In general, there are two important types of curvature: Extrinsic Curvature and Intrinsic Curvature. The Extrinsic Curvature of curves in 2- and 3-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 2- and 3-D curves was studied, attention turned to the curvature of surfaces in 3-space. The main curvatures which emerged from this scrutiny are the Mean Curvature, Gaussian Curvature, and the Weingarten Map. 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 2-dimensional ``inhabitants'' of the surface, whereas Mean Curvature and the Weingarten Map are not detectable to someone who can't study the 3-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, 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 2-D,
let a Plane Curve be given by Cartesian parametric equations and .
Then the curvature is defined by

(1) |

(2) |

(3) |

(4) |

Combining (2) and (4) gives

(5) |

For a 2-D curve written in the form , the equation of curvature becomes

(6) |

(7) |

(8) |

(9) |

Now consider a parameterized Space Curve in 3-D for which the Tangent Vector is defined as

(10) |

(11) |

(12) |

(13) |

(14) |

(15) |

The curvature of a 2-D curve is related to the Radius of Curvature of the curve's Osculating Circle.
Consider a Circle specified parametrically by

(16) |

(17) |

(18) |

(19) |

so and the equations of the Circle can be rewritten as

(20) |

(21) |

(22) |

(23) |

(24) |

as expected.

Four very important derivative relations in differential geometry related to the Frenet Formulas are

(25) | |||

(26) | |||

(27) | |||

(28) |

where

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),

(29) |

(30) |

The curvature is sometimes called the First Curvature and the Torsion the Second Curvature. In addition, a Third Curvature (sometimes called Total
Curvature)

(31) |

**References**

Coxeter, H. S. M. *Introduction to Geometry, 2nd ed.* New York: Wiley, 1969.

Fischer, G. (Ed.). Plates 79-85 in
*Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume.*
Braunschweig, Germany: Vieweg, pp. 74-81, 1986.

Gray, A. ``Curvature of Curves in the Plane,'' ``Drawing Plane Curves with Assigned Curvature,'' and
``Drawing Space Curves with Assigned Curvature.'' §1.5, 6.4, and 7.8 in
*Modern Differential Geometry of Curves and Surfaces.* Boca Raton, FL: CRC Press, pp. 11-13,
68-69, 113-118, and 145-147, 1993.

Kreyszig, E. ``Principal Normal, Curvature, Osculating Circle.'' §12 in
*Differential Geometry.* New York: Dover, pp. 34-36, 1991.

Yates, R. C. ``Curvature.'' *A Handbook on Curves and Their Properties.* Ann Arbor, MI: J. W. Edwards, pp. 60-64, 1952.

© 1996-9

1999-05-25