Previous: Distance Functions to Plane Curves


Decomposition of Acceleration

Because we defined the unit normal to create an orthonormal basis for R2, we know that the acceleration vector X''( t) can be expressed in terms of T(t) and U(t) . There must be functions a(t) and b(t) such that

    X''( t)=a(t) T(t)+b (t)U(t).
By differentiating the expression for the velocity vector
    X'( t)=s'(t)T(t)
we obtain
    X''( t)=s''( t)T( t)+s'(t)T'( t)
Now T'( t) itself is already expressed in terms of the unit tangent and the unit normal, and since T(t) is a unit vector, T'( t) is perpendicular to T(t) ( T(t) ·T(t) =1 , taking the derivative of both sides yields 2T(t) ·T'( t)=0 ) so T'( t) must be a multiple of U(t) . We define T'( t) = kg(t) s'(t) U(t) . The function kg(t) is the curvature or geodesic curvature of the curve at the point X(t) . It follows that:
    X''( t)=s''( t)T( t)+kg(t)(s' (t)) 2U(t)

Demonstration 6: Acceleration and Curvature (demo under construction)
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Observe the relationship between the acceleration vector and the unit normal vector at the current position on the curve. The projection of the acceleration vector on the unit normal vector is related to the curvature of the curve. Watch how the change in this projection varies while the light blue point marking the current value of the geodesic curvature in the Graph of geodesic curvature window moves along the graph.

This demo shows the acceleration vectors of a plane curve emanating from the points on the curve with which they are associated and the graph of the geodesic curvature over the domain of the curve.

    Examine the acceleration vectors and curvature of the parabola.

    Examine the acceleration vectors and curvature of the ellipse.

    Examine the acceleration vectors and curvature of the exponential spiral.

We can approach curvature in terms of the circular images of the vectors T(t) and U(t), where we define the angle q(t) by the condition T(t)=[cos(q(t)), sin(q(t))]. Note that q(t) is only defined up to a multiple of 2p. However the difference q(b)-q(a) is well-defined, for any interval a≤t≤b. Also, for this choice of q, we have U(t) =[-sin(q(t)), cos(q(t))]. Then

    T'(t)=[-sin(q(t))q'(t), cos(q(t))q'(t)]
    =U(t)q'(t)
and q'(t) is independent of the multiple of 2p used in the definition of q(t). If we write X'(t)=s'T(t), then X''(t)=s''T(t)+s'T'(t), so X''(t)=s''T(t)+s'(q'U'(t)) so from the "kinematic definition" we have s'q'(t) = (s')2k(t), and therefore k(t) = q'(t)s'(t). In terms of an arclength parametrization, we then have k(s)=q'(s). That relates the decomposition approach to the approach in terms of the normal images.


Next: Normalized Parallel Curves