Periodic Curves & Surfaces

A periodic knot vector can be either uniform or non-uniform.

A periodic degree d NURBS curve has (d-1) continuous derivatives at the start/end point.

The differences between successive knot values are equal near the start and end of the spline; that is, the differences repeat themselves and hence the term “periodic”.

Specifically, when -degree < i < degree, a periodic knot vector satisfies:

k[(degree-1)+i+1] - k[(degree-1)+i] = k[(cv_count-1)+i+1] - k[(cv_count)+i]

For example a cubic periodic knot vector looks like:

{a,b,c,d,e, ...,  p+a,p+b,p+c,p+d,p+e}

with a < b < c < d < e and e < p+a.

Chapter 12 of The NURBS Book has a few pages discussing periodic NURBS (look in the index), but the discussion is limited. Chapter 14 of DeBoor’s Practical Guide to Splines provides a few more details.