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 is described by a unit tangent vector, . A section of a curve is also called an arc, and length along the curve is
arc length, . 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 . For a general curve which might have a varying curvature along its length, the curvature at a point on the curve is the
limit of the curvature of sections containing 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.
Arc-length parametrization A curve that is
parametrized by arc length is a
vector-valued function that is denoted by the Greek letter
gamma with an overbar, , that describes the position of a point on the curve, , in terms of its arc-length distance, along the curve from some other reference point on the curve. Thus for some interval in \mathbb{R}, {{math|:
I → \mathbb{R}
n}} with \boldsymbol{P}(s) = \boldsymbol{\bar\gamma}(s). If is a differentiable curve, then the first derivative of , is a unit tangent vector, , and \left\| \boldsymbol{\bar\gamma}'(s) \right\| = 1 \boldsymbol{T}(s) = \boldsymbol{\bar\gamma}'(s). If is twice differentiable, the second derivative of is , which is also the curvature vector, . \boldsymbol{K}(s) = \boldsymbol{T}'(s) = \boldsymbol{\bar\gamma}''(s) Curvature is the magnitude of the second derivative of . \kappa(s) = \left\| \boldsymbol{K}(s) \right\| = \left\| \boldsymbol{T}'(s) \right\| = \left\| \boldsymbol{\bar\gamma}''(s) \right\| The parameter can also be interpreted as a time parameter. Then 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.
General parametrization A twice differentiable curve, {{math|
γ: [
a,
b] → \mathbb{R}
n}}, that is not parametrized by arc length can be re-parametrized by arc length provided that is everywhere not zero, so that is always a finite positive number. The arc-length parameter, , is defined by s(t) ~{=}~ \int_a^t \left\| \boldsymbol{\gamma}'(x) \right\| \, \mathrm{d}{x}, which has an inverse function . The arc-length parametrization is the function which is defined as \boldsymbol{\bar\gamma}(s) ~{=}~ \boldsymbol{\gamma}(t(s)). Both and trace the same path in \mathbb{R}^n and so have the same curvature vector and curvature at each point on the curve. For a given and its corresponding , point and its unit tangent vector, , curvature vector, , and curvature, , are: \boldsymbol{P} = \boldsymbol{\bar\gamma}(s) = \boldsymbol{\gamma}(t) \boldsymbol{T} = \boldsymbol{\bar\gamma}'(s) = \frac {\boldsymbol{\gamma}'(t)} {\left\| \boldsymbol{\gamma}'(t) \right\| } \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) \begin{align} \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{align} The curvature vector, , is the
perpendicular component of relative to the tangent vector . This is also reflected in the second expression for the curvature: the expression inside the parentheses is , where is the angle between the vectors and , so that the square root produces . If is twice continuously differentiable, then so is and , while is continuously differentiable, and and are continuous. Often it is difficult or impossible to express the arc-length parametrization, , in
closed form even when is given in closed form. This is typically the case when it is difficult or impossible to express or its inverse in closed form. However curvature can be expressed only in terms of the first and second derivatives of , without direct reference to .
Curvature vector The curvature vector, denoted with an upper-case , is the derivative of the unit tangent vector, , with respect to arc length, : \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 , 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 : \boldsymbol{K}(s) = \boldsymbol{\bar\gamma}''(s). • The curvature vector of a general parametrization, , is the
perpendicular component of relative to the tangent vector : \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, , 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, : \boldsymbol{N} = \frac {K} {\left\| K \right\|}. • The curvature vector is a
differential-geometric property of the curve at ; it does not depend on how the curve is parametrized.
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 on a curve, every other point of the curve defines a circle (or sometimes a line) passing through and
tangent to the curve at . The osculating circle is the
limit, if it exists, of this circle when tends to . Then the
center of curvature and the
radius of curvature of the curve at are the center and the radius of the osculating circle. The radius of curvature, , is the
reciprocal of the curvature, provided that the curvature is not zero: R = \frac{1}{\kappa}~. For a curve , since a non-zero
curvature vector, , points from the point towards the center of curvature, but the magnitude of is the curvature, , the center of curvature, 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".
Curvature from arc and chord length Given two points and on a curve , let be the arc length of the portion of the curve between and and let denote the length of the line segment from to . The curvature of at 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 approaches on . The denominator can equally well be taken to be . The formula is valid in any dimension. The formula follows by verifying it for the osculating circle.
Exceptional cases 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 at a point if the
one-sided derivatives for exist but are different values, or likewise for . 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 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.
Examples The following are examples of curves with application of the relevant concepts and formulas.
