In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or surface is contained in a larger space, curvature can be defined extrinsically relative to the ambient space. Curvature of Riemannian manifolds of dimension at least two can be defined intrinsically without reference to a larger space.

For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle — that is, the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number.

For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean 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 of maximal curvature, minimal curvature, and mean curvature.

History

This section needs expansion. You can help by adding to it. (October 2019)

In Tractatus de configurationibus qualitatum et motuum,^[1] the 14th-century philosopher and mathematician Nicole 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 a differentiable curve was originally defined through osculating circles. In this setting, Augustin-Louis Cauchy showed that the center of curvature is the intersection point of two infinitely close normal lines to the curve.^[3]

Plane curves

Intuitively, the curvature describes for any part of a curve how much the curve direction changes over a small distance travelled (e.g. angle in rad/m), so it is a measure of the instantaneous rate of change of direction 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 point p rotates^[4] when point p moves at unit speed along the curve. In fact, it can be proved that this instantaneous rate of change is exactly the curvature. More precisely, suppose that the point is moving on the curve at a constant speed of one unit, that is, the position of the point P(s) is a function of the parameter s, which may be thought as the time or as the arc length from a given origin. Let T(s) be a unit tangent vector of the curve at P(s), which is also the derivative of P(s) with respect to s. Then, the derivative of T(s) with respect to s 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 is continuously differentiable near P, for having a tangent that varies continuously; it requires also that the curve is twice differentiable at P, for insuring the existence of the involved limits, and of the derivative of T(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 in kinematics, this characterization is often given as a definition of the curvature.

Osculating circle

Historically, the curvature of a differentiable curve was defined through the osculating circle, which is the circle that best approximates the curve at a point. More precisely, given a point P on a curve, every other point Q of the curve defines a circle (or sometimes a line) passing through Q and tangent to the curve at P. The osculating circle is the limit, if it exists, of this circle when Q tends to P. Then the center and the radius of curvature of the curve at P are the center and the radius of the osculating circle. The curvature is the reciprocal of radius of curvature. That is, the curvature is

${\displaystyle \kappa ={\frac {1}{R}},}$

where R is the radius of curvature^[5] (the whole circle has this curvature, it can be read as turn 2π over the length 2πR).

This definition is difficult to manipulate and to express in formulas. Therefore, other equivalent definitions have been introduced.

In terms of arc-length parametrization

Every differentiable curve can be parametrized with respect to arc length.^[6] In the case of a plane curve, this means the existence of a parametrization γ(s) = (x(s), y(s)), where x and y are real-valued differentiable functions whose derivatives satisfy

${\displaystyle \|{\boldsymbol {\gamma }}'\|={\sqrt {x'(s)^{2}+y'(s)^{2}}}=1.}$

This means that the tangent vector

${\displaystyle \mathbf {T} (s)={\bigl (}x'(s),y'(s){\bigr )}}$

has a length equal to one and is thus a unit tangent vector.

If the curve is twice differentiable, that is, if the second derivatives of x and y exist, then the derivative of T(s) exists. This vector is normal to the curve, its length is the curvature κ(s), and it is oriented toward the center of curvature. That is,

${\displaystyle {\begin{aligned}\mathbf {T} (s)&={\boldsymbol {\gamma }}'(s),\\\|\mathbf {T} (s)\|^{2}&=1\ {\text{(constant)}}\implies \mathbf {T} '(s)\cdot \mathbf {T} (s)=0,\\\kappa (s)&=\|\mathbf {T} '(s)\|=\|{\boldsymbol {\gamma }}''(s)\|={\sqrt {x''(s)^{2}+y''(s)^{2}}}\end{aligned}}}$

Moreover, because the radius of curvature is (assuming 𝜿(s) ≠ 0)

${\displaystyle R(s)={\frac {1}{\kappa (s)}},}$

and the center of curvature is on the normal to the curve, the center of curvature is the point

${\displaystyle \mathbf {C} (s)={\boldsymbol {\gamma }}(s)+{\frac {1}{\kappa (s)^{2}}}\mathbf {T} '(s).}$

(In case the curvature is zero, the center of curvature is not located anywhere on the plane R² and is often said to be located "at infinity".)

If N(s) is the unit normal vector obtained from T(s) by a counterclockwise rotation of π/2, then

${\displaystyle \mathbf {T} '(s)=k(s)\mathbf {N} (s),}$

with k(s) = ± κ(s). The real number k(s) is called the oriented curvature or signed 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, the change of variable s → –s provides another arc-length parametrization, and changes the sign of k(s).

In terms of a general parametrization

Let γ(t) = (x(t), y(t)) be a proper parametric representation of a twice differentiable plane curve. Here proper means that on the domain of definition of the parametrization, the derivative dγ/dt is defined, differentiable and nowhere equal to the zero vector.

With such a parametrization, the signed curvature is

${\displaystyle k={\frac {x'y''-y'x''}{{\bigl (}{x'}^{2}+{y'}^{2}{\bigr )}{\vphantom {'}}^{3/2}}},}$

where primes refer to derivatives with respect to t. The curvature κ is thus

${\displaystyle \mathrm{\kappa <!--\; \kappa \; -->}=\frac{\left|x^{\prime }{y}^{″}-<!--\; -\; -->y^{\prime }{x}^{″}\right|}{({x^{\prime }}^2+{y^{\prime }}^2}}$Zdroj:https://en.wikipedia.org?pojem=Curvature
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