Denote such a subinterval by Ii. Then, the family of subrectangles of the form

is a partition of T that is, the subrectangles C are non-overlapping and their union is T. The diameter of a subrectangle C is by definition, the largest of the lengths of the intervals whose product is C, and the diameter of a given partition of T is defined as the largest of the diameters of the subrectangles in the partition.Let f : T R be a function defined on a rectangle T. Consider a partition

of T defined as above, where m is a positive integer. A Riemann sum is a sum of the form

where for each k the point Pk is in Ck and m(Ck) is the product of the lengths of the intervals whose Cartesian product is Ck.The function f is said to be Riemann integrable if the limit

exists, where the limit is taken over all possible partitions of T f is Riemann integrable, S is called the Riemann integral of f over T and is denoted

The Riemann integral of a function defined over an arbitrary bounded n-dimensional set can be defined by extending that function to a function defined over a half-open rectangle whose values are zero outside the domain of the original function. Then, the integral of the original function over the original domain is defined to be the integral of the extended function over its rectangular domain, if it exists.In what follows the Riemann integral in n dimensions will be called multiple integral.[edit] PropertiesMultiple integrals have many of the same properties of integrals of functions of one variable (linearity, additivity, monotonicity, etc.. Moreover, just as in one variable, one can use the multiple integral to find the average of a function over a given set. More specifically, given a set D Rn and an integrable function f over D, the average value of f over its domain is given by

where m(D) is the measure of D.

[edit] Particular casesIn the case of T R2, the integral

is the double integral of f on T, and if T R3 the integral

is the triple integral of f on T.Notice that, by convention, the double integral has two integral signs, and the triple integral has three; this is just notational convenience, and comes handy when computing a multiple integral as an iterated integral (as shown later in the article).

[edit] Methods of integrationThe resolution of problems with multiple integrals consists in most of cases in finding the way to reduce the multiple integral to a series of integrals of one variable, each being directly solvable.[edit] Direct examinationSometimes, it is possible to obtain the result of the integration without any direct calculations.[edit] ConstantsIn the case of a constant function, the result is straightforward: simply multiply the measure by the constant function c. If c = 1, and is integrated over a subregion of R2 the product gives the area of the region, while in R3 it is the volume of the region.

For example:

and Let us integrate f over D, with respect to x first:

[edit] Use of the possible symmetriesIn the case of a domain where there are symmetries respecting at least one of the axes and where the function has at least one parity in respect to a variable, the integral becomes null (the sum of opposite and equal values is null).It is sufficient that - in functions on Rn - the dependent variable is odd with the symmetric axis.

Example (1):

Given f(x, y) = 2 sin x y3 + 5 and T = x2 + y2 radius 1 centered in the origin of the axes, boundary included). Using the property of linearity, the integral can be decomposed in three pieces:

2 sin x and 3y3 are both odd functions and moreover it is evident that the T disc has a symmetry for the x and even the y axis; therefore the only contribution to the final result of the integral is that of the constant function 5 because the other two pieces are null.

Example (2):

Consider the function f(x, y, z) = x exp(y2 + z2) and as integration region the sphere with radius 2 centered in the origin of the axes T = x2 + y2 + z2to x-axis to show that the integral is 0, because the function is an odd function of that variable. [edit] Formulae of reductionFormulae of reduction use the concept of simple domain to make possible the decomposition of the multiple integral as a product of other one-variable integrals. These have to be solved from the right to the left considering the other variables as constants (which is the same procedure as the calculus of partial derivatives).[edit] Normal domains on R2See also: Order of integration (calculus) [edit] x-axisIf D is a measurable domain perpendicular to the x-axis and is a continuous function x x) (defined in the [a,b] interval) are the two functions that determine D. Then:

[edit] y-axisIf D is a measurable domain perpendicular to the y-axis and is a continuous function; then y) (defined in the [a,b] interval) are the two functions that determine D. Then:

[edit] Example

Example: D region for integral by reduction's formulas.Consider this region: (please see the graphic in the example). Calculate

This domain is perpendicular to both the x and to the y axes. To apply the formulas you have to find the functions that determine D and its definition's interval. In this case the two functions are: and while the interval is given from the intersections of the functions with , so the interval is (normality has been chosen with respect to the x axis for a better visual understanding). It's now possible to apply the formulas:

(at first the second integral is calculated considering x as a constant). The remaining operations consist of applying the basic techniques of integration:

If we choose the normality in respect to the y axis we could calculate

and obtain the same value.

Example of a normal domain in R3 (xy-plane).[edit] Normal domains on R3The extension of these formulae to triple integrals should be apparent:T is a domain perpendicular to the xy x,y,z x,y,z) functions. Then:

(this definition is the same for the other five normality cases on R3).[edit] Change of variablesThe limits of integration are often not easily interchangeable (without normality or with complex formulae to integrate), one makes a change of variables to rewrite the integral in a more "comfortable" region, which can be described in simpler formulae. To do so, the function must be adapted to the new coordinates.Example (1-a): The function is ; if one adopts this substitution therefore one obtains the new function .

Similarly for the domain because it is delimited by the original variables that were transformed before (x and y in example).

the differentials dx and dy transform via the determinant of the Jacobian matrix containing the partial derivatives of the transformations regarding the new variable (consider, as an example, the differential transformation in polar coordinates).

There exist three main "kinds" of changes of variable (one in R2, two in R3); however, a suitable substitution can be found using the same principle in a more general way.[edit] Polar coordinatesSee also: Polar coordinate system

Transformation from cartesian to polar coordinates.In R2 if the domain has a circular "symmetry" and the function has some "particular" characteristics you can apply the transformation to polar coordinates (see the example in the picture) which means that the generic points P(x,y) in cartesian coordinates switch to their respective points in polar coordinates. That allows one to change the "shape" of the domain and simplify the operations.The fundamental relation to make the transformation is the following:

Example (2-a):The function is and applying the transformation one obtains

Example (2-b):The function is In this case one has:

using the Pythagorean trigonometric identity (very useful to simplify this operation). The transformation of the domain is made by defining the radius' crown length and the amplitude of the described angle to define

x, y.

