# Lecture 11—Discrete Curves

This lecture presents the discrete counterpart of the previous lecture on smooth curves. Here we also arrive at a discrete version of the fundamental theorem for plane curves: a discrete curve is completely determined by its discrete parameterization (a.k.a. edge lengths) and its discrete curvature (a.k.a. exterior angles). Can you come up with a discrete version of the fundamental theorem for space curves? If we think of torsion as the rate at which the binormal is changing, then a natural analogue might be to (i) associate a binormal $B_i$ with each vertex, equal to the normal of the plane containing $f_{i-1}$, $f_{i}$, and $f_{i+1}$, and (ii) associate a torsion $\tau_{ij}$ to each edge $ij$, equal to the angle between $B_i$ and $B_{i+1}$. Using this data, can you recover a discrete space curve from edge lengths $\ell_{ij}$, exterior angles $\kappa_i$ at vertices, and torsions $\tau_{ij}$ associated with edges? What’s the actual algorithm? (If you find this problem intriguing, leave a comment in the notes! It’s not required for class credit.)