Question

Find curvature and Normal vector at (theta) = pi for the hypocycloid x = 4cos(theta) + cos(4theta), y = 4sin(theta) - sin(4theta)

Answer 1

\[\large x = 4cos(\theta) + cos(4\theta)\]
\[\large y = 4sin(\theta) - sin(4\theta)\]

Answer 2

So Lets see...our position vector will be

\[\large R <4cos(\theta) + cos(4\theta), 4sin(\theta) - sin(4\theta)>\]
So \(\Large V = \frac{dR}{d\theta} = <-4(sin\theta + sin(4\theta), 4(cos\theta - cos(4\theta)>\)
And \(\Large A =\frac{dV}{d\theta}<-4(cos\theta + 4cos(4\theta) , -4(sin\theta - 4sin(4\theta)>\)

So I should use the formula \(\Large K = \frac{|\vec V \times \vec a|}{|V|^3}\)

Answer 3

Lot of simplifying I can see, but is there an easier way to do it?