» Required Reading » Table of Contents » Chapter 15 15.3 Curvature and Radius of Curvature
	Previous Section	Next Section

The next important feature of interest is how much the curve differs from being a straight line at position s. We measure this by the curvature (s), which is defined by

,

which is, the change in tangent direction per unit change in distance along the curve.
The vector T being a unit vector has no dimension; that is, it is unaffected by a uniform change in scale of all coordinates. s on the other hand,  is a length; therefore dT/ds has the dimension of the reciprocal of a length that is, of a distance.

There is a second way to describe the information represented by the curvature: the center of curvature of the curve at parameter t or s is the point q(t) such that a circle centered at q which meets our curve at r(t), will have the same slope and curvature as our curve has there. The radius of that circle is called the radius of curvature of our curve at argument t. We will see that the radius of curvature, which is a length is exactly .
In the applet in the next section you can enter your favorite parametrized curve and see the circle of curvature.

The center of curvature will lie in the plane determined by the tangent vector T(t) and its derivative with respect to t (or s). This plane is called the plane of curvature at argument t. The condition that the circle through r(t) have the slope of T(t) implies that q(t) lies on the line perpendicular to T through the point r(t) in the plane of curvature. The statement above about radius of curvature is the fact that the curvature of a circle is the reciprocal of its radius. We define the unit vector N(t) to point from r(t) toward the center of curvature.

If we define a(t) to be the derivative with respect to t of v(t), then N(t) points in the direction of the projection of a(t) normal to T(t). We denote a unit normal to the plane of curvature to be B(t). B(t) is then normal to both v(t) and a(t) and hence can be described as .
The "Frenet Frame" defined by a curve at a point is the set of unit vectors, T(t), N(t) and B(t).