Unit Tangent Vector and Unit Normal Vector I was trying to compute the unit normal vector of $\alpha(t)=(cos^3(t),sin^3(t))$ and was finding that it gets particularly unpleasant after the curve is reparameterized by unit length.  Are there any substitutions or tricks I could use to avoid making the unit normal vector so unpleasant?

You don't need to find a unit speed parametrisation in order to compute the normal vector.  Just find the tangent vector of the curve, normalize it to get a unit vector, and then rotate the resulting vector by the matrix $\pmatrix{0 & -1 \\ 1 & 0}$ (namely, switch the coordinates and change the sign of one of them).
