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Curvature

From Wikipedia, the free encyclopedia
This article is about mathematics and related concepts in geometry. For other uses, seeCurvature (disambiguation).
Mathematical measure of how much a curve or surface deviates from flatness

A migrating wild-typeDictyostelium discoideum cell whose boundary is colored by curvature. Scale bar: 5 μm.

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, curvature describes how sharply the curve bends. The canonical examples arecircles: smaller circles bend more sharply and hence have higher curvature. For a point on a general curve, the direction of the curve is described by its tangent line. How sharply the curve is bending at that point can be measured by how much that tangent line changes direction per unit distance along the curve.

Curvature measures the angular rate of change of the direction of the tangent line, or the unit tangent vector, of the curve per unit distance along the curve. Curvature is expressed in units ofradians per unit distance. For a circle, that rate of change is the same at all points on the circle and is equal to thereciprocal of the circle'sradius. Straight lines don't change direction and have zero curvature. The curvature at a point on a twicedifferentiable curve is the magnitude of itscurvature vector at that point and is also the curvature of itsosculating circle, which is the circle that best approximates the curve near that point.

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.

History

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The history of curvature began with theancient Greeks' basic distinction between straight and circular lines, with the concept later developed by figures likeAristotle andApollonius. The development of calculus in the 17th century, particularly by Newton and Leibniz, provided tools to systematically calculate curvature for curves. Euler then extended the study to surfaces, followed by Gauss's crucial insight of "intrinsic" curvature, which is independent of how a surface is embedded in space, and Riemann's generalization to higher dimensions.[1]

InTractatus de configurationibus qualitatum et motuum,[2] 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.[3]

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.[4]

Curves

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Intuitively, curvature describes for any part of a curve how much the curve direction changes over a small distance along the curve. The direction of the curve at any pointP is described by a unit tangent vector,T. A section of a curve is also called an arc, and length along the curve isarc length,s. So the curvature for a small section of the curve is the angle of the change of the direction of the tangent vector divided by the arc lengthΔs. For a general curve which might have a varying curvature along its length, the curvature at a pointP on the curve is the limit of the curvature of sections containingP as the length of the sections approaches zero. For a twice differentiable curve, that limit is the magnitude of thederivative of the unit tangent vector with respect to arc length. Using the lowercase Greek letterkappa to denote curvature:

κ=‖dTds‖.{\displaystyle \kappa =\left\|{\frac {\mathrm {d} {\boldsymbol {T}}}{\mathrm {d} s}}\right\|.}

Curvature is a differential-geometric property of the curve; it does not depend on the parametrization of the curve. In particular, it does not depend on the orientation of the parametrized curve, i.e. which direction along the curve is associated with increasing parameter values.

Arc-length parametrization

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See also:Differentiable curve § Length and natural parametrization

A curve that is parametrized by arc length is avector-valued function that is denoted by the Greek lettergamma with an overbar,–γ, that describes the position of a point on the curve,P, in terms of its arc-length distance,s along the curve from some other reference point on the curve. Thus for some intervalI = [a,b] inR{\displaystyle \mathbb {R} },–γ:I →R{\displaystyle \mathbb {R} }n with

P(s)=γ¯(s).{\displaystyle {\boldsymbol {P}}(s)={\boldsymbol {\bar {\gamma }}}(s).}

If–γ is a differentiable curve, then the first derivative of–γ,–γ′(s) is a unit tangent vector,T(s), and

‖γ¯′(s)‖=1{\displaystyle \left\|{\boldsymbol {\bar {\gamma }}}'(s)\right\|=1}

T(s)=γ¯′(s).{\displaystyle {\boldsymbol {T}}(s)={\boldsymbol {\bar {\gamma }}}'(s).}

If–γ is twice differentiable, the second derivative of–γ isT′(s), which is also the curvature vector,K(s).

