Calyx

silviana amethyst

Equation ¶

\[ x^2+y^2 z^3 - z^4 \]

Properties: ¶

unbounded
degree 5
one singular curve, of degree 1, containing a single singular point.

Some story ¶

I decomposed this surface using projection onto \(x\) and \(y\). I attempted to match the bounding sphere used on Herwig Hauser’s gallery, so I used a sphere centered at the origin, with radius 10.

Initial decomposition efforts using a random projection and the default sphere largely failed. Bertini_real could be improved in terms of its ability to identify two points as the same. So, I switched to the x-y projection used in these models and pictures.

Bertini_real input files ¶

input

CONFIG
randomseed: 2;
tracktype: 1;
finaltol: 1e-15;
ampmaxprec: 3096;
ampsafetydigits1: 30;
ampsafetydigits2: 30;
maxnumbersteps: 50000;
pathtruncationthreshold: 1e30;
endpointfinitethreshold: 1e12;
sharpendigits: 60;
endgamenum: 2;
numsamplepoints: 20;
odepredictor: 2;
condnumthreshold: 1e300;
tracktolbeforeeg: 1e-8;
maxnewtonits: 2;
maxstepsbeforenewton: 0;
securitylevel: 1;
imagthreshold: 1e-7;
END;
INPUT

variable_group x, y, z;
function f1;
f1 = x^2+y^2*z^3 - z^4;
END;

projection

sphere