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(?):
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<img src="http://cs.smith.edu/~orourke/MathOverflow/HerringboneRectCirc.jpg" alt="HerringboneRectCirc" />
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A number of questions suggest themselves:

> <b>Q1</b>. 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 $n$?

> <b>Q2</b>. 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 fewer than four regions?

> <b>Q3</b>. 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),
when the pattern is drawn by parallels to a geodesic,
but it already
seems interesting in $\mathbb{R}^2$.
Thanks for any insights and/or pointers to relevant literature!

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<img src="http://cs.smith.edu/~orourke/MathOverflow/herrinbgone.jpg" alt="Cloth pattern" />
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<b>Update</b>. Sergei's idea, as articulated by Cristi, leads to a 3-region herringbone partition
of a circle.  Here his Cristi's illustration from his comment:
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;![Cristi][1]


  [1]: https://i.sstatic.net/eME2X.gif