Part 1: Curvature and the Unit Normal

In the last section, we explored those ideas related to velocity-namely, distance, speed, and the unit tangent vector. In this section, we do the same for acceleration by exploring the concepts of linear acceleration, curvature, and the unit normal vector.

Thoughout this section, we will assume that r( t) parameterizes a smooth curve and is second differentiable in each component. Thus, its unit tangent T satisfies ||T(t) || = 1, which implies that T( t)·T( t) = 1. Differentiation yields

d dt ( T( t) · T( t))   =  d dt  1 dT dt ·T + T· dT dt  =  0 2T·  dT dt  =  0

That is, the derivative of T is orthogonal to T.  Indeed, the unit normal vector is the unit vector  

N = 1 || dT/dt|| dT dt

(1)

That is, N is a unit vector which is orthogonal to T. alternatively, T'  = || T' ||  N.   

EXAMPLE 1    Find the unit normal N to the helix r( t) = á 4cos( t), 4sin(t) ,3t ñ       

Solution: Since the velocity is v = á -4sin( t) ,4cos( t) ,3 ñ , the speed is

v =  16sin2(t) +16cos2(t)+9  =  16+9 = 5

and consequently the unit tangent vector is

T =  1 v v =  -4 5 sin(t) , 4 5 cos( t) , 3 5

The derivative of the unit tangent vector is

dT dt  =  d dt -4 5 sin(t) , 4 5 cos( t) , 3 5   =   -4 5 cos( t) , -4 5 sin(t) ,0

which has a length of

dT dt  =   16 25 cos2(t) +  16 25 sin2( t) +02  =  4 5

Thus, the unit normal is

N =  1 || dT/dt|| dT dt  =  1 4/5 -4 5 cos( t) ,  -4 5 sin( t) ,0

which simplifies to N = á -cos( t) ,-sin( t) ,0 ñ .  For example, if t=p /2, then N = á 0 ,-1 ,0 ñ , as is shown below along with the unit tangent T at the same point.

       

If a curve r(s) is parameterized by its arclength variable s, then the curvature of the curve is defined

k  =     dT ds

For any other parameterization r(t) , we notice that

dT dt  =    dT ds    ds dt      = kv

(2)

since v = ds/dt.  Thus, in general the curvature of a curve is given by

k =  1 v     dT dt

Since T'  = ||T' || N, equation (2) implies that

dT dt = κ v  N

(3)

However, as we will soon see, these formulas only define k; they are not necessarily the best means of computing k.    

Check your Reading: Is N the only unit vector orthogonal to T at a given point on the curve?