Example of a domain transformation from cartesian to polar.Example (2-c):The domain is

Example (2-d):The domain is , that is the circular crown in the positive y

rectangle: . The Jacobian determinant of that transformation is the following:

which has been obtained by inserting the partial derivatives of x ydx dy d d

Once the function is transformed and the domain evaluated, it is possible to define the formula for the change of variables in polar coordinates:

Example (2-e):The function is and as the domain the same in 2-d example. From the previous analysis of Dfunction:

finally let's apply the integration formula:

Once the intervals are known, you have

[edit] Cylindrical coordinates

Cylindrical coordinates.In R3 the integration on domains with a circular base can be made by the passage in cylindrical coordinates; the transformation of the function is made by the following relation:

The domain transformation can be graphically attained, because only the shape of the base varies, while the height follows the

shape of the starting region.Example (3-a):The region is (that is the "tube" whose base is the circular crown of the 2-d example and whose height is 5); if the transformation is applied, this region is obtained: (that is the parallelepiped whose base is the rectangle in 2-d example and whose height is 5). Because the z component is unvaried during the transformation, the dx dy dz differentials vary as in the passage in polar coordinates: therefore, they become .Finally, it is possible to apply the final formula to cylindrical coordinates:

This method is convenient in case of cylindrical or conical domains or in regions where is easy to individuate the z interval and even transform the circular base and the function.Example (3-b):The function is and as integration domain this cylinder: . The transformation of D in cylindrical coordinates is the following:

while the function becomes

Finally you can apply the integration's formula:

developing the formula you have

[edit] Spherical coordinates

Spherical coordinates.In R3 some domains have a spherical symmetry, so it's possible to specify the coordinates of every point of the integration region by two angles and one distance. It's possible to use therefore the passage in spherical coordinates; the function is transformed by this relation:

Note that points on z axis do not have a precise characterization in spherical coordinates, so The better integration domain for this passage is obviously the sphere.Example (4-a):The domain is (sphere with radius 4 and center in the origin); applying the transformation you get this region: The Jacobian determinant of this transformation is the following:

The dx dy dz 2 d d dFinally you obtain the final integration formula:

It's better to use this method in case of spherical domains and in case of functions that can be easily simplified, by the first fundamental relation of trigonometry, extended in R3 (please see example 4-b); in other cases it can be better to use cylindrical coordinates (please see example 4-c).

Note that the extra 2 and come from the Jacobian.

Example (4-b):D is the same region of the 4-a example and is the function to integrate. Its transformation is very easy:

while we know the intervals of the transformed region T from D:

Let's therefore apply the integration's formula:

and, developing, we get

Example (4-c):The domain D is the ball with center in the origin and radius 3a () and is the function to integrate. Looking at the domain, it seems convenient to adopt the passage in spherical coordinates, in fact, the intervals of the variables that delimit the new T region are obviously:

However, applying the transformation, we get . Applying the formula for integration we would obtain:

which is very hard to solve. This problem will be solved by using the passage in cylindrical coordinates. The new T intervals are

the z interval has been obtained by dividing the ball in two hemispheres simply by solving the inequality from the formula of D (and then directly transforming x2 + y2 in 2). The new function is simply 2. Applying the integration formula . Then we get

Now let's apply the transformation

(the new intervals become ). We get

because , we get

after inverting the integration's bounds and multiplying the terms between parenthesis, it is possible to decompose the integral in two parts that can be directly solved:

Thanks to the passage in cylindrical coordinates it was possible to reduce the triple integral to an easier one-variable integral. See also the differential volume entry in nabla in cylindrical and spherical coordinates.

[edit] Example of mathematical applications - Computing a volumeThanks to the methods previously described it is possible to demonstrate the value of the volume of some solid volumes.

Cylinder: Consider the domain as the circular base of radius R and the function as a constant of the height h. It is possible to write this in polar coordinates like so:

Verification: Volume = base area * height =

Sphere: Is a ready demonstration of applying the passage in spherical coordinates of the integrated constant function 1 on the sphere of the same radius R:

Tetrahedron (triangular pyramid or 3-simplex): The volume of the tetrahedron with apex in the origin and chines of length l carefully lay down to you on the three cartesian axes can be calculated through the reduction formulas considering, as an example, normality regarding the plan xy and to axis x and like function constant 1.

Verification: Volume = base area × height/3 =

Example of an improper domain.

[edit] Multiple improper integralIn case of unbounded domains or functions not bounded near the boundary of the domain, we have to introduce the double improperintegral or the triple improper integral.

[edit] Multiple integrals and iterated integralsSee also: Order of integration (calculus) Fubini's theorem states that if

that is, the integral is absolutely convergent, then the multiple integral will give the same result as the iterated integral,

In particular this will occur if | f(x,y) | is a bounded function and A and B are bounded sets.If the integral is not absolutely convergent, care is needed not to confuse the concepts of multiple integral and iterated integral, especially since the same notation is often used for either concept. The notation

means, in some cases, an iterated integral rather than a true double integral. In an iterated integral, the outer integral

is the integral with respect to x of the following function of x:

A double integral, on the other hand, is defined with respect to area in the xy-plane. If the double integral exists, then it is equal to each of the two iterated integrals (either "dy dx" or "dx dy") and one often computes it by computing either of the iterated integrals. But sometimes the two iterated integrals exist when the double integral does not, and in some such cases the two iterated integrals are different numbers, i.e., one has

This is an instance of rearrangement of a conditionally convergent integral.The notation

may be used if one wishes to be emphatic about intending a double integral rather than an iterated integral.