K(s)=T′(s)=γ¯″(s){\displaystyle {\boldsymbol {K}}(s)={\boldsymbol {T}}'(s)={\boldsymbol {\bar {\gamma }}}''(s)}

Curvature is the magnitude of the second derivative of–γ .κ(s)=‖K(s)‖=‖T′(s)‖=‖γ¯″(s)‖{\displaystyle \kappa (s)=\left\|{\boldsymbol {K}}(s)\right\|=\left\|{\boldsymbol {T}}'(s)\right\|=\left\|{\boldsymbol {\bar {\gamma }}}''(s)\right\|}

The parameters can also be interpreted as a time parameter. Then–γ(s) describes the path of a particle that moves along the curve at a constant unit speed. Curvature can then be understood as a measure of how fast the direction of the particle rotates.[5]

General parametrization

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See also:Differentiable curve § Length and natural parametrization

A twice differentiable curve,γ: [a,b] →R{\displaystyle \mathbb {R} }n, that is not parametrized by arc length can be re-parametrized by arc length provided thatγ′(t) is everywhere not zero, so that1/‖γ′(t)‖ is always a finite positive number.

The arc-length parameter,s, is defined by

s(t) = ∫at‖γ′(x)‖dx,{\displaystyle s(t)~{=}~\int _{a}^{t}\left\|{\boldsymbol {\gamma }}'(x)\right\|\,\mathrm {d} {x},}

which has an inverse functiont(s).The arc-length parametrization is the function–γ which is defined as

γ¯(s) = γ(t(s)).{\displaystyle {\boldsymbol {\bar {\gamma }}}(s)~{=}~{\boldsymbol {\gamma }}(t(s)).}

Bothγ and–γ trace the same path inRn{\displaystyle \mathbb {R} ^{n}} and so have the same curvature vector and curvature at each pointP on the curve. For a givens and its correspondingt =t(s), pointP and its unit tangent vector,T, curvature vector,K, and curvature,κ, are:

P=γ¯(s)=γ(t){\displaystyle {\boldsymbol {P}}={\boldsymbol {\bar {\gamma }}}(s)={\boldsymbol {\gamma }}(t)}

T=γ¯′(s)=γ′(t)‖γ′(t)‖{\displaystyle {\boldsymbol {T}}={\boldsymbol {\bar {\gamma }}}'(s)={\frac {{\boldsymbol {\gamma }}'(t)}{\left\|{\boldsymbol {\gamma }}'(t)\right\|}}}

K=γ¯″(s)=γ″(t)‖γ′(t)‖2−T(T⋅γ″(t)‖γ′(t)‖2){\displaystyle {\boldsymbol {K}}={\boldsymbol {\bar {\gamma }}}''(s)={\frac {{\boldsymbol {\gamma }}''(t)}{\left\|{\boldsymbol {\gamma }}'(t)\right\|^{2}}}-{\boldsymbol {T}}\left({\boldsymbol {T}}\cdot {\frac {{\boldsymbol {\gamma }}''(t)}{\left\|{\boldsymbol {\gamma }}'(t)\right\|^{2}}}\right)}

κ=‖γ¯″(s)‖=‖γ′(t)‖′2‖γ″(t)‖′2−(γ′(t)⋅γ″(t))′2‖γ′(t)‖′3=‖γ″(t)‖‖γ′(t)‖21−(T⋅γ″(t)‖γ″(t)‖)2   .{\displaystyle {\begin{aligned}\kappa =\left\|{\boldsymbol {\bar {\gamma }}}''(s)\right\|&={\frac {\sqrt {{\bigl \|}{\boldsymbol {\gamma }}'(t){\bigr \|}{\vphantom {'}}^{2}{\bigl \|}{\boldsymbol {\gamma }}''(t){\bigr \|}{\vphantom {'}}^{2}-{\bigl (}{\boldsymbol {\gamma }}'(t)\cdot {\boldsymbol {\gamma }}''(t){\bigr )}{\vphantom {'}}^{2}}}{{\bigl \|}{\boldsymbol {\gamma }}'(t){\bigr \|}{\vphantom {'}}^{3}}}\\\\&={\frac {\left\|{\boldsymbol {\gamma }}''(t)\right\|}{\left\|{\boldsymbol {\gamma }}'(t)\right\|^{2}}}{\sqrt {1-\left({\boldsymbol {T}}\cdot {\frac {{\boldsymbol {\gamma }}''(t)}{{\bigl \|}{\boldsymbol {\gamma }}''(t){\bigr \|}}}\right)^{2}~}}~~.\\\end{aligned}}}

The curvature vector,K, is the perpendicular component ofγ′′(t) /‖γ′(t)‖2relative to the tangent vectorγ′(t).This is also reflected in the second expression for the curvature: the expression inside the parentheses iscosθ, whereθ is the angle between the vectorsT andγ′′(t), so that the square root producessinθ.

