Inmathematics,curvature is any of several strongly related concepts ingeometry that intuitively measure the amount by which acurve deviates from being astraight line or by which asurface deviates from being aplane. If a curve or surface is contained in a larger space, curvature can be definedextrinsically relative to the ambient space.Curvature of Riemannian manifolds of dimension at least two can be definedintrinsically without reference to a larger space.
For curves, the canonical example is that of acircle, which has a curvature equal to thereciprocal of itsradius. Smaller circles bend more sharply, and hence have higher curvature. The curvatureat a point of adifferentiable curve is the curvature of itsosculating circle — that is, the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to thetangent, which is a vector quantity, the curvature at a point is typically ascalar quantity, that is, it is expressed by a singlereal number.
For surfaces (and, more generally for higher-dimensionalmanifolds), that areembedded in aEuclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or manifold. This leads to the concepts ofmaximal curvature,minimal curvature, andmean curvature.
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InTractatus de configurationibus qualitatum et motuum,[1] the 14th-century philosopher and mathematicianNicole Oresme introduces the concept of curvature as a measure of departure from straightness; for circles he has the curvature as being inversely proportional to the radius; and he attempts to extend this idea to other curves as a continuously varying magnitude.[2]
The curvature of adifferentiable curve was originally defined throughosculating circles. In this setting,Augustin-Louis Cauchy showed that the center of curvature is the intersection point of two infinitely closenormal lines to the curve.[3]
Intuitively, the curvature describes for any part of a curve how much the curve direction changes over a small distance travelled (e.g. angle per length inrad/m), so it is a measure of theinstantaneous rate of change ofdirection of a point that moves on the curve: the larger the curvature, the larger this rate of change. In other words, the curvature measures how fast the unit tangent vector to the curve at pointP rotates[4] when pointP moves at unit speed along the curve. In fact, it can be proved that this instantaneous rate of change of direction is exactly the curvature.
More precisely, suppose that the point is moving on the curve at a constant speed of one unit per time, that is, the position of the pointP(s) is a unit-speed function of the parameters. (The parameter may be thought as time or asarc length from a given origin.) LetT(s) be aunit tangent vector of the curve atP(s), which is also thederivative ofP(s) with respect tos. Then, the derivative ofT(s) with respect tos is a vector that is normal to the curve and whose length is the curvature.
To be meaningful, the definition of the curvature and its different characterizations require that the curve iscontinuously differentiable nearP, for having a tangent that varies continuously; it requires also that the curve is twice differentiable atP, for insuring the existence of the involved limits, and of the derivative ofT(s).
The characterization of the curvature in terms of the derivative of the unit tangent vector is probably less intuitive than the definition in terms of the osculating circle, but formulas for computing the curvature are easier to deduce. Therefore, and also because of its use inkinematics, this characterization is often given as a definition of the curvature.
Historically, the curvature of a differentiable curve was defined through theosculating circle, which is the circle that best approximates the curve at a point. More precisely, given a pointP on a curve, every other pointQ of the curve defines a circle (or sometimes a line) passing throughQ andtangent to the curve atP. The osculating circle is thelimit, if it exists, of this circle whenQ tends toP. Then thecenter and theradius of curvature of the curve atP are the center and the radius of the osculating circle. The curvature is thereciprocal of radius of curvature. That is, the curvature is
whereR is the radius of curvature[5] (the whole circle has this curvature, it can be read as turn2π over the length2πR).
This definition is difficult to manipulate and to express in formulas. Therefore, other equivalent definitions have been introduced.
Everydifferentiable curve can beparametrized with respect toarc length.[6] In the case of a plane curve, this means the existence of a parametrizationγ(s) = (x(s),y(s)), wherex andy are real-valued differentiable functions whose derivatives satisfy
This means that the tangent vector
has a length equal to one and is thus aunit tangent vector.
If the curve is twice differentiable, that is, if the second derivatives ofx andy exist, then the derivative ofT(s) exists. This vector is normal to the curve, its length is the curvatureκ(s), and it is oriented toward the center of curvature. That is,
Moreover, because the radius of curvature is (assuming𝜿(s) ≠ 0)
and the center of curvature is on the normal to the curve, the center of curvature is the point
(In case the curvature is zero, the center of curvature is not located anywhere on the planeR2 and is often said to be located "at infinity".)