[edit] Some practical applicationsThese integrals are used in many applications in physics.In mechanics the moment of inertia is calculated as volume integral (that is a triple integral) of the density weighed with the square of the distance from the axis:

In electromagnetism, Maxwell's equations can be written by means of multiple integrals to calculate the total magnetic and electric fields. In the following example, the electric field produced by a distribution of charges is obtained by a triple integral of a vector function:

[edit] See also

Main analysis theorems that relate multiple integrals: Divergence theorem Stokes' theorem Green's theorem

[edit] References

Robert A. Adams - Calculus: A Complete Course (5th Edition)

Fundamental theorem of curves

In differential geometry, the fundamental theorem of curves states that any regular curve with non-zero curvature has its shape (and size) completely determined by its curvature and torsion.A curve can be described, and thereby defined, by a pair of scalar fields: curvature and torsion , both of which depend on some parameter which parametrizes the curve but which can ideally be the arc length of the curve. From just the curvature and torsion, the vector fields for the tangent, normal, and binormal vectors can be derived using the Frenet-Serret formulas. Then, integration of the tangent field (done numerically, if not analytically) yields the curve.If a pair of curves are in different positions but have the same curvature and torsion, then they are congruent to each other.

Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and in the Euclidean space by methods of differential and integral calculus.

Starting in antiquity, many concrete curves have been thoroughly investigated using synthetic approach. Differential geometry takesanother path: curves are represented in a parametrized form, and their geometric properties and various quantities associated with them, such as 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 a regular curve in an Euclidean space has no intrinsic geometry. Any regular curve may be parametrized by the arc length (the natural parametrization) and from the point of view of a bug 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 the way in which they bend and twist. Quantitatively, this 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.

In mathematics, curvature refers to any of a number of loosely

related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, but this is defined in different ways depending on the context. There is a key distinction between extrinsic curvature, which is defined for objects embedded in another space (usually a Euclidean space) in a way that relates to the radius of curvature of circles that touch the object, an

MathWorld Contributors > Budney >Interactive Entries > Interactive Demonstrations >Less... CurvatureIn general, there are two important types of curvature: extrinsic curvature and intrinsic curvature. The extrinsic curvature of curves in two- and three-space was the first type of curvature to be studied historically, culminating in the Frenet formulas, which describe a space curve entirely in terms of its "curvature," torsion, and the initial starting point and direction. After the curvature of two- and three-dimensional curves was studied, attention turned to the curvature of surfaces in three-space. The main curvatures that emerged from this scrutiny are the mean curvature, Gaussian curvature, and the Weingarten map. Mean curvature was the most important for applications at the time and was the most studied, but Gauss was the first to recognize the importance of the Gaussian curvature. Because Gaussian curvature is "intrinsic," it is detectable to two-dimensional "inhabitants" of the surface, whereas mean curvature and the Weingarten map are not detectable to someone who can't study the three-dimensional space surrounding the surface on which he resides. The importance of Gaussian curvature to an inhabitant is that it controls the surface area of spheres around the inhabitant. Riemann and many others generalized the concept of curvature to sectional curvature, scalar curvature, the Riemann tensor, Ricci curvature, and a host of other intrinsic and extrinsic curvatures. General curvatures no longer need to be numbers, and can take the form of a map, group, groupoid, tensor field, etc. The simplest form of curvature and that usually first encountered in calculus is an extrinsic curvature. In two dimensions, let a plane curve be given by Cartesian parametric equations and . Then the curvature is defined by

(1) (2) (3) (4) where is the tangential angle and is the arc length. As can readily be seen from the definition, curvature therefore has units of inverse distance. The derivative in the above equation can be found using the identity (5) (6) (7) so

(8) and

(9) (10) (11) (12) Combining (9), (10), and (12) then gives

(13) For a two-dimensional curve written in the form , the equation of curvature becomes (14) If the two-dimensional curve is instead parameterized in polar coordinates, then (15) where (Gray 1997, p. 89). In pedal coordinates, the curvature is given by (16) The curvature for a two-dimensional curve given implicitly by is given by (17) (Gray 1997). Now consider a parameterized space curve in three dimensions for which the tangent vector is defined as (18) Therefore,

(19) (20) (21) where is the normal vector. But (22) (23) (24) so taking norms of both sides gives

(25) Solving for then gives (26) (27) (28) (Gray 1997, p. 192). The curvature of a two-dimensional curve is related to the radius of curvature of the curve's osculating circle. Consider a circle specified parametrically by

(29) (30) which is tangent to the curve at a given point. The curvature is then

(31) or one over the radius of curvature. The curvature of a circle can also be repeated in vector notation. For the circle with , the arc length is (32) (33) (34) so and the equations of the circle can be rewritten as

(35) (36) The position vector is then given by

(37) and the tangent vector is (38) (39) so the curvature is related to the radius of curvature by (40) (41) (42) (43) as expected. Four very important derivative relations in differential geometry related to the Frenet formulas are (44) (45) (46) (47) where is the tangent vector, is the normal vector, is the binormal vector, and is the torsion (Coxeter 1969, p. 322). The curvature at a point on a surface takes on a variety of values as the plane through the normal varies. As varies, it achieves a minimum and a maximum (which are in perpendicular directions) known as the principal curvatures. As shown in Coxeter (1969, pp. 352-353),

(48) (49) where is the Gaussian curvature, is the mean curvature, and det denotes the determinant. The curvature is sometimes called the first curvature and the torsion the second curvature. In addition, a third curvature (sometimes called total curvature) (50) is also defined. A signed version of the curvature of a circle appearing in the Descartes circle theorem for the radius of the fourth of four mutually tangent circles is called the bend. Bottom of FormREFERENCES: Casey, J. Exploring Curvature. Wiesbaden, Germany: Vieweg, 1996. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. Fischer, G. (Ed.). Plates 79-85 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, pp. 74-81, 1986. Gray, A. "Curvature of Curves in the Plane," "Drawing Plane Curves with Assigned Curvature," and "Drawing Space Curves with Assigned Curvature." §1.5, 6.4, and 10.2 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 14-17, 140-146, and 222-224, 1997. Kreyszig, E. "Principal Normal, Curvature, Osculating Circle." §12 in Differential Geometry. New York: Dover, pp. 34-36, 1991. Yates, R. C. "Curvature." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 60-64, 1952.

