I'd like to approximate an arc with a fixed number of line segments in $L1$ or $L2$ sense. Let's define the problem as follows:
Given a smooth function $f: [0,1]\to \mathbb{R}, f(0)=f(1)=0$ and $N\in \mathbb{N}$ find a sequence $\{(x_1,y_1), ..., (x_N,y_N)\}$ to minimize the area between the graph of $f$ and the polygon formed by $\{(x_0=0,y_0=0), (x_1,y_1), ..., (x_N,y_N), (x_{N+1}=1,y_{N+1}=0)\}$. For example, for $N=1$ this we are approximating an arc with a triangle that minimizes area between the arc and the triangle.
In $L1$, $\Sigma_{n=0}^N\int_{x_n}^{x_{n+1}} |f(x)-(x-x_n)\frac{y_n-y_{n+1}}{x_n-x_{n+1}}|dx \to \min$.
In $L2$, $\Sigma_{n=0}^N\int_{x_n}^{x_{n+1}} (f(x)-(x-x_n)\frac{y_n-y_{n+1}}{x_n-x_{n+1}})^2dx \to \min$.
Is there some theory I could look up for that? For approximating with a polynomial we have Chebyshev polynomials, but what's available for approximating with a polygon?
"Approximation of Curves by Line Segments" by Henry Stone resolved that 60+ years ago.