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Given the following vector, can someone tell me what geometric role the constant b plays? Assume I can measure the constants a (amplitude) and v_0 (velocity of midpoints) directly. I have been told that b represents the 'curviness', but when using a curvature definition for b, the graph looks nothing like it should.

Here is a link to the graph if that helps

$$ \vec{s} = (v_{0}t+b\cos(2t))\hat{x}+a\sin(t)\hat{y} $$

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    It looks like 'curviness' to me. – 2017-02-03
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    How can this be expressed quantitatively. The best I can come up with would be treating the amplitude like the radius of a circle. Then the curviness should be b=1/a, but this is too small to produce the curving effect. – 2017-02-03

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For a better understanding, I'd suggest to study the curve for $v_0=0$: $$ x=b\cos(2t)=b(1-2\sin^2t)=b\left(1-2{y^2\over a^2}\right)=b-{2b\over a^2}y^2. $$ This is the equation of a parabola, intersecting the $y$-axis at $y=\pm a/\sqrt2$ and with vertex $V=(b,0)$. Its radius of curvature at the vertex is $\rho=a^2/4b$, therefore $b$ concurs with $a$ in determining the parabola curvature.