Previous: Binormal Vectors
5. The Acceleration in the Frenet Frame and the Curvature of a Space CurveAny vector in space can be expressed in terms of the tangential , principal normal , and binormal vectors T(t) , P(t) and B(t) at any point of a smooth curve. In particular, we can express the acceleration vector in terms of this frame, so there must be functions a(t) , b(t) and c(t) such that | |
As in the case of plane curves, we can determine the first of these functions
by differentiating the expression for the velocity vector
| |
We proceed in a similar way to find
P'(
t)
. First, since
P·P
=1
,
2P'(
t)·P(
t)=0
. Similarly, since
P·T
=0
, we find that
P'(
t)·T(
t)=-s'(t) | |
Another definition for the torsion which many books give and which is equivalent to the one given above is through the relationship: Verify the equivalence of these definitions. | |
Demonstration 8: Demo on Curvature and Torsion In this demonstration, there are two windows that show the graph of the curvature function and the torsion function in terms of the parameter t . Consider this graph for various curves. See what happens when the space curve is actually a plane curve. Take a look at the space cardioid family given by | |
A useful way to write the relationships between
T, |