IfN(s) is theunit normal vector obtained fromT(s) by a counterclockwise rotation ofπ/2, then
withk(s) = ±κ(s). The real numberk(s) is called theoriented curvature orsigned curvature. It depends on both the orientation of the plane (definition of counterclockwise), and the orientation of the curve provided by the parametrization. In fact, thechange of variables → –s provides another arc-length parametrization, and changes the sign ofk(s).
With the above, the center of curvature can be expressed as:
Letγ(t) = (x(t),y(t)) be a properparametric representation of a twice differentiable plane curve. Hereproper means that on thedomain of definition of the parametrization, the derivativedγ/dtis defined, differentiable and nowhere equal to the zero vector.
With such a parametrization, the signed curvature is
where primes refer to derivatives with respect tot. The curvatureκ is thus
These can be expressed in acoordinate-free way as
These formulas can be derived from the special case of arc-length parametrization in the following way. The above condition on the parametrisation imply that the arc lengths is a differentiablemonotonic function of the parametert, and conversely thatt is a monotonic function ofs. Moreover, by changing, if needed,s to–s, one may suppose that these functions are increasing and have a positive derivative. Using notation of the preceding section and thechain rule, one has
and thus, by taking the norm of both sides
where the prime denotes differentiation with respect tot.
The curvature is the norm of the derivative ofT with respect tos. By using the above formula and the chain rule this derivative and its norm can be expressed in terms ofγ′ andγ″ only, with the arc-length parameters completely eliminated, giving the above formulas for the curvature.
Thegraph of a functiony =f(x), is a special case of a parametrized curve, of the form
As the first and second derivatives ofx are 1 and 0, previous formulas simplify to
for the curvature, and to
for the signed curvature.
In the general case of a curve, the sign of the signed curvature is somewhat arbitrary, as it depends on the orientation of the curve. In the case of the graph of a function, there is a natural orientation by increasing values ofx. This makes significant the sign of the signed curvature.
The sign of the signed curvature is the same as the sign of the second derivative off. If it is positive then the graph has an upward concavity, and, if it is negative the graph has a downward concavity. If it is zero, then one has aninflection point or anundulation point.
When theslope of the graph (that is the derivative of the function) is small, the signed curvature is well approximated by the second derivative. More precisely, usingbig O notation, one has
It is common inphysics andengineering to approximate the curvature with the second derivative, for example, inbeam theory or for deriving thewave equation of a string under tension, and other applications where small slopes are involved. This often allows systems that are otherwisenonlinear to be treated approximately as linear.
If a curve is defined inpolar coordinates by the radius expressed as a function of the polar angle, that isr is a function ofθ, then its curvature is
where the prime refers to differentiation with respect toθ.
This results from the formula for general parametrizations, by considering the parametrization
For a curve defined by animplicit equationF(x,y) = 0 withpartial derivatives denotedFx ,Fy ,Fxx ,Fxy ,Fyy ,the curvature is given by[7]
The signed curvature is not defined, as it depends on an orientation of the curve that is not provided by the implicit equation. Note that changingF into–F would not change the curve defined byF(x,y) = 0, but it would change the sign of the numerator if theabsolute value were omitted in the preceding formula.
A point of the curve whereFx =Fy = 0 is asingular point, which means that the curve is not differentiable at this point, and thus that the curvature is not defined (most often, the point is either a crossing point or acusp).
The above formula for the curvature can be derived from the expression of the curvature of the graph of a function by using theimplicit function theorem and the fact that, on such a curve, one has
It can be useful to verify on simple examples that the different formulas given in the preceding sections give the same result.
A common parametrization of acircle of radiusr isγ(t) = (r cost,r sint). The formula for the curvature gives
It follows, as expected, that the radius of curvature is the radius of the circle, and that the center of curvature is the center of the circle.
The circle is a rare case where the arc-length parametrization is easy to compute, as it is
It is an arc-length parametrization, since the norm of
is equal to one. This parametrization gives the same value for the curvature, as it amounts to division byr3 in both the numerator and the denominator in the preceding formula.
The same circle can also be defined by the implicit equationF(x,y) = 0 withF(x,y) =x2 +y2 –r2. Then, the formula for the curvature in this case gives
Consider theparabolay =ax2 +bx +c.
It is the graph of a function, with derivative2ax +b, and second derivative2a. So, the signed curvature is
It has the sign ofa for all values ofx. This means that, ifa > 0, the concavity is upward directed everywhere; ifa < 0, the concavity is downward directed; fora = 0, the curvature is zero everywhere, confirming that the parabola degenerates into a line in this case.
