## Appendix A Details of Equation-Based Point-Clouds  
Let \(P_{0}\) = (\(P_{0x}\), \(P_{0y}\)) and \(P_{1}\) = (\(P_{1x}\), \(P_{1y}\)) be the starting and end points of a straight-line in the \(x\)-\(y\) plane. Let \(\forall t\in[0,\,1]\) be the parameter. An arbitrary point, \(P(t)\) = (\(P_{x}(t)\), \(P_{y}(t)\)) between \(P_{0}\) and \(P_{1}\) is given as follows:  
\[P_{x}(t)=P_{0x}(1-t)+P_{1x}(t)\qquad P_{y}(t)=P_{0y}(1-t)+P_{1y}(t) \tag{1}\]  
While creating a point-cloud using Equation (1), the parameter is varied as follows: \(t\) = 0, \(\Delta t\), 2\(\Delta t\),..., 1. Thus, the value of \(\Delta t\) decides the number of points in the point-cloud. In Figure 4, the point-cloud representing a straight-line is created using the abovementioned formulation.  
A straight-line given by the point-cloud defined by Equation (1) can be used as an edge of the face of a shape (e.g., a hexagon). In this case, one can rotate the point-cloud around the center of the shape to create other edges, as required. To be more specific, let \(P_{r}\) = (\(P_{rx}\), \(P_{ry}\)) be the center of the shape and \(P^{\prime}(t)\) = (\(P^{\prime}_{x}(t)\), \(P^{\prime}_{y}(t)\)) be the position of the point, \(P(t)\) = (\(P_{x}(t)\), \(P_{y}(t)\)) after its rotation around \(P_{r}\) = (\(P_{rx}\), \(P_{ry}\)) in the counter-clockwise direction by an angle \(\theta\). Thus, the following formulation holds:  
\[P^{\prime}_{x}(t)=P_{rx}+(P_{x}(t)-P_{rx})\cos\theta-\left(P_{y}(t)-P_{ry} \right)\sin\theta \tag{2}\]  
\[P^{\prime}_{y}(t)=P_{ry}+(P_{x}(t)-P_{rx})\sin\theta+\left(P_{y}(t)-P_{ry} \right)\cos\theta \tag{3}\]  
The formulation defined by Equations (2) and (3) is used five times to create the five point-clouds of the five edges from the point-cloud of the first edge for \(\theta\) = \(\pi/3\), 2\(\pi/3\), \(\pi\), 4\(\pi/3\), and 5\(\pi/3\). This way one can create the face of any shape in terms of point-clouds.  
Let \(P_{c}\) = (\(P_{cx}\), \(P_{cy}\)) be a center of ellipse in the \(x\)-\(y\) plane. Let \(\forall t\in[0,1]\) be the parameter. Consider, \(a\) and \(b\) are major and minor radius of the ellipse, respectively. For ellipse, \(a\), \(b\) > 0 and \(a\) > \(b\). An arbitrary point, \(P(t)\) = (\(P_{x}(t)\), \(P_{y}(t)\)) is given by  
\[P_{x}(t)=P_{cx}+a\cos(2\pi t)\ P_{y}(t)=P_{cy}+b\cos(2\pi t) \tag{4}\]  
While creating a point-cloud using Equation (4), the parameter is varied as follows: \(t\) = \(-1\), \(-1\) + \(\Delta t\), \(-1\) + 2\(\Delta t\),..., 1. Thus, the value of \(\Delta t\) decides the number of points in the point-cloud. In Figure 4, the point-cloud representing the ellipse is created using the abovementioned formulation.  
Let \(P(t)\) = (\(P_{x}(t)\), \(P_{y}(t)\)) be an arbitrary point in the \(x\)-\(y\) plane. Let \(\forall t\in[-1,\,1]\) be the parameter. A curve called Bicorn curve is given as follows:  
\[P_{x}(t)=\sin(\pi t)\qquad\quad P_{y}(t)=\frac{\cos^{2}(\pi t)(2+\cos(\pi t))} {3+\sin^{2}(\pi t)} \tag{5}\]While creating a point-cloud using Equation (A5), the parameter is varied as follows: \(t=-1\), \(-1+\Delta t\), \(-1+2\Delta t\),..., 1. Thus, the value of \(\Delta t\) decides the number of points in the point-cloud. In Figure 4, the point-cloud representing the Bicorn curve is created using the abovementioned formulation.