Curvature

Curvature is classified into two principal types: extrinsic curvature and intrinsic curvature. The extrinsic curvature of curves in two- and three-space was the first type of curvature to be studied historically, culminating in the Frenet formulas. For a three times continuously differentiable space curve with positive curvature, these determine the curve from its curvature and torsion as functions of arc length, together with its initial point and unit tangent vector, normal vector, and binormal vector.

After the curvature of two- and three-dimensional curves was studied, attention turned to the curvature of surfaces in three-space. The main curvatures that emerged from this scrutiny are the mean curvature, Gaussian curvature, and the shape operator. Mean curvature was the most important for applications at the time and was the most studied, but Gauss was the first to recognize the importance of the Gaussian curvature.

Because Gaussian curvature is "intrinsic," it is detectable to two-dimensional "inhabitants" of the surface, whereas mean curvature and the shape operator are not detectable to someone who can't study the three-dimensional space surrounding the surface on which he resides. The importance of Gaussian curvature to an inhabitant is that it controls the surface area of spheres around the inhabitant.

Riemann and many others generalized the concept of curvature to sectional curvature, scalar curvature, the Riemann tensor, Ricci curvature tensor, and a host of other intrinsic and extrinsic curvatures. General curvatures no longer need to be numbers, and can take the form of a map, group, groupoid, tensor field, etc.

The simplest form of curvature and that usually first encountered in calculus is an extrinsic curvature. In two dimensions, let a plane curve be given by Cartesian parametric equations and , with twice continuously differentiable coordinates and . The signed curvature , sometimes also called the "first curvature" (Kreyszig 1991, p. 47), is defined by

					(1)
					(2)
					(3)
					(4)

where is the tangential angle, measured counterclockwise, and is the arc length increasing with . The unsigned curvature is . Curvature therefore has units of inverse distance. Where , the derivative can be found using the identity

			(5)
			(6)
			(7)

so

(8)

and

			(9)
			(10)
			(11)
			(12)

Combining these expressions gives

(13)

For a two-dimensional curve written in the form and oriented in the direction of increasing , the signed curvature becomes

(14)

If the two-dimensional curve is instead parameterized in polar coordinates with increasing , then

(15)

where (Gray 1997, p. 89). In pedal coordinates, the unsigned curvature is given locally by

(16)

where , , and is a differentiable function of . The unsigned curvature of a twice continuously differentiable implicit curve , where , is

(17)

(Gray 1997).

Now consider a twice continuously differentiable space curve with nonzero velocity in three dimensions. Here denotes the nonnegative curvature, and the unit tangent vector is defined as

(18)

Therefore,

			(19)
			(20)
			(21)

where is the principal unit normal vector where . But

			(22)
			(23)

so taking norms of both sides gives

(24)

Solving for then gives

			(25)
			(26)
			(27)

(Gray 1997, p. 192).

The curvature of a two-dimensional curve is related to the radius of curvature of the curve's osculating circle. Consider a circle of radius specified parametrically by

(28)

(29)

which is tangent to the curve at a given point. The curvature is then

(30)

or one over the radius of curvature. The curvature of a circle can also be repeated in vector notation. For the circle with , the arc length is

			(31)
			(32)
			(33)

so and the equations of the circle can be rewritten as

(34)

(35)

The radius vector is then given by

(36)

and the tangent vector is

			(37)
			(38)

so the curvature is related to the radius of curvature by

			(39)
			(40)
			(41)
			(42)

as expected.

For a three times continuously differentiable space curve parameterized by arc length with , four derivative relations following from the Frenet formulas are

			(43)
			(44)
			(45)
			(46)

where is the tangent vector, is the normal vector, is the binormal vector, and is the torsion. In these four relations, dots denote derivatives with respect to (Coxeter 1969, p. 322).

The curvature at a point on a surface takes on a variety of values as the plane through the normal varies. As varies, it achieves a minimum and a maximum (which are in perpendicular directions) known as the principal curvatures. As shown in Coxeter (1969, pp. 352-353),

(47)

(48)

where is the Gaussian curvature, is the mean curvature, and det denotes the determinant.

The curvature is sometimes called the first curvature and the torsion the second curvature. The line element associated with the third curvature (sometimes called total curvature) is

(49)

where , , and are the arc length elements of the curves traced by the unit tangent vector, normal vector, and binormal vector, respectively. A signed version of the curvature of a circle appearing in the Descartes circle theorem for the radius of the fourth of four mutually tangent circles is called the bend.