The (unsigned) curvature is maximal forx = –b/2a, that is at thestationary point (zero derivative) of the function, which is thevertex of the parabola.
Consider the parametrizationγ(t) = (t,at2 +bt +c) = (x,y). The first derivative ofx is1, and the second derivative is zero. Substituting into the formula for general parametrizations gives exactly the same result as above, withx replaced byt. If we use primes for derivatives with respect to the parametert.
The same parabola can also be defined by the implicit equationF(x,y) = 0 withF(x,y) =ax2 +bx +c –y. AsFy = –1, andFyy =Fxy = 0, one obtains exactly the same value for the (unsigned) curvature. However, the signed curvature is meaningless here, as–F(x,y) = 0 is a valid implicit equation for the same parabola, which gives the opposite sign for the curvature.
The expression of the curvatureIn terms of arc-length parametrization is essentially thefirst Frenet–Serret formula
where the primes refer to the derivatives with respect to the arc lengths, andN(s) is the normal unit vector in the direction ofT′(s).
As planar curves have zerotorsion, the second Frenet–Serret formula provides the relation
For a general parametrization by a parametert, one needs expressions involving derivatives with respect tot. As these are obtained by multiplying byds/dt the derivatives with respect tos, one has, for any proper parametrization
Acurvature comb[8] can be used to represent graphically the curvature of every point on a curve. If is a parametrised curve its comb is defined as the parametrized curve
where are the curvature and normal vector respectively, and is a scaling factor (to be chosen as to enhance the graphical representation).
Curvature combs are useful when combining two different curves in CAD environments. They provide a visual representation of the continuity between the curves. The continuity can be defined as being in one of four levels.
G0 : The 2 curvature combs are at an angle at the junction.
G1 : The teeth of the 2 combs are parallel at the junction but are of different length.
G2 : The teeth are parallel and of the same length. However the tangents of the 2 combs are not the same.
G3 : The teeth are parallel and of the same length and the tangents of the 2 combs are the same.
The above image shows a G2 continuity at the 2 junctions.
As in the case of curves in two dimensions, the curvature of a regularspace curveC in three dimensions (and higher) is the magnitude of the acceleration of a particle moving with unit speed along a curve. Thus ifγ(s) is the arc-length parametrization ofC then the unit tangent vectorT(s) is given by
and the curvature is the magnitude of the acceleration:
The direction of the acceleration is the unit normal vectorN(s), which is defined by
The plane containing the two vectorsT(s) andN(s) is theosculating plane to the curve atγ(s). The curvature has the following geometrical interpretation. There exists a circle in the osculating plane tangent toγ(s) whoseTaylor series to second order at the point of contact agrees with that ofγ(s). This is theosculating circle to the curve. The radius of the circleR(s) is called theradius of curvature, and the curvature is the reciprocal of the radius of curvature:
The tangent, curvature, and normal vector together describe the second-order behavior of a curve near a point. In three dimensions, the third-order behavior of a curve is described by a related notion oftorsion, which measures the extent to which a curve tends to move as a helical path in space. The torsion and curvature are related by theFrenet–Serret formulas (in three dimensions) andtheir generalization (in higher dimensions).
For a parametrically-defined space curve in three dimensions given in Cartesian coordinates byγ(t) = (x(t),y(t),z(t)), the curvature is
where the prime denotes differentiation with respect to the parametert. This can be expressed independently of thecoordinate system by means of the formula[9]
where × denotes thevector cross product. The following formula is valid for the curvature of curves in a Euclidean space of any dimension:
Given two pointsP andQ onC, lets(P,Q) be the arc length of the portion of the curve betweenP andQ and letd(P,Q) denote the length of the line segment fromP toQ. The curvature ofC atP is given by the limit[citation needed]
where the limit is taken as the pointQ approachesP onC. The denominator can equally well be taken to bed(P,Q)3. The formula is valid in any dimension. Furthermore, by considering the limit independently on either side ofP, this definition of the curvature can sometimes accommodate a singularity atP. The formula follows by verifying it for the osculating circle.
The curvature of curves drawn on asurface is the main tool for the defining and studying the curvature of the surface.