Ifγ is twice continuously differentiable, then so iss(t) and–γ, whileT(t) is continuously differentiable, andK(t) andκ(t) are continuous.

Often it is difficult or impossible to express the arc-length parametrization,–γ, in closed form even whenγ is given in closed form. This is typically the case when it is difficult or impossible to expresss(t) or its inverset(s) in closed form. However curvature can be expressed only in terms of the first and second derivatives ofγ, without direct reference to–γ.

Curvature vector

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The curvature vector, denoted with an upper-caseK, is the derivative of the unit tangent vector,T, with respect to arc length,s:

K=dTds.{\displaystyle {\boldsymbol {K}}={\frac {\mathrm {d} {\boldsymbol {T}}}{\mathrm {d} s}}.}

The curvature vector represents both the direction towards which the curve is turning as well as how sharply it turns.

The curvature vector has the following properties:

  • If the curvature vector is not zero:
    • The curvature vector points from the point on the curve,P, in the direction of the center of the osculating circle.
    • The curvature vector and the tangent vector are perpendicular vectors that span the osculating plane, the plane containing the osculating circle.
    • The curvature vector scaled to unit length is the unit normal vector,N:N=K‖K‖.{\displaystyle {\boldsymbol {N}}={\frac {K}{\left\|K\right\|}}.}

Osculating circle

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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 of curvature and theradius of curvature of the curve atP are the center and the radius of the osculating circle.

The radius of curvature,R, is thereciprocal of the curvature[6], provided that the curvature is not zero:R=1κ .{\displaystyle R={\frac {1}{\kappa }}~.}

For a curveγ, since a non-zero curvature vector,K(t), points from the pointP =γ(t) towards the center of curvature, but the magnitude ofK(t) is the curvature,κ(t), the center of curvature,C(t) is

C(t)=γ(t)+K(t)κ(t)2 .{\displaystyle {\boldsymbol {C}}(t)={\boldsymbol {\gamma }}(t)+{\frac {{\boldsymbol {K}}(t)}{\kappa (t)^{2}}}~.}

When the curvature is zero, for example on a straight line or at a point of inflection, the radius of curvature is infinite and the center of curvature is indeterminate or "at infinity".

Curvature from arc and chord length

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Given two pointsP andQ on a curveγ, 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 ofγ atP is given by the limit[citation needed]

κ(P)=limQ→P24(s(P,Q)−d(P,Q))s(P,Q)Q3 ,{\displaystyle \kappa ({\boldsymbol {P}})=\lim _{{\boldsymbol {Q}}\to {\boldsymbol {P}}}{\sqrt {\frac {24{\bigl (}s({\boldsymbol {P}},{\boldsymbol {Q}})-d({\boldsymbol {P}},{\boldsymbol {Q}}){\bigr )}}{s({\boldsymbol {P}},{\boldsymbol {Q}}){\vphantom {Q}}^{3}}}}~,}

where the limit is taken as the pointQ approachesP onγ. The denominator can equally well be taken to bed(P,Q)3. The formula is valid in any dimension. The formula follows by verifying it for the osculating circle.

Exceptional cases

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There may be some situations where the preconditions for the above formulas do not apply, but where it is still appropriate to apply the concept of curvature.

It can be useful to apply the concept of curvature to a curveγ at a pointP =γ(t0) if theone-sided derivatives forγ′(t0) exist but are different values, or likewise forγ′′(t0). In such a case, it could be useful to describe the curve with curvature at each side. Such might be the case of a curve that is constructed piecewise.