Radius of Curvature Optical Testing Applications Radius of Curvature is a critical design parameter in optical manufacturing. Many of ZYGO's optical interferometer products can be used to make high-accuracy radius of curvature measurements by measuring the distance between "cats eye" and confocal positions. This requires the purchase of a position measurement system consisting of either a linear encoder scale or, for extremely high accuracy measurements, ZYGO's distance measuring interferometer system. ZYGO's proprietary Radius of Curvature application for the MetroPro analysis software is also required. PTI 250 - Both manual and semi-automated systems are available. Please consult the PTI 250 Accessories Guide for details. GPI & VeriFire - Several configurations are available, horizontal or vertical orientation, using either linear encoder scale or ZYGO's distance measuring interferometer. Please consult the GPI/VeriFire Accessories Guide for details. MetroCell - The MetroCell vertical workstation enables high accuracy radius of curvature measurements in a vertical downward looking configuration. The MetroCell can be configured with either a linear encoder or displacement interferometer for precision measurement of the distance between the catseye and confocal positions. The VeriFire Asphere system offers the ability to perform semi-automated radius measurements with 5-axes of motorized staging and a dual axis displacement interferometer.

The History of Curvature

by Dan Margalit

Sometimes nature is too beautiful for words. This is one of the reasons why mathematicians have been so useful over the centuries. While the origins of mathematics lie in mundane processes like counting, the field has been steadily expanding since that time. One of the most significant changes resulting from the growth of mathematics is that it has become less and less focused on the practical and more and more focused on the theoretical. This has been a very slow process, as mathematics is still somewhere between the two extremes. Regardless, this glacier-like revolution has spawned a lot of beautiful mathematics that might not have otherwise come about. One such invention is the study of curvature. Many curves in the plane and in space are simply beautiful. Since words cannot do them justice, mathematicians have developed several ways of describing them. The most common method of describing a curve is to give its parameterization. Another way, however, is to say how much the curve "bends" at each point. This measure of bending is known by the technical word "curvature". It may surprise the reader that curvature is all that is needed to define a curve (up to rigid motions). For example, a curve that has constant curvature must be part or all of a circle (for these are the only curves that have the same amount of bending at every point). The study of this twisting property of curves goes back to ancient times, but few of its goals were realized until the invention of the calculus in the seventeenth century. Throughout the history of mathematics, the analysis of the curvature of curves has been a prime illustration of the beauty of mathematics and an indicator of its progress. The history of geometry in general has three very distinct stages: ancient geometry (Ancient Greeks), analytic geometry (Fermat and Descartes in the seventeenth century), and finally differential geometry (modern times) (Gamkrelidze 14). All along, curvature has held the attention of many great mathematicians. In Ancient Greece, for instance, there was a clear distinction between the curvatures of the classical Greek curves, the line and the circle. Simply, the distinction was that lines don’t bend, and circles bend the same at every point (McCleary 66). According to Proclus’ history of geometry, Aristotle expanded upon these notions, declaring that there were three kinds of loci: straight, circular, and mixed (Coolidge 375). These monumental notions were the germination of what eventually became the study of curvature. As stated in the Encyclopedia of Mathematical Sciences, the main result in the development of geometry from Ancient Greece to the present is the move from "archaic synthetic form into modern differential form" (Gamkrelidze 22). The process began with Aristotle and his contemporaries, but generalization and rigorization of this would come centuries later. The next in the line of landmark geometers is Apollonius of Perga, who lived in the third century B.C. Among his more significant findings was that at each point of a conic section there is exactly one normal line (Coolidge 376). This fact is one that would show its relevance to the study of curvature at a much later time. More importantly, in the fifth book of his Conic Sections are found the "germs of the subject of evolutes and centers of osculation", subjects intimately relatedto curvature (Cajori 49). According to the mathematical historian D.T. Whiteside, Apollonius applied methods for finding the radius of curvature which were amazingly similar to the methods used by Huygens and Newton two thousand years later (Whiteside 175). Unfortunately, neither Apollonius nor his contemporaries could take their geometrical ideas beyond the limits of their meager methods (such as exhaustion). Regardless, the work of these mathematicians was indispensable in the progess towards the study of curvature. As the first one and a half millennia of the Common Era produced few helpful methods in the area of curvature-related geometry, there were few advances in the field. Finally, in the fourteenth century, Nicole Oresme made a relevant contribution. Oresme is generally esteemed to be the fist person to draw a graph, and his work is viewed by some scholars to be an early attempt at coordinate geometry (Struik 1986 133). He also seems to be the first writer to hint at a definition of curvature. In fact he assumed the existence of a measure of twist called "curvitas" (Coolidge 376). He also indicated his awareness that, as in Figure 1, if two curves have the same tangent at a point and one is "inside" the other, then it has greater curvature (Coolidge 376). As was obvious to Oresme, it bends more. Additionally, Oresme remarked that the curvature of a circle is "uniformus" (Coolidge 376). He went on to propose that the curvature of a circle is proportional to the multiplicative inverse of its radius (Coolidge 376). Part of the importance of Oresme’s work is that it helped rekindle the fire that became the mathematicians’ drive to find the curvature of a general curve.