For a curve drawn on a surface (embedded in three-dimensionalEuclidean space), several curvatures are defined, which relates the direction of curvature to the surface's unitnormal vector, including the:
Any non-singular curve on a smooth surface has its tangent vectorT contained in thetangent plane of the surface. Thenormal curvature,kn, is the curvature of the curve projected onto the plane containing the curve's tangentT and the surface normalu; thegeodesic curvature,kg, is the curvature of the curve projected onto the surface's tangent plane; and thegeodesic torsion (orrelative torsion),τr, measures the rate of change of the surface normal around the curve's tangent.
Let the curve bearc-length parametrized, and lett =u ×T so thatT,t,u form anorthonormal basis, called theDarboux frame. The above quantities are related by:
All curves on the surface with the same tangent vector at a given point will have the same normal curvature, which is the same as the curvature of the curve obtained by intersecting the surface with the plane containingT andu. Taking all possible tangent vectors, the maximum and minimum values of the normal curvature at a point are called theprincipal curvatures,k1 andk2, and the directions of the corresponding tangent vectors are calledprincipal normal directions.
Curvature can be evaluated along surfacenormal sections, similar to§ Curves on surfaces above (see for example theEarth radius of curvature).
Some curved surfaces, such as those made from a smooth sheet of paper, can be flattened down into the plane without distorting their intrinsic features in any way. Suchdevelopable surfaces have zero Gaussian curvature (see below).[10]
In contrast to curves, which do not have intrinsic curvature, but do have extrinsic curvature (they only have a curvature given an embedding), surfaces can have intrinsic curvature, independent of an embedding. TheGaussian curvature, named afterCarl Friedrich Gauss, is equal to the product of the principal curvatures,k1k2. It has a dimension of length−2 and is positive forspheres, negative for one-sheethyperboloids and zero for planes andcylinders. It determines whether a surface islocallyconvex (when it is positive) or locally saddle-shaped (when it is negative).
Gaussian curvature is anintrinsic property of the surface, meaning it does not depend on the particularembedding of the surface; intuitively, this means that ants living on the surface could determine the Gaussian curvature. For example, an ant living on a sphere could measure the sum of the interior angles of a triangle and determine that it was greater than 180 degrees, implying that the space it inhabited had positive curvature. On the other hand, an ant living on a cylinder would not detect any such departure fromEuclidean geometry; in particular the ant could not detect that the two surfaces have different mean curvatures (see below), which is a purely extrinsic type of curvature.
Formally, Gaussian curvature only depends on theRiemannian metric of the surface. This isGauss's celebratedTheorema Egregium, which he found while concerned with geographic surveys and mapmaking.
An intrinsic definition of the Gaussian curvature at a pointP is the following: imagine an ant which is tied toP with a short thread of lengthr. It runs aroundP while the thread is completely stretched and measures the lengthC(r) of one complete trip aroundP. If the surface were flat, the ant would findC(r) = 2πr. On curved surfaces, the formula forC(r) will be different, and the Gaussian curvatureK at the pointP can be computed by theBertrand–Diguet–Puiseux theorem as
Theintegral of the Gaussian curvature over the whole surface is closely related to the surface'sEuler characteristic; see theGauss–Bonnet theorem.
The discrete analog of curvature, corresponding to curvature being concentrated at a point and particularly useful forpolyhedra, is the(angular) defect; the analog for theGauss–Bonnet theorem isDescartes' theorem on total angular defect.
Because (Gaussian) curvature can be defined without reference to an embedding space, it is not necessary that a surface be embedded in a higher-dimensional space in order to be curved. Such an intrinsically curved two-dimensional surface is a simple example of aRiemannian manifold.
The mean curvature is anextrinsic measure of curvature equal to half the sum of theprincipal curvatures,k1 +k2/2. It has a dimension of length−1. Mean curvature is closely related to the first variation ofsurface area. In particular, aminimal surface such as asoap film has mean curvature zero and asoap bubble has constant mean curvature. Unlike Gauss curvature, the mean curvature is extrinsic and depends on the embedding, for instance, acylinder and a plane are locallyisometric but the mean curvature of a plane is zero while that of a cylinder is nonzero.
The intrinsic and extrinsic curvature of a surface can be combined in the second fundamental form. This is aquadratic form in the tangent plane to the surface at a point whose value at a particular tangent vectorX to the surface is the normal component of the acceleration of a curve along the surface tangent toX; that is, it is the normal curvature to a curve tangent toX (seeabove). Symbolically,
whereN is the unit normal to the surface. For unit tangent vectorsX, the second fundamental form assumes the maximum valuek1 and minimum valuek2, which occur in the principal directionsu1 andu2, respectively. Thus, by theprincipal axis theorem, the second fundamental form is
Thus the second fundamental form encodes both the intrinsic and extrinsic curvatures.