Another situation occurs when the limit of a ratio results in an indeterminate 0 / 0 value for the curvature, for example when both derivatives exist but are both zero. In such a case, it might be possible to evaluate the underlying limit usingl'Hôpital's rule.

Examples

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The following are examples of curves with application of the relevant concepts and formulas.

Circle

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For an arc on a circle of radius R, the ratio of the size of the angle between the arc endpoint tangents, measured in radians, divided by the arc length L is (L/R)/L = 1/R.

A geometric explanation for why the curvature of a circle of radiusR at any pointP is1/R is partially illustrated by the diagram to the right.

The length of the red arc isL and the measure in radians of the arc's central angle, angle ACB, isL/R. The angle between the arc endpoint tangents is angle BDE, which is the same size as the central angle, because both angles are supplementary to angle BDE.

The ratio of the angle between the arc endpoint tangents, measured in radians, divided by the arc lengthL is(L/R)/L = 1/R.

Since the ratio is1/R for any arc of the circle that is less than a half circle, for arcs containing any given pointP on the circle, the limit of the ratio as arc length approaches zero is also1/R. Hence the curvature of the circle at any pointP is1/R.


A common parametrization of acircle of radiusr isγ(t) = (r cost,r sint). Thenγ′(t)=(−r sin t,r cos t)‖γ′(t)‖=rγ″(t)=(−r cos t,−r sin t)‖γ″(t)‖=rγ′(t)⋅γ″(t)=0 .{\displaystyle {\begin{array}{lll}{\boldsymbol {\gamma }}'(t)=(-r~\mathrm {sin} ~t,r~\mathrm {cos} ~t)&\qquad &\left\|{\boldsymbol {\gamma }}'(t)\right\|=r\\{\boldsymbol {\gamma }}''(t)=(-r~\mathrm {cos} ~t,-r~\mathrm {sin} ~t)&\qquad &\left\|{\boldsymbol {\gamma }}''(t)\right\|=r\\{\boldsymbol {\gamma }}'(t)\cdot {\boldsymbol {\gamma }}''(t)=0~.\\\end{array}}}The general formula for curvature givesκ(t)=r2r2−02r3=1r .{\displaystyle \kappa (t)={\frac {\sqrt {r^{2}\,r^{2}-0^{2}\,}}{r^{3}}}={\frac {1}{r}}~.}and the formula for a plane curve givesκ(t)=r2sin2⁡t+r2cos2⁡t(r2cos2⁡t+r2sin2⁡t)′3/2=1r.{\displaystyle \kappa (t)={\frac {r^{2}\sin ^{2}t+r^{2}\cos ^{2}t}{{\bigl (}r^{2}\cos ^{2}t+r^{2}\sin ^{2}t{\bigr )}{\vphantom {'}}^{3/2}}}={\frac {1}{r}}.}

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 isγ¯(s)=(rcos⁡sr,rsin⁡sr).{\displaystyle {\boldsymbol {\bar {\gamma }}}(s)=\left(r\cos {\frac {s}{r}},\,r\sin {\frac {s}{r}}\right).}It is an arc-length parametrization, since the norm ofγ¯′(s)=(−sin⁡sr,cos⁡sr){\displaystyle {\boldsymbol {\bar {\gamma }}}'(s)=\left(-\sin {\frac {s}{r}},\,\cos {\frac {s}{r}}\right)}is equal to one. Thenκ(s)=‖γ¯″(s)‖=‖(−1rcos⁡sr,−1rsin⁡sr)‖=1r{\displaystyle \kappa (s)=\left\|{\boldsymbol {\bar {\gamma }}}''(s)\right\|=\left\|\left(-{\frac {1}{r}}\cos {\frac {s}{r}},\,-{\frac {1}{r}}\sin {\frac {s}{r}}\right)\right\|={\frac {1}{r}}}gives the same value for the curvature.

