Curvature of Curves
Curvature of a curve
Suppose a smooth oriented curve in is parametrized by a smooth vector-valued function with and a positive orientation. Then the unit tangent vector is defined as follows:
where is the tangent vector of the curve at the point with position vector .
We want to define a quantity that describes how "sharp" the bending of the curve is, or more precisely, how much the curve direction changes over a small distance along the curve. Moreover, we would like to define this quantity such that it is independent of the parametrization. Hence, we consider the reparametrization by arc length and define this quantity, called curvature, as follows:
where is the arc length reparametrization i.e. . In other words, it is the magnitude of the rate of change of with respect to arc length. Moreover, we can rewrite the definition as follows:
(because by FTC, )
Therefore, i.e. the formula for curvature in terms of the given parametrization.
Example:
Consider a line parametrized by . Then i.e. a constant vector. Hence
and ,
which implies that i.e. zero curvature anywhere on the line.
Example:
Consider a circle on the plane , centered at with radius . It can be parametrized by with . Then . We have
,
which implies that .
Therefore, i.e. constant positive curvature on the circle.
An alternative curvature formula
Suppose . We have
Therefore, we have
where is the angle between and . Since , by previous exercise, and are orthogonal i.e. . Hence and we have
Exercise: Suppose a helix is parametrized by for some . Find its curvature in terms of and . Discuss the effect on the curvature if
Exercise: Consider with . It is a parametrization of a parabola on the xy-plane. Find such that maximum curvature occurs. Moreover, find the value of the maximum curvature.
Principal unit normal vector
The unit vector in the direction of is the principal unit normal vector, denoted by . Therefore, we have . Moreover, implies that and are orthogonal. Hence and are orthogonal.
Note: In practice, we can find the principal unit normal vector by computing the unit vector in the direction of because both and are pointing towards the same direction.
The plane containing and at a point on the curve is called the osculating plane at . And the circle contained in the osculating plane with center such that is in the same direction as and radius is called the osculating circle at . It is the circle that best approximates the curve at .
In the applet below, you can see how , osculating plane, osculating circle, and curvature change when you drag the slider to move the point along the curve.
Question: Why does the principal unit normal vector always point "inwards", the direction that the curve is bending?
Exercise: Given , a parametrization of a circular helix with . Find its principal normal unit vector.