Circle A geometric explanation for why the curvature of a circle of radius at any point is is partially illustrated by the diagram to the right. The length of the red arc is and the measure in radians of the arc's central angle, angle ACB, is . 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 length is . Since the ratio is for any arc of the circle that is less than a half circle, for arcs containing any given point on the circle, the limit of the ratio as arc length approaches zero is also . Hence the curvature of the circle at any point is . A common parametrization of a
circle of radius is . Then \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 gives \kappa(t)= \frac{\sqrt{r^2\, r^2 - 0^2\,}} {r^3} = \frac 1r~. and the formula for a plane curve gives \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 1r. 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 \boldsymbol\bar\gamma(s)= \left(r\cos \frac sr,\, r\sin \frac sr\right). It is an arc-length parametrization, since the norm of \boldsymbol{\bar\gamma}'(s) = \left(-\sin \frac sr,\, \cos \frac sr\right) is equal to one. Then \kappa(s) = \left\|\boldsymbol{\bar\gamma}''(s)\right\| = \left\|\left(-\frac 1r\cos \frac sr,\, -\frac 1r \sin \frac sr\right)\right\| = \frac 1r gives the same value for the curvature. The same circle can also be defined by the implicit equation with . Then, the formula for the curvature in this case gives \begin{align} \kappa &= \frac{\left|F_y^2F_{xx}-2F_xF_yF_{xy}+F_x^2F_{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}} =\frac1r.\end{align}
Parabola Consider the
parabola . It is the graph of a function, with derivative , and second derivative . So, the signed curvature is k(x)=\frac{2a}{ \bigl(1+\left(2ax+b\right)^2\bigr)\vphantom{)}^{3/2}}~. It has the sign of for all values of . This means that, if , the concavity is upward directed everywhere; if , the concavity is downward directed; for , the curvature is zero everywhere, confirming that the parabola degenerates into a line in this case. The (unsigned) curvature is maximal for , that is at the
stationary point (zero derivative) of the function, which is the
vertex of the parabola. Consider the parametrization . The first derivative of is , and the second derivative is zero. Substituting into the formula for general parametrizations gives exactly the same result as above, with replaced by and with primes referring to derivatives with respect to the parameter . The same parabola can also be defined by the implicit equation with . As , and , 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 Let 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 exists and is nowhere equal to the zero vector. The curvature 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 . 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 and 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.
Signed curvature For plane curves, it can be useful to express the curvature as a single scalar that can be positive or negative, called the
signed curvature or
oriented curvature and denoted with a lowercase k. The signed curvature formulas are similar to those for except that they omit taking the absolute value of the numerator: 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}~. Then . Whether is positive or negative depends on the orientation of the curve. Whether a positive 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 increasing , 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 of is dependent on the orientation of the parametrization, 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 be the
unit normal vector obtained from the unit tangent vector, , by a counterclockwise rotation of . Then is dependent on the orientation of the curve and points to the left when moving along the curve in the direction of increasing . However the curvature vector, is equal to the product of the signed curvature, and , because their orientation dependencies cancel: \boldsymbol{K} = k\,\boldsymbol{\bar N}. Similarly, the center of curvature can be expressed using the signed curvature and : \boldsymbol{C}(s)= \boldsymbol{\gamma}(s) + \frac{\mathbf{\bar N}(s)}{k(s)}.
Graph of a function The
graph of a function , is a special case of a parametrized curve, of the form \begin{align} x&=t\\ y&=f(t). \end{align} As the first and second derivatives of are 1 and 0, previous formulas simplify to \kappa = \frac{\left|y''\right|}{\bigl(1+{y'}^2\bigr)\vphantom{'}^{3/2}} for the curvature and to 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 of . 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 of . 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 an
inflection point or an
undulation point. When the
slope of the graph (that is the derivative of the function) is small, the signed curvature is well approximated by the second derivative. More precisely, using
big O notation, one has k(x)=y'' \Bigl(1 + O\bigl({\textstyle y'}^2\bigr) \Bigr). It is common in
physics and
engineering to approximate the curvature with the second derivative, for example, in
beam theory or for deriving the
wave equation of a string under tension, and other applications where small slopes are involved. This often allows systems that are otherwise
nonlinear to be treated approximately as linear.
Implicit curve For a curve defined by an
implicit equation with
partial derivatives denoted , , , , , the curvature is given by \kappa = \frac{\left|F_y^2F_{xx}-2F_xF_yF_{xy}+F_x^2F_{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 changing into would not change the curve defined by , but it would change the sign of the numerator if the
absolute value were omitted in the preceding formula. A point of the curve where is a
singular 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 a
cusp). The above formula for the curvature can be derived from the expression of the curvature of the graph of a function by using the
implicit function theorem and the fact that, on such a curve, one has \frac {dy}{dx}=-\frac{F_x}{F_y}.
Polar coordinates If a curve is defined in
polar coordinates by the radius expressed as a function of the polar angle, that is is a function of , then its curvature is \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 to . This results from the formula for general parametrizations, by considering the parametrization \begin{align} x&=r\cos \theta\\ y&=r\sin \theta. \end{align}
Curvature comb A
curvature comb can be used to represent graphically the curvature of every point on a curve. If is a curve parametrized , its comb is defined as the parametrized curve defined by \mathrm{Comb}(t) = \boldsymbol{\gamma}(t) - s\,\kappa(t)\boldsymbol{N}(t), where is the curvature, is the unit normal vector that points toward the center of curvature, and 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 The
first Frenet–Serret formula relates the unit tangent vector, curvature, and the normal vector of an
arc-length parametrization \mathbf T'(s) = \kappa(s) \mathbf N(s), where the primes refer to the derivatives with respect to the arc length , and is the normal unit vector in the direction of . As planar curves have zero
torsion, the second Frenet–Serret formula provides the relation \begin{align} \frac {d\mathbf{N}}{ds} &= -\kappa\mathbf{T},\\ &= -\kappa\frac{d\boldsymbol{\gamma}}{ds}. \end{align} For a general parametrization by a parameter , one needs expressions involving derivatives with respect to . As these are obtained by multiplying by the derivatives with respect to , one has, for any proper parametrization \mathbf{N}'(t) = -\kappa(t)\boldsymbol{\gamma}'(t).
Curves in three dimensions For a parametrically defined curve in three dimensions given in Cartesian coordinates by , 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 . Both the curvature and the curvature vector can be expressed using the
vector cross product and the unit tangent vector : \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 relative to . ==Surfaces==