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 givesκ=|Fy2Fxx−2FxFyFxy+Fx2Fyy|(Fx2+Fy2)′3/2=8y2+8x2(4x2+4y2)′3/2=8r2(4r2)′3/2=1r.{\displaystyle {\begin{aligned}\kappa &={\frac {\left|F_{y}^{2}F_{xx}-2F_{x}F_{y}F_{xy}+F_{x}^{2}F_{yy}\right|}{{\bigl (}F_{x}^{2}+F_{y}^{2}{\bigr )}{\vphantom {'}}^{3/2}}}\\&={\frac {8y^{2}+8x^{2}}{{\bigl (}4x^{2}+4y^{2}{\bigr )}{\vphantom {'}}^{3/2}}}\\&={\frac {8r^{2}}{{\bigl (}4r^{2}{\bigr )}{\vphantom {'}}^{3/2}}}={\frac {1}{r}}.\end{aligned}}}

Parabola

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Parabola y = x2, its curvature, its comb(see§ Curvature comb below) with scaling factor = 1, and the osculating circle at x = 0.

Consider theparabolay =ax2 +bx +c.

It is the graph of a function, with derivative2ax +b, and second derivative2a. So, the signed curvature isk(x)=2a(1+(2ax+b)2))3/2 .{\displaystyle k(x)={\frac {2a}{{\bigl (}1+\left(2ax+b\right)^{2}{\bigr )}{\vphantom {)}}^{3/2}}}~.}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 and with primes referring to 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 not defined for an implicit equation since the signed curvature depends on an orientation of the curve that is not provided by the implicit equation.

Plane curves

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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/dt⁠exists and is nowhere equal to the zero vector.

The curvatureκ of a plane curve can be expressed in ways that are specific to two dimensions, such as

κ=|x′y″−y′x″|(x′2+y′2)′3/2,{\displaystyle \kappa ={\frac {\left|x'y''-y'x''\right|}{{\bigl (}{x'}^{2}+{y'}^{2}{\bigr )}{\vphantom {'}}^{3/2}}},}

where primes refer to derivatives with respect tot.

This can be expressed in acoordinate-free way asκ=|det(γ′,γ″)|‖γ′‖3,{\displaystyle \kappa ={\frac {\left|\det \left({\boldsymbol {\gamma }}',{\boldsymbol {\gamma }}''\right)\right|}{\|{\boldsymbol {\gamma }}'\|^{3}}},}

where the numerator is the absolute value of the determinant of the 2-by-2 matrix withγ′ andγ′′ as the columns.

These formulas can be understood as an application of thecross product formula forcurvature in three dimensions. Since the operands have zeros in the third dimension, the cross product result will have zero values for the first two dimensions, so only the value in the third dimension is relevant to calculating the magnitude of the cross product. The formula for the value of the third dimension thus appears in the numerator of the above formulas.

Signed curvature

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For plane curves, it can be useful to express the curvature as a single scalar that can be positive or negative, called thesigned curvature ororiented curvature and denoted with a lowercase k. The signed curvature formulas are similar to those forκ except that they omit taking the absolute value of the numerator:

k=x′y″−y′x″(x′2+y′2)′3/2=det(γ′,γ″)‖γ′‖3 .{\displaystyle k={\frac {x'y''-y'x''}{{\bigl (}{x'}^{2}+{y'}^{2}{\bigr )}{\vphantom {'}}^{3/2}}}={\frac {\det \left({\boldsymbol {\gamma }}',{\boldsymbol {\gamma }}''\right)}{\|{\boldsymbol {\gamma }}'\|^{3}}}~.}

Thenk = ± κ. Whetherk is positive or negative depends on the orientation of the curve. Whether a positivek corresponds to clockwise or counterclockwise turning depends on the orientation of the curve and the orientation of the coordinate axes. With a standard orientation of the coordinate axes, when moving along the curve in the direction of increasingt,k is positive if the curve turns to the left, counterclockwise, and it is negative if the curve turns to the right, clockwise. This is consistent with the convention of treating counterclockwise rotations as rotations through a positive angle. However, since the sign ofk is dependent on the orientation of the parametrization,k is not differential-geometric property property of the curve.

Except for orientation issues, the signed curvature for a plane curve captures similar information as the curvature vector, which for a plane curve is constrained to just one dimension, the line that is perpendicular to the unit tangent vector.

