Problem 43648. Arc length of points interpolation

Given a n by m matrix representing m vectors in n dimensions. Calculate the arc length of the closed loop curve going though these points in the order that they are given. The parametric curve, c(t) , between points p(k) and p(k+1) is defined as,

c(t) = p(k-1) * (-t/2+t^2-t^3/2) + p(k) * (1-5/2*t^2+3/2*t^3) + p(k+1) * (t/2+2*t^2-3/2*t^3) + p(k+2) * (-t^2/2+t^3/2),

where t goes from 0 to 1. These interpolation polynomials can also be found using the constraints c(0)=p(k), c(1)=p(k+1), c'(0)=(p(k+1)-p(k-1))/2 and c'(1)=(p(k+2)-p(k))/2.

For example for the points

points = [[1; 0] [0; 1] [-1; 0] [0; -1]];

would yield to following curve: