Curvature

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, curvature describes how sharply the curve bends. The canonical examples are circles: 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 of radians per unit distance. For a circle, that rate of change is the same at all points on the circle and is equal to the reciprocal of the circle's radius. Straight lines don't change direction and have zero curvature. The curvature at a point on a twice differentiable curve is the magnitude of its curvature vector at that point and is also the curvature of its osculating circle, which is the circle that best approximates the curve near that point.

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.

The history of curvature began with the ancient Greeks' basic distinction between straight and circular lines, with the concept later developed by figures like Aristotle and Apollonius. 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.

In Tractatus de configurationibus qualitatum et motuum, 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.

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.

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 point P is described by a unit tangent vector, T. A section of a curve is also called an arc, and length along the curve is arc 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 length Δs. For a general curve which might have a varying curvature along its length, the curvature at a point P on the curve is the limit of the curvature of sections containing P as the length of the sections approaches zero. For a twice differentiable curve, that limit is the magnitude of the derivative of the unit tangent vector with respect to arc length. Using the lowercase Greek letter kappa to denote curvature:

\[\kappa = \left \| \frac {\mathrm{d}\boldsymbol{T}} {\mathrm{d}s} \right \| .\]

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.

A curve that is parametrized by arc length is a vector-valued function that is denoted by the Greek letter gamma 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 interval I = [a, b] in \(\mathbb{R}\), –γ: I → \(\mathbb{R}\) with

\[\boldsymbol{P}(s) = \boldsymbol{\bar\gamma}(s).\]

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

\[\left\| \boldsymbol{\bar\gamma}'(s) \right\| = 1\]

\[\boldsymbol{T}(s) = \boldsymbol{\bar\gamma}'(s).\]

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

Condensed: the full section is in Wikipedia.

A twice differentiable curve, γ: [a, b] → \(\mathbb{R}\), that is not parametrized by arc length can be re-parametrized by arc length provided that γ′(t) is everywhere not zero, so that 1/‖γ′(t)‖ is always a finite positive number.

The arc-length parameter, s, is defined by

\[s(t) ~{=}~ \int_a^t \left\| \boldsymbol{\gamma}'(x) \right\| \, \mathrm{d}{x},\]

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

\[\boldsymbol{\bar\gamma}(s) ~{=}~ \boldsymbol{\gamma}(t(s)).\]

Both γ and –γ trace the same path in \(\mathbb{R}^n\) and so have the same curvature vector and curvature at each point P on the curve. For a given s and its corresponding t = t(s), point P and its unit tangent vector, T, curvature vector, K, and curvature, κ, are:

Condensed: the full section is in Wikipedia.

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

\[\boldsymbol{K} = \frac {\mathrm{d}\boldsymbol{T}} {\mathrm{d}s}.\]

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:

• The magnitude of the curvature vector is the curvature: \[\kappa = \left\| \boldsymbol{K} \right\|.\]
• The curvature vector is perpendicular to the unit tangent vector T, or in terms of the dot product: \[\boldsymbol{K} \cdot \boldsymbol{T} = 0~.\]
• The curvature vector is the second derivative of an arc-length parametrization –γ : \[\boldsymbol{K}(s) = \boldsymbol{\bar\gamma}''(s).\]
• The curvature vector of a general parametrization, γ, is the perpendicular component of γ′′(t) / ‖γ′(t)‖ relative to the tangent vector γ′(t): \[\boldsymbol{K}(t) = \frac {\boldsymbol{\gamma}''(t)} {\left\| \boldsymbol{\gamma}'(t) \right\|^2} - \boldsymbol{T} \left ( \boldsymbol{T} \cdot \frac {\boldsymbol{\gamma}''(t)} {\left\| \boldsymbol{\gamma}'(t) \right\|^2} \right).\]If the curve is in \(\mathbb{R}^3\), then the curvature vector can also be expressed as: \[\boldsymbol{K}(t) = \boldsymbol{T} \times \frac {\boldsymbol{\gamma}''(t)} {\left\|\boldsymbol{\gamma}'(t)\right\|^2} \times \boldsymbol{T}\] where × denotes the vector cross product.

• 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: \[\boldsymbol{N} = \frac {K} {\left\| K \right\|}.\]
• The curvature vector is a differential-geometric property of the curve at P; it does not depend on how the curve is parametrized.

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 of curvature and the radius of curvature of the curve at P are the center and the radius of the osculating circle.