Figure 1 Almost three centuries later, Johannes Kepler (1571-1630) made indirect contributions to the theory of curvature. His first donation was his generalization of the Problem of Alhazin). Kepler generalized Alhazin’s solution and discovered a method for finding the image of a brilliant point when reflected off of a general curve. Part of Kepler’s genius was to approximate the general curve with a circle at the point of reflection, thus reducing the problem to that of Alhazin (Coolidge 377). In later times, this approximating circle would come to be known as the "circle of curvature" of a curve at a point; for the radius of the circle is inversely proportional to how much the curve bends at the point. Kepler’s work was instrumental in the development of curvature, for he seems to be the first to actually develop methods for investigating the degree of twisting in a curve. Another great advance in man’s pursuit of curvature was taken by the first analytic geometers. Most notably, this includes Pierre de Fermat and Rene Descartes. Their most important contribution was to describe general geometric curves with algebraic equations. This was obviously essential in order for curves to be treated by calculus (or for calculus to be invented, for that matter). Unfortunately, however, these mathematicians were still lacking one major ingredient useful inthe analysis of curves and circles--pi. This was a major hindrance in the geometrical work of the time. Descartes lamented, "the ratios between straight and curved lines are not known, and I believe cannot be discovered by human minds, and therefore no conclusion based upon such ratios can be accepted as rigorous and exact" (Katz 403). Thus, the growth of analytic geometry in the seventeenth century was stunted, and the explicit invention of curvature was preempted. In 1673, a mathematician named Christiaan Huygens published the influential book Horologium oscillatorium sive de motu pendulorum ad horologia aptato demonstrationes geometrica. Most significantly, this work deals with evolutes and involutes of curves. He describes a very specific method for finding the "involute" of a curve. As seen in the accompanying demo, in order to find the involute of a curve A, one should wrap a string tightly against A, on its convex side. Then, keeping one end of the string fixed, the other end is pulled away from the curve. If the string is kept taut, then the moving end of the string will trace out a curve, B. B is called the involute of A, and A is called the evolute of B (see demo)(Coolidge 377). What is important is that each step of the way, the string is tangent to the evolute and perpendicular to the involute. Huygens "attempt[ed]" to define the radius of curvature of the involute as the distance between the points of contact between the involute with the string and evolute with the string (Coolidge 377). This working definition of curvature is exactly what the curvature of an involute would come to be defined as. Huygens even specifically stated that the locus of the "centers of curves", one on each normal, is the evolute of a curve (Kline 556). This definition has also remained unchanged ever since. However, Huygens’ method required the evolute to be given in advance. This did not help if one wanted to find the radius of curvature of an arbitrary curve. In order to find what was happening in a "really really small" portion of a general curve, calculus would be needed. Calculus had already been invented by the time Huygens published Horologium oscillatorium, but Huygens was not yet cognizant of its methods. Therefore, Huygens had only a general method and could not find an exact solution to the problem of curvature.Figure 2One of Huygens’ most impressive accomplishments in Horologium oscillatorium was his work with the cycloid. Huygens showed geometrically that the evolute of the left half of the lower cycloid is the right half of the upper cycloid (Figure 2). He did this using "synthetic and infinitesimal ideas" (McCleary 78). This, combined with the fact that the cycloid is a tautochrone, led to Huygens’ genius invention, the cycloidal pendulum. The idea is that if a pendulum (bob on a string) is swung between the arches of a cycloid (see demo), then the bob of the pendulum will trace out the involute of a cycloid--a cycloid. Since the cycloid is a tautochrone, the pendulum will retain its period even as its amplitude decreases in time. Under ideal conditions (no friction or air resistance), this would make for a perfect time-keeper. Huygens' Pendulum is a very practical application of curvature, and it would prove to serve as inspiration for further investigation of curvature by later mathematicians. Finally, in the late seventeenth century, the Calculus was invented and turned the House of Math into a metropolis. More specifically, Calculus’ methods of infinitesimals and limits were the perfect tools for the problem of curvature. (In fact, Huygens had already tried to give a description of curvature using limit-like ideas.) Because most curves have a different degree of bending at every point, mathematicians needed a method to look at an "infinitely small" part of a curve in order to say what was happening at a particular point. Calculus became this prescribed method. Of the two "co-discoverers" of the calculus, Sir Isaac Newton and Gottfried Wilhelm Leibniz, Newton was the one who did more extensive (and more accurate) work with curvature. He seemed to be attracted to this subject primarily for its beauty. In Problem 5 of his Methods of series and fluxions (To find the curvature of any curve at a given point), he begins, "The problem has the mark of exceptional elegance and of being pre-eminently useful in the science of curves" (Whiteside 151). Newton started to investigate the problem of curvature by listing its most elementary properties:

41. A circle has a constant curvature which is inversely proportional to its radius. 2. The largest circle that is tangent to a curve (on its concave side) at a point has the same curvature as the curve at that

point. 3. The center of this circle is the "centre of curvature" of the curve at that point.

(Whiteside 151)

The second item on this list is the earliest definition of curvature that would allow both rigorization and generalization, and it is therefore very key. Newton had not at this point described how to find the "largest circle", but he said that it is the circle that touches a curve on its concave side and is of such a size that "no other tangent circle can be drawn between in the contact angles neighbouring that point" (Whiteside 151). Figure 3 demonstrates this. The blue circles are too small, the red circles are too big, and the purple one is just right—the osculating circle. Later on in this book, Newton does indeed rigorize and generalize these ideas, thus founding the field of differential geometry—the application of calculus to geometry.Figure 3Newton’s definition of center of curvature is significant because it is the first place in his work with curvature where he uses the concept of infinitesimals. About the center of curvature, he states, "It is the meet of normals at indefinitely small distances from [thepoint in question] on its either side" (see demo) (Whiteside 153). Newton used this very method to at long last find an equation for the radius of curvature. By finding the intersection of the normal at the point in question and an infinitely close normal, Newton derived the following formula for curvature:

