28 Feb
2007
28 Feb
'07
9:42 p.m.
Extending my earlier post... Standardizing so that the diameter is the unit interval, graph the circle y with y = sqrt(x(1-x)) Scale by c to get an ellipse y[c] with "circularity" c y[c](x) := c sqrt(x(1-x)) where c=1 is circular, c=0 is a line. This seems a natural parameterization since interpolating between the graphs of two ellipses is equivalent to interpolating their circularities: (1-x)y[c0](x) + x y[c1](x) = y[(1-x)c0 + x c1](x) Then without loss of generality we can set c0=1 to get a standard oval z[c] z[c](x) := (1 + x(c-1)) sqrt(x(1-x)) which is circular with c=1, a teardrop with c=0 and ovaltine for c between. The value c=0.5 is reasonably eggy; c=0.618... outputs an auric anser.