An encapsulation of surface curvature can be found in the shape operator,S, which is aself-adjointlinear operator from the tangent plane to itself (specifically, the differential of theGauss map).
For a surface with tangent vectorsX and normalN, the shape operator can be expressed compactly inindex summation notation as
(Compare thealternative expression of curvature for a plane curve.)
TheWeingarten equations give the value ofS in terms of the coefficients of thefirst andsecond fundamental forms as
The principal curvatures are theeigenvalues of the shape operator, the principal curvature directions are itseigenvectors, the Gauss curvature is itsdeterminant, and the mean curvature is half itstrace.
By extension of the former argument, a space of three or more dimensions can be intrinsically curved. The curvature isintrinsic in the sense that it is a property defined at every point in the space, rather than a property defined with respect to a larger space that contains it. In general, a curved space may or may not be conceived as being embedded in a higher-dimensionalambient space; if not then its curvature can only be defined intrinsically.
After the discovery of the intrinsic definition of curvature, which is closely connected withnon-Euclidean geometry, many mathematicians and scientists questioned whether ordinary physical space might be curved, although the success of Euclidean geometry up to that time meant that the radius of curvature must be astronomically large. In the theory ofgeneral relativity, which describesgravity andcosmology, the idea is slightly generalised to the "curvature ofspacetime"; in relativity theory spacetime is apseudo-Riemannian manifold. Once a time coordinate is defined, the three-dimensional space corresponding to a particular time is generally a curved Riemannian manifold; but since the time coordinate choice is largely arbitrary, it is the underlying spacetime curvature that is physically significant.
Although an arbitrarily curved space is very complex to describe, the curvature of a space which is locallyisotropic andhomogeneous is described by a single Gaussian curvature, as for a surface; mathematically these are strong conditions, but they correspond to reasonable physical assumptions (all points and all directions are indistinguishable). A positive curvature corresponds to the inverse square radius of curvature; an example is a sphere orhypersphere. An example of negatively curved space ishyperbolic geometry (see also:non-positive curvature). A space or space-time with zero curvature is calledflat. For example,Euclidean space is an example of a flat space, andMinkowski space is an example of a flat spacetime. There are other examples of flat geometries in both settings, though. Atorus or acylinder can both be given flat metrics, but differ in theirtopology. Other topologies are also possible for curved space(see also:Shape of the universe).
The mathematical notion ofcurvature is also defined in much more general contexts.[11] Many of these generalizations emphasize different aspects of the curvature as it is understood in lower dimensions.
One such generalization is kinematic. The curvature of a curve can naturally be considered as a kinematic quantity, representing the force felt by a certain observer moving along the curve; analogously, curvature in higher dimensions can be regarded as a kind oftidal force (this is one way of thinking of thesectional curvature). This generalization of curvature depends on how nearby test particles diverge or converge when they are allowed to move freely in the space; seeJacobi field.
Another broad generalization of curvature comes from the study ofparallel transport on a surface. For instance, if a vector is moved around a loop on the surface of a sphere keeping parallel throughout the motion, then the final position of the vector may not be the same as the initial position of the vector. This phenomenon is known asholonomy.[12] Various generalizations capture in an abstract form this idea of curvature as a measure of holonomy; seecurvature form. A closely related notion of curvature comes fromgauge theory in physics, where the curvature represents a field and avector potential for the field is a quantity that is in general path-dependent: it may change if an observer moves around a loop.
Two more generalizations of curvature are thescalar curvature andRicci curvature. In a curved surface such as the sphere, the area of a disc on the surface differs from the area of a disc of the same radius in flat space. This difference (in a suitable limit) is measured by the scalar curvature. The difference in area of a sector of the disc is measured by the Ricci curvature. Each of the scalar curvature and Ricci curvature are defined in analogous ways in three and higher dimensions. They are particularly important in relativity theory, where they both appear on the side ofEinstein's field equations that represents the geometry of spacetime (the other side of which represents the presence of matter and energy). These generalizations of curvature underlie, for instance, the notion that curvature can be a property of ameasure; seecurvature of a measure.
Another generalization of curvature relies on the ability tocompare a curved space with another space that hasconstant curvature. Often this is done with triangles in the spaces. The notion of a triangle makes senses inmetric spaces, and this gives rise toCAT(k) spaces.
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