Newton derived a similar equation for use with polar coordinates (Kline 364). One of Newton’s observations regarding his equation for curvature was that it will yield an undefined number at points of inflection(Coolidge 378). As he noted, curves behave like a straight line near a point of inflection (they don’t bend either way), and since the radius of curvature of a straight line is infinite, the radius of curvature at these points is infinite. Newton used the term "point of straightness" to describe inflection points, and gave the example ax3 = y4 (Figure 4) to illustrate his point.Figure 4Newton also set down that the points of greatest or least flexure (bending) of a curve can be found by setting the fluxion of the radius of curvature equal to zero (Whiteside 181). This is exactly his method of finding maxima and minima applied to radii of curvature. In modern differential geometry texts, such points are called "vertices". Also, Newton remarked that cusps can be found by setting the radius of curvature itself equal to zero (Whiteside 181). Newton also calculated the formula for radius of curvature for several curves, including the cycloid and the Archimedan spiral (Kline 556). In doing so, he reproduced Huygens’ result that the "locus of centre of curvature" (evolute) of a cycloid is a cycloid. Of course, Newton did it with his own analytic methods as opposed to Huygens’ geometrical means. Furthermore, Newton re-invented Huygens’ cycloidal pendulum (Whiteside 163). It is unclear whether or not Newton made these discoveries himself or was made aware of them though other mathematicians. What is clear is that Newton’s methods were his own. The ease with which he was able to derive these results is what is most impressive. Other mathematicians of Newton’s time also worked on the problem of curvature. However, none were as prolific, and none were as successful. For example, Leibniz (who also claimed to have invented the calculus), was the one who gave the circle of curvature the name which has stuck ever since--the "osculating circle" (Kline 378). One of the reasons why Leibniz’s work with curvature is not given very much recognition is because of a very fundamental error he made. In one of his works regarding curvature, Leibniz made the claim that an osculating circle has a four-point contact with a curve near the point for which the circle is drawn (Coolidge 377). James Bernoulli very promptly pointed this out this mistake (Coolidge 377). Incidentally, James and his brother, John Bernoulli, gave their own formula for curvature in 1691. James called it the "golden theorem" (Kline 382). This theorem stated that the radius of curvature of a curve was equal to two equivalent ratios:dx ds : d dy = dy ds : d dxJames also gave this result in polar coordinates. In the following century, many Calculus text books were written, and many dealt with the subject of curvature. Marquis de L’Hopital included Newton’s formula for the radius of curvature in his Analyse (Katz 446). In 1731, Alexis-Claude Clairaut (1713-1765) became the first geometer to publish on "curves of double curvature" (Kline 557). In modern language, such curve are simply space curves (curves in three-dimensional space). Clairaut’s "new" curvature, which he named "torsion" was a measure of how quickly a space curve pulls away from the plane of its osculating circle at a point (Kline 559). In order to study such curves, Clairaut borrowed one of Descartes’ methods and projected his space curves onto two perpendicular planes, then treating these projections as regular plane curves (Kline 557). Clairaut’s work was significant, because it paved the way for later geometers, such as Gauss, to study the curvature of surfaces, which also

bend in two directions. Thomas Simpson, in his book on the method of fluxions, also dealt with the subject of curvature. In Section V (The Use of Fluxions in Determining the Radii of Curvature, and the Evolute of Curves), he described the string-construction of evolutes which was developed by Huygens. Simpson also used an interesting description of curvature. He stated that the degree of curvature is equivalent to the second fluxion of the curve, and this is a measure of "deflection from [the] tangent" (Simpson 53). This is an important observation because it directly links the first and second fluxions of a curve. Leonhard Euler is the mathematician responsible for the important theorem that the magnitude of curvature equals the magnitude of the second derivative of a parameterization at a point (McCleary 67). As parameterized curves would eventually become central to differential geometry, this was a monumental discovery. In 1774, Euler made more revolutionary statements about curvature. Most importantly, he devised a new way of defining curvature. To each tangent vector of a curve he assigned a point on the unit circle which corresponds to the direction of that tangent vector (see demo). Then, he defined curvature as ds’/ds, the change in angle of the tangent divided by the change in arc length (in an infinitely small locale) (see Figure 5) (Kline 559).. On an intuitive level, one can see that a large change of angle in a short distance will produce a large curvature, as Euler’s equation indicates. In addition Euler produced an analytical expression for the radius of curvature: (Kline 559). Even though Euler was describing concepts that were a century old, it seemed like he was inventing new concepts, for his methods of studying curvature were very innovative.Figure 5In 1826, Augustin-Louis Cauchy made a significant improvement to differential geometry. In his Lecons sur les applications du calcul infinitesimal a la geometrie, he cast off the idea of constant infinitesimals (Kline 560). He rightly regarded infinitesimals as quantities which were approaching zero. He also "straightened out the confusion" between differentials and increments (Kline 560). He noted that ds2 = dx2 + dy2 + dz2 should be more properly written as (ds/dt)2 = (dx/dt)2 + (dy/dt)2 + (dz/dt)2 (Kline 560). These were important distinctions not only for differential geometry, but for calculus in general as well. Thomas Hill wrote another text on curvature, called An Elementary Treatise On Curvature. Much of this work is regurgitation of his predecessors’ findings. However, Simpson does offer something which was a fairly novel finding, whether of not he originated the idea. In the chapter concerning the classification of curves he states, "An equation between the radius of curvature . . . and the angle it makes with a given direction, implies all the conditions of the form of the curve, though not of its position" (Hill 15). Eventually, this (slightly modified) would come to be known as the Fundamental Theorem of the Local Theory of Plane Curves. In modern differential geometry textbooks, this theorem is not attributed to a specific mathematician, so it just seems to have been a "product of the times". Hill can be thought of as a representative of the mid-nineteenth century, as his textbook encompasses most of the differential geometry and applications of the time. Karl Friedrich Gauss brought differential geometry to a whole new level. He is known primarily for his work on the theoryof surfaces. He produced a formula for curvature which was very similar to Euler’s, and from this he derived the following profound analogy: arc : amplitude : curvature :: time : motion : velocity (Gauss 83). In order to understand this, one should think of a point moving along a curve with uniform velocity with respect to arc length. Then curvature is a measure of the amplitude of the curve. Gauss realized an ingenious method for measuring the curvature of a two-dimensional surface in three-dimensional space. The so-called Gaussian curvature of a surface at a point is the product of the greatest and least curvatures of all curves passing though that point in the surface. Most of Gauss’ findings deal with complicated properties of surfaces and are beyond the scope of an introductory exploration of curvature. Gauss did add a new dimension to the study of curvature and will always hold a place in its history. Mathematical treatment of curvature underwent a drastic metamorphosis over the history of the problem. Differential geometry started with vague definitions and simple concepts and developed into the well-oiled machine that it is today. Clearly, the most dramatic leap came when the Calculus was invented. Differential geometry and curvature were natural applications for the Calculus because they provided words to its music--practical applications (map making, light ray travel, etc...) to the theory. Since then, mathematicians have become more proficient with the concept of infinitesimals--one no longer speaks of "adjacent points" in the same way that Newton did. In any case, this author is certain that the journey has only just begun. Gauss made the leap from curvature of one-dimensional curves to the curvature of two-dimensional surfaces in three-dimensional space. It will not be long before common mathematicians are proficient with the three curvatures of a three-dimensional surface lying in four-dimensional space. As Gauss said, "The subject is still so far from being exhausted" (Katz 693). And so are the mathematiciansGlobal curvature, ideal knots and models of DNA packing

