On a Class of Polar Log-Aesthetic Curves

Curves are essential concepts that enable compounded aesthetic curves, e.g., to assemble complex silhouettes, match a specific curvature profile in industrial design, and construct smooth, comfortable, and safe trajectories in vehicle-robot navigation systems. New mechanisms able to encode, generate, evaluate, and deform aesthetic curves are expected to improve the throughput and the quality of industrial design. In recent years, the study of (log) aesthetic curves have attracted the community’s attention due to its ubiquity in natural phenomena such as bird eggs, butterfly wings, falcon flights, and manufactured products such as Japanese swords and automobiles. A (log) aesthetic curve renders a logarithmic curvature graph approximated by a straight line, and polar aesthetic curves enable to mode user-defined dynamics of the polar tangential angle in the polar coordinate system. As such, the curvature profile often becomes a by-product of the tangential angle. In this paper, we extend the concept of polar aesthetic curves and establish the analytical formulations to construct aesthetic curves with user-defined criteria. In particular, we propose the closed-form analytic characterizations of polar log-aesthetic curves meeting user-defined criteria of curvature profiles and dynamics of polar tangential angles. We present numerical examples portraying the feasibility of rendering the logarithmic curvature graphs represented by a straight line. Our approach enables the seamless characterization of aesthetic curves in the polar co-ordinate system, which can model aesthetic shapes with desirable aesthetic curvature profiles.

Topographical curvature is sufficient to control epithelium elongation

How biophysical cues can control tissue morphogenesis is a central question in biology and for the development of efficient tissue engineering strategies. Recent data suggest that specific topographies such as grooves and ridges can trigger anisotropic tissue growth. However, the specific contribution of biologically relevant topographical features such as cell-scale curvature is still unclear. Here we engineer a series of grooves and ridges model topographies exhibiting specific curvature at the ridge/groove junctions and monitored the growth of epithelial colonies on these surfaces. We observe a striking proportionality between the maximum convex curvature of the ridges and the elongation of the epithelium. This is accompanied by the anisotropic distribution of F-actin and nuclei with partial exclusion of both in convex regions as well as the curvature-dependent reorientation of pluricellular protrusions and mitotic spindles. This demonstrates that curvature itself is sufficient to trigger and modulate the oriented growth of epithelia through the formation of convex “topographical barriers” and establishes curvature as a powerful tuning parameter for tissue engineering and biomimetic biomaterial design.

The ultimate shape of pebbles, natural and artificial

Flint pebbles that have been worn down enough to be symmetrical may be spherical, or may approximate to prolate or oblate spheroids or to ellipsoids. When, however, the section is at all elongated, it is not as a rule accurately elliptical, but except at the axial points it lies entirely outside an ellipse adjusted to the same axes. Thus, if one of the axes is much smaller than the other, the pebble is much flatter than an ellipsoid. Considering the quasi-spheroidal pebbles, whether prolate or oblate, these are always flattened at the poles, and the very oblate ones become tabular or even concave at the poles. These statements hold for flint pebbles, but a large quartzite pebble is figured which is pretty accurately an oblate spheroid, the meridional sections being truly elliptical. Experiments are described with chalk pebbles, initially shaped as prolate or oblate spheroids, and the change of figure under abrasion is observed. This depends in some degree on what is the abrasive. Steel nuts, nails ('tin tacks’) and small shot were used. In general the axes tend to approach equality, but not rapidly enough for the spherical form to be attained before the pebble has disappeared. The form initially spheroidal becomes flattened at the poles just like the natural flint pebbles, and may become concave, as flints sometimes do. The circumstances determining the rate of abrasion at any point are considered, and it is shown that this abrasion cannot be merely a function of the local specific curvature. The figure at points other than the one under consideration comes into question, so that the problem in this form has no determinate answer. It is shown by simple mechanical considerations how the concave form arises.