Using a standard orientation of the coordinate axes, let—N be theunit normal vector obtained from the unit tangent vector,T, by a counterclockwise rotation of⁠π/2⁠.Then—N is dependent on the orientation of the curve and points to the left when moving along the curve in the direction of increasingt. However the curvature vector,K is equal to the product of the signed curvature,k and—N, because their orientation dependencies cancel:

K=kN¯.{\displaystyle {\boldsymbol {K}}=k\,{\boldsymbol {\bar {N}}}.}

Similarly, the center of curvature can be expressed using the signed curvature and—N:C(s)=γ(s)+N¯(s)k(s).{\displaystyle {\boldsymbol {C}}(s)={\boldsymbol {\gamma }}(s)+{\frac {\mathbf {\bar {N}} (s)}{k(s)}}.}

Graph of a function

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Thegraph of a functiony =f(x), is a special case of a parametrized curve, of the formx=ty=f(t).{\displaystyle {\begin{aligned}x&=t\\y&=f(t).\end{aligned}}}As the first and second derivatives ofx are 1 and 0, previous formulas simplify toκ=|y″|(1+y′2)′3/2{\displaystyle \kappa ={\frac {\left|y''\right|}{{\bigl (}1+{y'}^{2}{\bigr )}{\vphantom {'}}^{3/2}}}}for the curvature and tok=y″(1+y′2)′3/2{\displaystyle k={\frac {y''}{{\bigl (}1+{y'}^{2}{\bigr )}{\vphantom {'}}^{3/2}}}}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 gives additional significance to 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 hask(x)=y″(1+O(y′2)).{\displaystyle k(x)=y''{\Bigl (}1+O{\bigl (}{\textstyle y'}^{2}{\bigr )}{\Bigr )}.}

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.

Implicit curve

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For a curve defined by animplicit equationF(x,y) = 0 withpartial derivatives denotedFx ,Fy ,Fxx ,Fxy ,Fyy ,the curvature is given by[7]κ=|Fy2Fxx−2FxFyFxy+Fx2Fyy|(Fx2+Fy2)′3/2.{\displaystyle \kappa ={\frac {\left|F_{y}^{2}F_{xx}-2F_{x}F_{y}F_{xy}+F_{x}^{2}F_{yy}\right|}{{\bigl (}F_{x}^{2}+F_{y}^{2}{\bigr )}{\vphantom {'}}^{3/2}}}.}

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 hasdydx=−FxFy.{\displaystyle {\frac {dy}{dx}}=-{\frac {F_{x}}{F_{y}}}.}

Polar coordinates

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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 isκ(θ)=|r2+2r′2−rr″|(r2+r′2)′3/2,{\displaystyle \kappa (\theta )={\frac {\left|r^{2}+2{r'}^{2}-r\,r''\right|}{{\bigl (}r^{2}+{r'}^{2}{\bigr )}{\vphantom {'}}^{3/2}}},}where the prime refers to differentiation with respect toθ.

This results from the formula for general parametrizations, by considering the parametrizationx=rcos⁡θy=rsin⁡θ.{\displaystyle {\begin{aligned}x&=r\cos \theta \\y&=r\sin \theta .\end{aligned}}}

Curvature comb

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Curvature comb
Curvature comb

Acurvature comb[8] can be used to represent graphically the curvature of every point on a curve. Ifγ is a curve parametrizedt, its comb is defined as the parametrized curve defined byComb(t)=γ(t)−sκ(t)N(t),{\displaystyle \mathrm {Comb} (t)={\boldsymbol {\gamma }}(t)-s\,\kappa (t){\boldsymbol {N}}(t),}whereκ is the curvature,N is the unit normal vector that points toward the center of curvature, ands is a scaling factor that is chosen 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.

Frenet–Serret formulas for plane curves

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The vectorsT andN at two points on a plane curve, a translated version of the second frame (dotted), andδT the change inT. Hereδs is the distance between the points. In the limit⁠dT/ds⁠ will be in the directionN. The curvature describes the rate of rotation of the frame.