Global curvature characterizes knot tightness.

Overview and acknowledgementsDuring a lunch-time discussion on the ideal shapes of knots and such, the concept of the "global curvature function" of a space curve was born. This function is related to, but distinct from, standard local curvature, and is connected to various physically appealing properties of a curve. Global curvature provides a concise characterization of curve thickness, and of certain ideal shapes of knots as have been investigated within the context of DNA. Moreover, global curvature is connected to the writhing number of a space curve and has applications in the study of self-contact and packing problems for rods. Some of the figures here are based on numerical data from Katritch, V., Bednar, J., Michoud, D., Scharein, R.G., Dubochet, J. & Stasiak, A. (1996) Nature 384, 142-145. Katritch, V., Olson, W.K., Pieranski, P., Dubochet, J. & Stasiak, A. (1997) Nature 388, 148-151. It is a pleasure to thank A. Stasiak, V. Katritch and P. Pieranski for making their knot data available.

Related articlesO. Gonzalez & R. de la Llave, "Existence of ideal knots," Journal of Knot Theory and Its Ramifications 12 (2003) 123-133.

J.R. Banavar, O. Gonzalez, J.H. Maddocks & A. Maritan, ``Self-interactions of strands and sheets,'' Journal of StatisticalPhysics 110 (2003) 35-50.

O. Gonzalez, J.H. Maddocks & J. Smutny, ``Curves, circles and spheres,'' in Physical Knots: Knotting, Linking and Folding Geometric Objects in R^3, J.A. Calvo, K.C. Millett & E.J. Rawdon, Eds., Contemporary Mathematics 304 (2002) 195-215.

O. Gonzalez, J.H. Maddocks, F. Schuricht & H. von der Mosel, ``Global curvature and self-contact of nonlinearly elastic curves and rods,'' Calculus of Variations and Partial Differential Equations, 14 (2002) 29-68.

O. Gonzalez & J.H. Maddocks, "Global Curvature, Thickness and the Ideal Shapes of Knots," The Proceedings of the National Academy of Sciences, USA, 96 (1999) 4769-4773.

Table of contents

Motivation Definition of global curvature Applications of global curvature Ideal knots

definition a theorem Yes, but is it sharp?

Self-contact problems Optimal packing problems

MotivationAny smooth, non-self-intersecting curve can be thickened into a smooth, non-self-intersecting tube of constant radius centered on the curve, as illustrated below.

If the curve is a straight line there is no upper bound on the tube radius, but for non-straight curves there is a critical radius above which the tube either ceases to be smooth or exhibits self-contact. This critical radius is an intrinsic property of the curve called its thickness or normal injectivity radius. Of the two examples shown above, the first has a critical radius determined by smoothness

and the second has a critical radius determined by self-contact.

If one considers the class of smooth, non-self-intersecting closed curves of a prescribed knot type and unit length, one may ask which curve in this class is the thickest. The thickness of such a curve is an intrinsic property of the knot, and the curve itself provides a certain ideal shape or representation of the knot type. For example, the two curves shown below are of the same knot type (K31), but the second is substantially thicker than the first. In fact, the second is considered to be of maximum thickness for its knot type -- it is considered to be an ideal shape for the K31 knot.

Approximations of ideal shapes in above sense have been found via a series of computer experiments by Katritch and coworkers. These shapes were seen to have intriguing physical features, and even a correspondence to time-averaged shapes of knotted DNA molecules in solution. So how does one mathematically define curve thickness and the ideal shape of a knot? What are necessary and sufficient conditions for a knotted curve to be ideal? Answering these questions has led to a new geometrical quantity for space curves: the global radiusof curvature function. This function provides a simple characterization of curve thickness, and provides an elementary necessary condition that any ideal shape of a knotted curve must satisfy. Moreover, global radius of curvature is connected to the writhing number of a space curve and has applications in the study of self-contact problems for rods.

Definition of global curvatureOur definition of global radius of curvature is based on the elementary facts that any three non-collinear points x, y and z in three-dimensional space define a unique circle (the circumcircle), and the radius of this circle (the circumradius) can be written as

where A(x,y,z) is the area of the triangle with vertices x, y and z, |x-y| is the Euclidean distance between the points x and y, and so on. When the points x, y and z are distinct, but collinear, the circumcircle degenerates into a straight line and we assign a value of infinity to r(x,y,z). When x, y and z are points on a simple, smooth curve C, the domain of the function r(x,y,z) can be extended by continuous limits to all points on C.

For example, the limit of r(x,y,z) as y,z approach x along C is just the standard local radius of curvature at x, and the limit circumcircle is just the osculating circle to C at x. If one holds x and y fixed, and takes the limit as z approaches y along C, then the limiting value of r(x,y,z) is the radius of that circle which passes through x and is tangent to C at y. Given a simple, smooth curve C we define the global radius of curvature at each point x by

which can actually be shown to be a continuous function of x on C. Global radius of curvature can be interpreted as a generalization, indeed a globalization, of the standard local radius of curvature. In fact, since the points y=x and z=x are competitorsin the minimization, global radius of curvature is bounded by local radius of curvature, that is,

The figure below shows plots of global (black) and local (grey) radius of curvature versus arclength for some example space curves C. The space curves are represented by their critical tubes, which are colored by global radius of curvature. The blue regions are where global radius of curvature is minimal.

