OpenSCAD "polygircle" based on superellipse / superformula equation

polygircle.scad

// Number of sides
m = 3; // [1:1:8]
// Pointedness
p = 1;  // [0:0.1:4]
// Inradius
r = 1;

// Larger m requires larger n1 for comparable pointedness
// scaling with m squared seems about right
// p=1 corresponds with canonical squircle (superellipse with n=4)
n1 = p/4*m*m; 

// Ratios determined to give the flattest edges for a given m value
n_ratio = [16.01, 4, 1.779, 1, 0.632, 0.432, 0.311, 0.233][m-1];
// If ratio is too large, the x coordinate can exceed the inradius then dip inward to form "dimples".
// Criteria for table values was the largest ratio which never exceeded r by some epsilon
// Tested with: p=40, $fn=8000, r=1 (high "pointedness", outside of typical range, makes dimples more apparent)

// https://en.wikipedia.org/wiki/Superformula
function superformula(m=4,n1=4,n2=4,n3=1,a=1,b=1) = [for(i=[0:1:$fn-1])  
  let(th = i*360/$fn, R = (abs(cos(m/4*th)/a)^n2+abs(sin(m/4*th)/b)^n3)^(-1/n1)) 
  //echo(th, R*cos(th)) 
  //assert((th>20 && th<340) || R*cos(th)<=1.000001) // used to find n_ratio's
  R*[cos(th), sin(th)]
];

// subset of superformula, where a=b, and n2=n3
function polygircle(m=4,n1=4,n=4,r=1) = superformula(m,n1,n,n,r,r);

polygon(polygircle(m=m,n1=n1,n=n1*n_ratio,r,$fn=200));