Advanced elasticity/Curvature

It helps to a know a bit about curvature when you start learning how to do buckling analysis. The following discussion goes through the derivation of some useful elementary results relating to curvature. You have already learned these in your introductory calculus course. However, you may have forgotten the details. So this is a refresher lesson.

Tangent vector to a curve edit

Let   be a vector valued function (curve) of the parameter  . The unit tangent vector to the curve traced by the function   is given by


Note that the "velocity" of a point on the curve is in the direction of the tangent. Therefore, the unit tangent vector and the unit velocity vector have the same value


A straight line has the equation


Taking the derivative with respect to   we see that the tangent vector is constant, i.e., it does not change direction. Alternatively, we may say that the condition   implies that the unit tangent vector does not change direction.

If the curve is not a straight line, then the quantity   measures the tendency of the curve to change direction.

Normal vector to a curve edit

The unit normal to the curve is defined as


Curvature vector of a curve edit

The curvature vector is defined as the rate of change of the unit tangent vector with respect to the arc length. If   measures the arc length, then the curvature vector is given by  . Now, the "velocity" is given by




Therefore the curvature vector has the same direction at the unit normal vector.

Curvature edit

The curvature ( ) of the curve is the length of the curvature vector. That means,


Radius of curvature edit

To get a feel for the radius of curvature, consider the equation of a circle


where   is the radius of the circle and   are the unit basis vectors in the   directions. Then the "velocity" is given by


and the unit tangent vector is


Differentiating with respect to  ,


Therefore, the curvature of the circle is


This shows that the radius of the circle is the reciprocal of the curvature of the circle. The radius of curvature of any curve is defined in an analogous manner as the reciprocal of the curvature of the curve at a point.

Curvature of plane curves edit

Let us now consider a curve in a plane  . Let   be the angle that the tangent vector to the curve makes with the positive  -axis. Then we can write


where   are the unit basis vectors in the   directions.

Taking the derivative we have




Using the chain rule


The curvature can then be expressed as


Useful relation for the curvature of plane curves edit

If the plane curve is parameterized as


the curvature of curve can also be expressed as


If, in addition,  , we have

Curvature of a plane curve


Proof: The tangent vector to the curve is given by




Differentiating both sides with respect to  ,




Plugging (2) back into (1) we get


The curvature is given by






Plugging (3) and (5) into (4) gives


For the situation where   we can parameterize the curve using   to get  . Then,


Bibliography edit

  • Varberg and Parcell, Calculus, 7th edition, Prentice Hall, 1997.
  • Apostol, T. M., Calculus Vol. I, 2nd edition, Wiley, 1967.