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, 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
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.
Curves
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: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, withIf is a differentiable curve, then the first derivative of, is a unit tangent vector,, and
If is twice differentiable, the second derivative of is, which is also the curvature vector,.
Curvature is the magnitude of the second derivative of.
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,, 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
which has an inverse function.
The arc-length parametrization is the function which is defined as
Both and trace the same path in 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:
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, :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:
- The curvature vector is perpendicular to the unit tangent vector, or in terms of the dot product:
- The curvature vector is the second derivative of an arc-length parametrization :
- The curvature vector of a general parametrization,, is the perpendicular component of relative to the tangent vector : If the curve is in, then the curvature vector can also be expressed as: 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, :
- The curvature vector is a differential-geometric property of the curve at ; it does not depend on how the curve is parametrized.
Osculating circle
The radius of curvature,, is the reciprocal of the curvature, provided that the curvature is not zero:
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
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 limitwhere 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.