Thefirst Frenet–Serret formula relates the unit tangent vector, curvature, and the normal vector of anarc-length parametrizationT′(s)=κ(s)N(s),{\displaystyle \mathbf {T} '(s)=\kappa (s)\mathbf {N} (s),}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 relationdNds=−κT,=−κdγds.{\displaystyle {\begin{aligned}{\frac {d\mathbf {N} }{ds}}&=-\kappa \mathbf {T} ,\\&=-\kappa {\frac {d{\boldsymbol {\gamma }}}{ds}}.\end{aligned}}}

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 parametrizationN′(t)=−κ(t)γ′(t).{\displaystyle \mathbf {N} '(t)=-\kappa (t){\boldsymbol {\gamma }}'(t).}

Curves in three dimensions

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See also:Frenet–Serret formulas
See also:Differentiable curve § Special Frenet vectors and generalized curvatures
See also:Differential geometry of curves
Animation of the curvature and the acceleration vector'T'′(s)

For a parametrically defined curve in three dimensions given in Cartesian coordinates byγ(t) = (x(t),y(t),z(t)), the curvature is

κ=(z″y′−y″z′)′2+(x″z′−z″x′)′2+(y″x′−x″y′)′2(x′2+y′2+z′2)′3/2,{\displaystyle \kappa ={\frac {\sqrt {{\bigl (}z''y'-y''z'{\bigr )}{\vphantom {'}}^{2}+{\bigl (}x''z'-z''x'{\bigr )}{\vphantom {'}}^{2}+{\bigl (}y''x'-x''y'{\bigr )}{\vphantom {'}}^{2}}}{{\bigl (}{x'}^{2}+{y'}^{2}+{z'}^{2}{\bigr )}{\vphantom {'}}^{3/2}}},}

where the prime denotes differentiation with respect to the parametert. Both the curvature[9] and the curvature vector can be expressed using thevector cross product and the unit tangent vectorT:

K=T×γ″‖γ′‖2×T{\displaystyle {\boldsymbol {K}}={\boldsymbol {T}}\times {\frac {{\boldsymbol {\gamma }}''}{\left\|{\boldsymbol {\gamma }}'\right\|^{2}}}\times {\boldsymbol {T}}}

κ=‖γ′×γ″‖‖γ′‖′3 .{\displaystyle \kappa ={\frac {{\bigl \|}{\boldsymbol {\gamma }}'\times {\boldsymbol {\gamma }}''{\bigr \|}}{{\bigl \|}{\boldsymbol {\gamma }}'{\bigr \|}{\vphantom {'}}^{3}}}~.}

These formulas are related to the general formulas for curvature and the curvature vector, except that they use the vector cross product instead of the scalar dot product to express the perpendicular component ofγ′′ / ‖γ′‖^2 relative toγ′.

Surfaces

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For broader coverage of this topic, seeDifferential geometry of surfaces.

The curvature of curves drawn on asurface is the main tool for the defining and studying the curvature of the surface.

Curves on surfaces

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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:

(T′t′u′)=(0κgκn−κg0τr−κn−τr0)(Ttu){\displaystyle {\begin{pmatrix}\mathbf {T} '\\\mathbf {t} '\\\mathbf {u} '\end{pmatrix}}={\begin{pmatrix}0&\kappa _{\mathrm {g} }&\kappa _{\mathrm {n} }\\-\kappa _{\mathrm {g} }&0&\tau _{\mathrm {r} }\\-\kappa _{\mathrm {n} }&-\tau _{\mathrm {r} }&0\end{pmatrix}}{\begin{pmatrix}\mathbf {T} \\\mathbf {t} \\\mathbf {u} \end{pmatrix}}}

Principal curvature

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Saddle surface with normal planes in directions of principal curvatures
Main article:Principal curvature

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.

Normal sections

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Curvature can be evaluated along surfacenormal sections, similar to§ Curves on surfaces above (see for example theEarth radius of curvature).

Developable surfaces

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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]

Gaussian curvature

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Main article:Gaussian curvature

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

K=limr→0+3(2πr−C(r)πr3).{\displaystyle K=\lim _{r\to 0^{+}}3\left({\frac {2\pi r-C(r)}{\pi r^{3}}}\right).}

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.