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

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

\[\boldsymbol{C}(t) = \boldsymbol{\gamma}(t) + \frac {\boldsymbol{K}(t)} {\kappa(t)^2}~.\]

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".

Given two points P and Q on a curve γ, let s(P,Q) be the arc length of the portion of the curve between P and Q and let d(P,Q) denote the length of the line segment from P to Q. The curvature of γ at P is given by the limit

\[\kappa(\boldsymbol{P}) = \lim_{\boldsymbol{Q}\to \boldsymbol{P}}\sqrt\frac{24\bigl(s(\boldsymbol{P},\boldsymbol{Q})-d(\boldsymbol{P},\boldsymbol{Q})\bigr)}{s(\boldsymbol{P},\boldsymbol{Q})\vphantom{Q}^3}~,\]

where the limit is taken as the point Q approaches P on γ. The denominator can equally well be taken to be d(P,Q). The formula is valid in any dimension. The formula follows by verifying it for the osculating circle.

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 curve γ at a point P = γ(t₀) if the one-sided derivatives for γ′(t₀) exist but are different values, or likewise for γ′′(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 using l'Hôpital's rule.

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 exists and is nowhere equal to the zero vector.

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

\[\kappa = \frac{\left|x'y''-y'x''\right|}{\bigl({x'}^2+{y'}^2\bigr)\vphantom{'}^{3/2}},\]

where primes refer to derivatives with respect to t.

This can be expressed in a coordinate-free way as \[\kappa = \frac{\left|\det\left(\boldsymbol{\gamma}',\boldsymbol{\gamma}''\right)\right|}{\|\boldsymbol{\gamma}'\|^3},\]

where the numerator is the absolute value of the determinant of the 2-by-2 matrix with γ′ and γ′′ as the columns.

These formulas can be understood as an application of the cross product formula for curvature 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.

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

\[\kappa=\frac{\sqrt{\bigl(z''y'-y''z'\bigr)\vphantom{'}^2+\bigl(x''z'-z''x'\bigr)\vphantom{'}^2+\bigl(y''x'-x''y'\bigr)\vphantom{'}^2}} {\bigl({x'}^2+{y'}^2+{z'}^2\bigr)\vphantom{'}^{3/2}},\]

where the prime denotes differentiation with respect to the parameter t. Both the curvature and the curvature vector can be expressed using the vector cross product and the unit tangent vector T:

\[\boldsymbol{K} = \boldsymbol{T} \times \frac {\boldsymbol{\gamma}''} {\left\|\boldsymbol{\gamma}'\right\|^2} \times \boldsymbol{T}\]

\[\kappa = \frac{\bigl\|\boldsymbol{\gamma}' \times \boldsymbol{\gamma}''\bigr\|}{\bigl\|\boldsymbol{\gamma}'\bigr\|\vphantom{'}^3}~.\]

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 of γ′′ / ‖γ′‖^2 relative to γ′.

For a curve drawn on a surface (embedded in three-dimensional Euclidean space), several curvatures are defined, which relates the direction of curvature to the surface's unit normal vector, including the:

• normal curvature
• geodesic curvature
• geodesic torsion

Any non-singular curve on a smooth surface has its tangent vector T contained in the tangent plane of the surface. The normal curvature, k_n, is the curvature of the curve projected onto the plane containing the curve's tangent T and the surface normal u; the geodesic curvature, k_g, is the curvature of the curve projected onto the surface's tangent plane; and the geodesic torsion (or relative torsion), τ_r, measures the rate of change of the surface normal around the curve's tangent.

Let the curve be arc-length parametrized, and let t = u × T so that T, t, u form an orthonormal basis, called the Darboux frame. The above quantities are related by:

\(\begin{pmatrix} \mathbf{T}'\\ \mathbf{t}'\\ \mathbf{u}' \end{pmatrix} = \begin{pmatrix} 0&\kappa_\mathrm{g}&\kappa_\mathrm{n}\\ -\kappa_\mathrm{g}&0&\tau_\mathrm{r}\\ -\kappa_\mathrm{n}&-\tau_\mathrm{r}&0 \end{pmatrix} \begin{pmatrix} \mathbf{T}\\ \mathbf{t}\\ \mathbf{u} \end{pmatrix}\)

Curvature can be evaluated along surface normal sections, similar to § Curves on surfaces above (see for example the Earth 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. Such developable surfaces have zero Gaussian curvature (see below).