Question

define carvature and compute k(t) for the circular Helix

Answer 1

curvature is defined as the rate of change of angle of the tangent with respect to the arc length of the curve.
\[ \kappa = \frac{d\theta}{ds}\]
it is not difficult to show that this equals
\[ \kappa = \frac{d\theta}{ds} = \left|\frac{dt}{ds} \right |=\left|\frac{dr'}{ds} \right | = \frac{\left| r' \times r''\right|}{|r'|^3}\]
where t is tangent vector and r is position vector.
for helix, you have parametric equation
\[ r(t) = (a \cos t, b\sin t, c t)\]
use the above formula to calculate curvature.

Answer 2

a, b, and c are constants