Mean curvature

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Main article:Mean curvature

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.

Second fundamental form

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Main article:Second fundamental form

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,

II⁡(X,X)=N⋅(∇XX){\displaystyle \operatorname {I\!I} (\mathbf {X} ,\mathbf {X} )=\mathbf {N} \cdot (\nabla _{\mathbf {X} }\mathbf {X} )}

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

II⁡(X,X)=k1(X⋅u1)2+k2(X⋅u2)2.{\displaystyle \operatorname {I\!I} (\mathbf {X} ,\mathbf {X} )=k_{1}\left(\mathbf {X} \cdot \mathbf {u} _{1}\right)^{2}+k_{2}\left(\mathbf {X} \cdot \mathbf {u} _{2}\right)^{2}.}

Thus the second fundamental form encodes both the intrinsic and extrinsic curvatures.

Shape operator

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Further information:Shape operator

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

∂aN=−SbaXb.{\displaystyle \partial _{a}\mathbf {N} =-S_{ba}\mathbf {X} _{b}.}

(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

S=(EG−F2)−1(eG−fFfG−gFfE−eFgE−fF).{\displaystyle S=\left(EG-F^{2}\right)^{-1}{\begin{pmatrix}eG-fF&fG-gF\\fE-eF&gE-fF\end{pmatrix}}.}

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.

Curvature of space

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Further information:Curvature of Riemannian manifolds andCurved space
"Curvature of space" redirects here; not to be confused withCurvature of space-time.

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).

Generalizations

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Moving a vector along a curve from A → N → B → A produces another vector. The inability to return to the initial vector is measured by the holonomy of the surface. In a space with no curvature, the angle α is 0 degrees, and in a space with curvature, the angle α is greater than 0 degrees. The more space is curved, the greater the magnitude of the angle α.

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.

See also

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Notes

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  1. ^Dan Margalit,"The History of Curvature",University of Georgia:Villanova University: robert.jantzen, retrieved2025-10-14
  2. ^Clagett, Marshall (1968),Nicole Oresme and the Medieval Geometry of Qualities and Motions, Madison, WI: University of Wisconsin Press,ISBN 978-0-299-04880-8
  3. ^Serrano, Isabel M.;Suceavă, Bogdan D. (2015), "A Medieval Mystery: Nicole Oresme's Concept of Curvitas",Notices of the AMS,62 (9):1030–1034,doi:10.1090/noti1275
  4. ^Borovik, Alexandre;Katz, Mikhail G. (2011), "Who Gave You the Cauchy–Weierstrass Tale? The Dual History of Rigorous Calculus",Foundations of Science,17 (3):245–276,arXiv:1108.2885,doi:10.1007/s10699-011-9235-x
  5. ^Pressley, Andrew (2001),Elementary Differential Geometry, London: Springer, p. 29,ISBN 978-1-85233-152-8
  6. ^Kline 1998, p. 458
  7. ^Goldman, Ron (2005), "Curvature formulas for implicit curves and surfaces",Computer Aided Geometric Design,22 (7):632–658,CiteSeerX 10.1.1.413.3008,doi:10.1016/j.cagd.2005.06.005
  8. ^Farin, Gerald (Nov 2016), "Curvature combs and curvature plots",Computer-Aided Design,80:6–8,doi:10.1016/j.cad.2016.08.003
  9. ^A proof of this can be found atthe article on curvature atWolfram MathWorld.
  10. ^developable surface, Mathworld. (Retrieved 11 February 2021)
  11. ^Kobayashi, Shōshichi;Nomizu, Katsumi (1963), "2–3",Foundations of Differential Geometry, New York: Interscience,ISBN 978-0-470-49647-3{{citation}}:ISBN / Date incompatibility (help)
  12. ^Henderson, David W.;Taimin̦a, Daina (2005),Experiencing Geometry: Euclidean and Non-Euclidean with History (3rd ed.), Upper Saddle River, NJ: Pearson Prentice Hall, pp. 98–99,doi:10.3792/euclid/9781429799850,ISBN 978-0-13-143748-7

References

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External links

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