]> 15.3 Curvature and Radius of Curvature

15.3 Curvature and Radius of Curvature

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 κ ( s ) , which is defined by

κ = | d T ^ ( t ( s ) ) d s |

which is, the magnitude of the change in unit tangent vector 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 d T ^ d s has the dimension of the reciprocal of a length that is, of the reciprocal 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 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 1 κ ( t ) , the reciprocal of the curvature.

In the applet in the next section you can enter your favorite parametrized curve and see the circle of curvature.

The center of curvature and the tangent vector to the curve, T ^ ( t ) , determine a plane called the plane of curvature.

Since the radius of a circle is always normal to a vector tangent to it, a line from r ( t ) toward the center of curvature will be normal to T ^ . The vector N ^ ( t ) , called the normal vector to the curve, is a unit vector pointing from r ( t ) toward the center of curvature.

B ^ ( t ) , the "binormal vector" is a unit vector normal to both T ^ and N ^ , that is to the plane of curvature. By convention its direction is that of T ^ × N ^ .

We define a ( t ) (the acceleration) to be the derivative with respect to t of v ( t ) , (the velocity). In these terms, T ^ is a unit vector in the direction of v , N ^ is a unit vector in the direction of the projection of a normal to v , while B ^ is a unit vector in the direction of v × a .

The "Frenet Frame" defined by a curve at "time" t is the set of unit vectors, T ^ ( t ) , N ^ ( t ) and B ^ ( t ) .