Previous: Distance Functions to Plane Curves
Decomposition of AccelerationBecause 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
Demonstration 6: Acceleration and Curvature (demo under construction) 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 |