Page 316 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 25: Curvature

In a plane, this is a scalar quantity, but in three or more dimensions it is described by a curvature Notes
vector that takes into account the direction of the bend as well as its sharpness. The curvature of
more complex objects (such as surfaces or even curved n-dimensional spaces) is described by
more complex objects from linear algebra, such as the general Riemann curvature tensor. The
remainder of this article discusses, from a mathematical perspective, some geometric examples
of curvature: the curvature of a curve embedded in a plane and the curvature of a surface in
Euclidean space. See the links below for further reading.

25.1 Curvature of Plane Curves

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.

One way is geometrical. It is natural to define the curvature of a straight line to be identically
zero. The curvature of a circle of radius R should be large if R is small and small if R is large.
Thus, the curvature of a circle is defined to be the reciprocal of the radius:

1
k= .
R
Given any curve C and a point P on it, there is a unique circle or line which most closely
approximates the curve near P, the osculating circle at P. The curvature of C at P is then defined
to be the curvature of that circle or line. The radius of curvature is defined as the reciprocal of the
curvature.
Another way to understand the curvature is physical. Suppose that a particle moves along the
curve with unit speed. Taking the time s as the parameter for C, this provides a natural
parametrization for the curve. The unit tangent vector T (which is also the velocity vector, since
the particle is moving with unit speed) also depends on time. The curvature is then the magnitude
of the rate of change of T. Symbolically,

dT
k= .
ds
This is the magnitude of the acceleration of the particle. Geometrically, this measures how fast
the unit tangent vector to the curve rotates. If a curve keeps close to the same direction, the unit
tangent vector changes very little and the curvature is small; where the curve undergoes a tight
turn, the curvature is large.

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