On a formula for sets of constant width in 2d

Figure 1.  Two sets of constant width with a boundary of circular arcs. On the left the Reuleaux triangle. The segments connect boundary points with opposite normal directions for which the constant width is attained. The points denote the centers of rotation

Figure 2.  The relation between the directional width and the support function

Figure 3.  Graphs of the functions $ a $ in Recipe 3.1 with the corresponding sets of constant width for $ r = \left\Vert a\right\Vert _{\infty } $

Figure 4.  $ G $ and $ G_{\varepsilon } $ with the auxiliary circles

Figure 5.  The dark (red) part contains the values $\int_\alpha ^\beta {\rho \left( s \right)} {e^{is}}ds$ for all allowed $\rho $ . The bounding curves $j_{0}(\cdot )$ , $j_{1}(\cdot )$ correspond to the extreme cases in (33) as function of $ \theta $ : $j_{0}\left( \theta \right) =\int_ \alpha ^{\alpha +\theta \left( \beta - \alpha \right) }e^{is}ds$ and $j_{1}\left( \theta \right) = \int_{\beta -\theta \left( \beta -\alpha \right) }^{\beta }e^{is}ds$ .