## Square-to-Circle

A square obviously has curvature gradients, with high curvature at the corners (with the corners thought of as being suitably smoothed so that a curvature is analytically definable), and zero curvature along the faces. The equilibrium shape for such an object, when the surface tension is isotropic, is a circle. Fig. 5a shows a digital square, 47 pixels on a side, with the solid and air surfaces highlighted. Fig. 5b shows the resulting circle after 300 cycles, with 1 pixel being moved each cycle, and the curvature measured with a b=13 box. Sintering is nearly complete, and equilibrium is nearly achieved. Fig. 5c shows the near-equilibrium shape obtained after 200 cycles using a b=3 box. Clearly the b=3 box is too small, since the shape shown has equal curvatures as measured by this box. The resolution of the b=3 box is not large enough to distinguish the true differences in curvature that still exist at this point.

Figure 5a: Pictures of the square-to-circle problem: (a) beginning square, 47 pixels on a side, with surface highlighted.

Figure 5b: Pictures of the square-to-circle problem: (b) sintered result after 300 cycles with a b=13 box.

Figure 5c: Pictures of the square-to-circle problem: (b) sintered result after 200 cycles with a b=3 box.

Fig. 6 shows the standard deviation S of the radial vector for the sintering square, defined in the following way. The origin is taken at the center of mass of the object, and a vector ri is drawn from the origin to the center of a solid surface pixel. The standard deviation S of the magnitude of r is calculated over the entire solid surface:

where the lower case s on the summation symbol indicates a summation over the solid surface. For a perfect continuum circle, S = 0. For a perfect digital circle, S has a finite value, due to the discrete nature of the surface. The value of S for a perfect digital circle of area π r2 47 x 47, equal to the original square, is 0.24, and is shown as a dashed line in Fig. 6. The abscissa, N, is the total number of pixels moved, and is equal to n, the number of pixels moved per cycle, times the total number of cycles. In this case n = 1. The different curves for the different values of b are easily explainable in analogy to Fig. 5c. The higher the value of b, the further sintering can proceed, as smaller and smaller curvature differences can be resolved. The b = 13 equilibrium result is almost at the perfect digital circle equilibrium shape indicated by the dashed line. However, it should be noted that if b is too large, some non-local effects are measured. That is, if the curvature changes rapidly on the scale of b, the surface dynamics at a point b may reflect the curvature from regions spatially removed from the point in question. In most cases, regions of rapidly changing curvature will disappear quickly enough so that these effects do not matter. Of course, increasing the resolution of the model by increasing the total number of pixels will help alleviate these problems, at the expense of computational speed.

Figure 6: Showing the standard deviation of the length of the radius vector for the square-to-circle problem shown in Fig. 5, after beign averaged around the perimeter of the sintered object. The origin is taken at the center-of-mass and the radius vector is drawn to the center of each solid surface pixel. Results are shown for various values of b. The dashed line is the result for a perfect 53-pixel diameter digital circle of equal area to the original 47 x 47 pixel square.

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