Applications of global curvatureA particularly interesting quantity for a simple, smooth curve C is the minimum value of global radius of curvature, namely

which is just the minimum value of the circumradius function r(x,y,z) over all triplets of points on C. This quantity has many physically appealing interpretations and applications:

Any spherical shell of radius less than Delta[C] cannot intersect C in three or more points (counting tangency points twice). In effect, a billiard ball of radius less than Delta[C] cannot find a stable resting place in C, for there is always enough room for it to pass through the curve, as illustrated in the figure below.

Delta[C] is the thickness of C in the sense that it is the radius of the thickest smooth tube that may be centered on C, as illustrated in the figure below.

Ideal knotsdefinitionWe can now use rhoG(x) and Delta[C] to define and characterize ideal shapes of knots. Let K denote the set of smooth, non-self-intersecting closed curves of a prescribed knot type and length. Then a curve C* in K is ideal if and only if

That is, among all curves in K, an ideal shape C* has maximal thickness. This definition corresponds precisely to the intuitive notion of the thickest tube of fixed length that can be tied into a given knot. a theoremSo what are the properties of curves C* that maximize Delta[C]? That is, what characterizes an ideal shape? Our study of this question led to the following theorem. Let C* be a curve in K with arclength parameter s in [0,L], and let J* be that subset of [0,L] for which the standard local curvature vanishes, that is

Then C* can be ideal only if there is a constant a>0 such that

This constant a>0 is actually the thickness of the ideal shape -- it is the radius of the thickest tube of fixed length that can be tied into the given knot. While we have defined everything here for smooth curves, there is a straightforward extension to discrete (piecewise linear) curves. Yes, but is it sharp?We tested our theorem on ideal shapes previously computed by A. Stasiak, V. Katritch and P. Pieranski. They had computed ideal shapes using hueristic, Monte Carlo type algorithms, and we had two main questions:

First, would their data satisfy our necessary condition? Second, how sharp was it? Was the inequality "rhoG > a" a real possibility, or was it just a weakness in our arguments?

Our tests (along with some other neat data) are summarized in the following figures. Results for non-ideal and ideal 3_1 knot

Frame a

Generic knot shape produced using a simple parametric representation. The red discs indicate points where rhoG can be associated with the local radius of curvature, and blue discs indicate points where rhoG can be associated with a distance of closest approach.

Frame b

Numerically computed ideal shape. Here rhoG corresponds to a distance of closest approach at all points.

Frame c

Same shape as in b , but with a different visualization of the global radius of curvature. Each of the several spokes emanating from a point on the curve represents the diameter of a disc that realizes rhoG at that point.

Frame d

Global and local radius of curvature plots for the shapes in a and b .

Light blue and light red curves are global and local radius of curvature for the non-ideal shape a . Dark blue and dark red curves are global and local radius of curvature for the ideal shape b .

(The light red curve is nearly periodic but its upper limits are not contained within the plot range.)

Details on Ideal Shape 160 points nearly uniformly spaced in arclength computed using a Metropolis Monte Carlo procedure (Katritch et al.) variation in rhoG: 0.1 PERCENT our necessary condition is satisfied!

Results for ideal composite 3_1#3_1 knot

Frame a

The tube shown here has a radius of Delta[C] where C is the centerline of the numerically computed ideal shape. The tube is colored by local curvature, where blue indicates near-zero values (straight portions).

Frame b

The sphere interpretation of Delta[C]. Any spherical shell of radius less than Delta[C] cannot intersect C in three or more points (counting tangency points twice). The spheres shown here have a radius of Delta[C].

Frame c

Global radius of curvature plot for shape in a . The light blue curve (partially obscured) corresponds to raw data from Katritch et al., and the dark blue curve corresponds to a corrected shape.

Frame d

Comparison of global radius of curvature (blue) and local radius of curvature (red) for the corrected shape.

Details on Ideal Shape 286 points nearly uniformly spaced in arclength computed using a Metropolis Monte Carlo procedure (Katritch et al.) original data (light blue) was slightly non-ideal corrected data (dark blue) yielded ideal variation in rhoG (excluding two upward spike regions): 0.4 PERCENT our necessary condition is satisfied, and rhoG > a seems to be real!

Self-contact problems(new developments coming soon!)

Optimal packing problems(new developments coming soon!)

Back to home pageBack Curvature Curvature is a measure of how much a curve curves.

Curvature is measured by fitting a circle into the curve, then taking the reciprocal of the circle’s radius. In this figure, at point x the curve is best described by a circle with radius r. At this point, the curvature is 1/r. (We use the reciprocal, 1/r, instead of just r because a flat line has an infinite radius. Taking the reciprocal gives us 0 instead of infinity.)

Continuity for NURBS curves and surfaces Continuity is a measure of how smoothly two curves or surfaces “flow” into each other at their meeting point. The type of curvature your curves and surfaces have may be important if you need to subsequently export your Maya NURBS surfaces to a CAD software application. Positional (G0)

The endpoints of the two curves or surfaces meet exactly. Note that the two curves or surfaces can meet at any angle and still have positional continuity.

Tangent (G1)

Curves or surfaces that have tangent continuity also have positional continuity, plus the end tangents match at the common endpoint The two curves will appear to be travelling in the same direction at the join, but they may still have very different apparent “speeds”(rate of change of the direction, also called curvature).

For example, in this figure, the two curves have the same tangent (the double-arrow line) at the join (the dot). But the curve to the left of the join has a slow (low) curvature at the join, while the curve to the right of the join has a fast (high) curvature at the join. Curvature (G2)

Curves or surfaces that are curvature continuous have tangent continuity, plus the curvature of the two curves matches at the common endpoint. The two curves appear to have the same “speed” at the join.