Let $R \subset \mathbb{R}^2$ be a region of the plane bounded by a Jordan curve. The boundary $\partial R$ could be a polygon, or a smooth curve—there are variations depending upon boundary assumptions. I would like to partition $R$ into regions $R_i$ that can be striped by parallel lines with the property that each stripe meets the boundary $\partial R_i$ at an angle that excludes the open interval $(\frac{1}{4}\pi,\frac{3}{4}\pi)$. In other words, the stripes meet the boundary of $R_i$ at $45^\circ$ or more sharply; they cannot meet the boundary nearly orthogonally. (Here the boundary is the boundary of $R_i$, which might include portions of the boundary of $R$.) And the ultimate goal is to partition $R$ into the minimum number of such regions.

For example, a rectangle can be partitioned into one region, but it seems a circle may need four regions(?):
           HerringboneRectCirc http://cs.smith.edu/~orourke/MathOverflow/HerringboneRectCirc.jpg

A number of questions suggest themselves:

Q1. For $R$ a polygon of $n$ vertices, what is the largest number of herringbone regions needed for a fixed $n$, and how many regions always suffice for a fixed $n$?

Q2. For $R$ a circle in $\mathbb{R}^2$, or the surface of a sphere in $\mathbb{R}^3$, what is the optimal (fewest regions) herringbone partition? Can the circle be herringbone-partitioned into three regions?

Q3. What is the optimal herringbone partition of a torus?

I'll stop here, as you can spin off these questions as easily as I. My original focus was on compact surfaces in $\mathbb{R}^3$ (the sphere and torus above), but it already seems interesting in $\mathbb{R}^2$. Thanks for any insights and/or pointers to relevant literature!