Explicit excluded volume of cylindrically symmetric convex bodies

We represent explicitly the excluded volume $V_{\mathrm{e}}\{{\mathcal{B}}_1,{\mathcal{B}}_2\}$ of two generic cylindrically symmetric, convex rigid bodies, ${\mathcal{B}}_1$ and ${\mathcal{B}}_2$, in terms of a family of shape functionals evaluated separately on ${\mathcal{B}}_1$ and ${\mathcal{B}}_2$. We show that $V_{\mathrm{e}}\{{\mathcal{B}}_1,{\mathcal{B}}_2\}$ fails systematically to feature a dipolar component, thus making illusory the assignment of any shape dipole to a tapered body in this class. The method proposed here is applied to cones and validated by a shape-reconstruction algorithm. It is further applied to spheroids (ellipsoids of revolution), for which it shows how some analytic estimates already regarded as classics should indeed be emended.

Figure 1

Sketch representing the unit vectors $\mathbf{\nu }$, ${\mathit{m}}_1$, and ${\mathit{m}}_2$. With $\mathbf{\nu }$ and ${\mathit{m}}_1$ fixed, ${\mathit{m}}_2$ as represented by (18b) describes a cone around $\mathbf{\nu }$ in the first step of the averaging process described in the text.

Figure 2

A circular cone with vertex in the origin $o$, semiamplitude $\alpha$, radius $R$, height $h$, and slant height $L$.

Figure 3

(a) For two congruent circular cones, ${\mathcal{C}}_1^{\alpha }$ and ${\mathcal{C}}_2^{\alpha }$, with slant height $L$ and semiamplitude $\alpha$, the graphs of $B_n$ scaled to $L^3$ are plotted against $0\leqq \alpha \leqq \frac{\pi }{2}$ for $n=2$ (solid line), $n=4$ (dashed line), and $n=6$ (dotted line), according to (A51). (b) For the same cones, ${\mathcal{C}}_1^{\alpha }$ and ${\mathcal{C}}_2^{\alpha }$, the graphs of $B_n$ scaled to $L^3$ are plotted against $0\leqq \alpha \leqq \frac{\pi }{2}$ for $n=1$ (thin solid line), $n=3$ (solid line), $n=5$ (dashed line), and $n=7$ (dotted line). In both panels, crosses represent the values computed numerically on the shape of the excluded body ${\mathcal{B}}_{\mathrm{e}}\{{\mathcal{C}}_1^{\alpha },{\mathcal{C}}_2^{\alpha }\}$ reconstructed with the algorithm recalled in Appendix pp2.

Figure 4

(a) The isotropic average $B_0$ as in (16) and (15) normalized to the cone's volume $V_{\mathrm{c}}$ delivered by (32c); it attains its minimum at $\alpha \doteq 0.14\pi$. (b) $B_0$ normalized to $L^3$ like all other coefficients $B_n$'s shown in Fig. 3; it attains its maximum at $\alpha \doteq 0.47\pi$. In both panels, crosses represent the volumes computed numerically to benchmark the shape-reconstruction algorithm described in Appendix pp2.

Figure 5

Excluded volume $V_{\mathrm{e}}$ of two congruent circular cones, ${\mathcal{C}}_1^{\alpha }$ and ${\mathcal{C}}_2^{\alpha }$, with slant height $L$ and semiamplitude ${\alpha }_0\doteq 0.14\pi$ corresponding to the minimum value of the scaled average $〈V_{\mathrm{e}}〉/V_{\mathrm{c}}$, where $V_{\mathrm{c}}$ is the volume of each cone. Two graphs, plotted against the angle $0\leqq \vartheta \leqq \pi$ made by the cones' axes, are delivered by (4) truncated at $n=3$ (solid line) and $n=9$ (dashed line). The third graph (dotted line) represents the octupolar approximation proposed in [18], which interpolates the excluded volumes of parallel ($\vartheta =0$) and antiparallel ($\vartheta =\pi$) configurations.

Figure 6

A spheroid is an ellipsoid of revolution. The symmetry axis is here denoted by $\mathit{m}$; $a$ and $b$ are the ellipsoid's semiaxes, in the direction of $\mathit{m}$ and in the direction orthogonal to $\mathit{m}$, respectively. This spheroid is said to be prolate (a) if the aspect ratio $\eta :=b/a$ is less than unity; it is said to be oblate (b) if $\eta >1$.

Figure 7

The plots of $B_2$ (solid line), $B_4$ (dashed line), and $B_6$ (dotted line) normalized to $B_0$ for $0\leqq \eta \leqq 1$. In the limit as $\eta \to 0$ (needle-shaped spheroids), the ratios shown here tend to $B_2/B_0=-5/8\doteq -0.63$, $B_4/B_0=-9/64\doteq -0.14$, and $B_6/B_0=-65/1024\doteq -0.06$.

