Curvature

In general, there are two important types of curvature: 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, which describe a space curve entirely in terms of its "curvature," torsion, and the initial starting point and direction.
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.
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 . Then the curvature , sometimes also called the "first curvature" (Kreyszig 1991, p. 47), is defined by

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

where is the tangential angle and is the arc length. As can readily be seen from the definition, curvature therefore has units of inverse distance. The derivative in the above equation can be found using the identity

			(5)
			(6)
			(7)

so

(8)

and

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

Combining equations (◇), 3), (10), and (12) then gives

(13)

For a two-dimensional curve written in the form , the equation of curvature becomes

(14)

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

(15)

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

(16)

The curvature for a two-dimensional curve given implicitly by is given by

(17)

(Gray 1997).
Now consider a parameterized space curve in three dimensions for which the tangent vector is defined as

(18)

Therefore,

			(19)
			(20)
			(21)

where is the normal vector. 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 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.
Four very important derivative relations in differential geometry related to 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 (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. In addition, a third curvature (sometimes called total curvature)

(49)

is also defined. 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.