Page 321 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Principal Curvature

All curves with the same tangent vector will have the same normal curvature, which is the same
as the curvature of the curve obtained by intersecting the surface with the plane containing T
and u. Taking all possible tangent vectors then the maximum and minimum values of the
normal curvature at a point are called the principal curvatures, k and k , and the directions of the
1
2
corresponding tangent vectors are called principal directions.
This is explained in detail in Unit 27 of this book.
25.4 Summary

Curvature refers to any of a number of loosely related concepts in different areas of

geometry. Intuitively, curvature is the amount by which a geometric object deviates from
being flat, or straight in the case of a line, but this is defined in different ways depending
on the context. There is a key distinction between extrinsic curvature, which is defined for
objects embedded in another space (usually a Euclidean space) in a way that relates to the
radius of curvature of circles that touch the object, and intrinsic curvature, which is defined
at each point in a Riemannian manifold. This article deals primarily with the first concept.
Cauchy defined the center of curvature C as the intersection point of two infinitely close
normals to the curve, the radius of curvature as the distance from the point to C, and the
curvature itself as the inverse of the radius of curvature.
Let C be a plane curve (the precise technical assumptions are given below). The curvature

of C at a point is a measure of how sensitive its tangent line is to moving the point to other
nearby points. There are a number of equivalent ways that this idea can be made precise.
The sign of the signed curvature k indicates the direction in which the unit tangent vector
rotates as a function of the parameter along the curve. If the unit tangent rotates
counterclockwise, then k > 0. If it rotates clockwise, then k < 0. The signed curvature
depends on the particular parametrization chosen for a curve. For example the unit circle
can be parametrised by (cos (), sin()) (counterclockwise, with k > 0), or by (cos(–),
sin(–)) (clockwise, with k < 0). As in the case of curves in two dimensions, the curvature
of a regular space curve C in three dimensions (and higher) is the magnitude of the
acceleration of a particle moving with unit speed along a curve. The tangent, curvature,
and normal vector together describe the second-order behavior of a curve near a point. In
three-dimensions, the third order behavior of a curve is described by a related notion of
torsion, which measures the extent to which a curve tends to perform a corkscrew in space.

25.5 Keywords

Curvature: Curvature refers to any of a number of loosely related concepts in different areas of
geometry.
Extrinsic curvature: Extrinsic curvature, which is defined for objects embedded in another space
(usually a Euclidean space) in a way that relates to the radius of curvature of circles that touch the
object
Intrinsic curvature: Intrinsic curvature, which is defined at each point in a Riemannian manifold.
This article deals primarily with the first concept.

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