Figure 8

(a) The plot of $B_0$ (normalized to $8V_{\mathrm{s}}$) for a prolate spheroid. It behaves like $3/32\eta$ as $\eta \to 0$. (b) The plot of $B_2$ (normalized to $V_{\mathrm{s}}$) for a prolate spheroid. It behaves like $-15\pi /32\eta$ as $\eta \to 0$. Both plots are easily extended to oblate spheroids by preserving their values under the transformation $\eta \mapsto 1/\eta$.

Figure 9

Sketch that describes how the radial mapping $\mathit{r}$ assigns to a unit vector $\mathbf{\nu }$ of ${\mathbb{S}}^2$ the translation that brings $o$ into the point on $\partial \mathcal{B}$ where $\mathbf{\nu }$ is the unit outward normal to $\partial \mathcal{B}$. The existence of such a mapping is guaranteed by the assumption that $\mathcal{B}$ belongs to ${\mathcal{K}}^{+}$.

Figure 10

Cross section of a cylindrically symmetric body $\mathcal{B}$ through a plane containing its axis of symmetry $\mathit{m}$. Both the outer unit normal $\mathit{n}$ and the principal direction of curvature ${\mathit{e}}_1$ along the local meridian are on this plane.

Figure 11

The plane curve $\mathit{y}$, which generates $\partial \mathcal{B}$ by a $2\pi$ rotation about $\mathit{m}$, is parametrized in the generic scalar $s$ as $\mathit{y}\left(s\right)=\mathit{x}(s,0)$, where $\mathit{x}(s,\phi )$ is given by (A23). The origin $o$ is taken to coincide with the uppermost pole.

Figure 12

The generating curve of a circular cone ${\mathcal{C}}^{\alpha }$ with semiamplitude $\alpha$, radius $R$, and height $h$, the two latter related to the slant height $L$ as in (A32). The parameter $s$ here represents the arclength along the slant side of the cone.

Figure 13

The half ellipse with semiaxes $a$ and $b$ whose $2\pi$ rotation about the symmetry axis $\mathit{m}={\mathit{e}}_x$ generates a spheroid with aspect ratio $\eta =b/a$. The parameter $\theta$ featured in (A35) designates the angle between $\mathit{y}\left(\theta \right)$ and ${\mathit{e}}_z$. The origin $o$ has been chosen in the center of the spheroid.

Figure 14

(a) Circular ridge $\mathfrak{R}$ of radius $R$ in the plane orthogonal to the symmetry axis $\mathit{m}$ at the distance $h$ from the origin $o$. The unit outward normal $\mathit{n}$ makes the angle ${\theta }_1$ with $\mathit{m}$ on one side and angle ${\theta }_2$ on the other side. (b) The sharp corner of $\mathfrak{R}$ is rounded off in a toroidal surface with meridian radius $\varepsilon$.

Figure 15

(a) Spherodisk ${\mathcal{D}}_{\varepsilon }$ defined as the Minkowski sum of the disk $\mathcal{D}$ and a ball ${\mathbb{B}}_{\varepsilon }^3$ of radius $\varepsilon$ and same center $o$ as $\mathcal{D}$. (b) The generating curve of ${\mathcal{D}}_{\varepsilon }$. The symmetry axis $\mathit{m}$ is orthogonal to $\mathcal{D}$.

Figure 16

The ratio $B_4/B_0=f_4/f_0$ is plotted against $\eta$ both according to the expressions in (A54a) and (39) (solid line) and to formula (60) of [9] (dashed line).

Figure 17

(a) A set of $30\mathrm{K}$ random points generated from $\partial {\mathcal{B}}_{\mathrm{e}}$ with the method adopted here. The sampling of the surface is clearly nonuniform. (b) The final configuration of 10 K reference vectors $\mathit{W}$ produced by the adaptive NG algorithm is more uniformly distributed. (c) and (d) From $\mathit{W}$, the ball-pivoting algorithm reconstructs the surface boundary $\partial {\mathcal{B}}_{\mathrm{e}}$ with no human intervention.

Figure 18

Gallery of reconstructed boundaries $\partial {\mathcal{B}}_{\mathrm{e}}$ for pairs of congruent circular cones of semiamplitude $\alpha$. Rows correspond to values of $\alpha$ equal to $\frac{\pi }{32}$, $\frac{\pi }{6}$, and $\frac{15}{32}\pi$, respectively, while columns correspond to values of the angle $\vartheta$ between the symmetry axes ${\mathit{m}}_1$ and ${\mathit{m}}_2$ equal to 0, $\frac{\pi }{2}$, and $\pi$, respectively. All figures are in the same scale and frame of reference.