Geisha

silviana amethyst

Equation ¶

\[ x^2yz+x^2z^2 = y^3z+y^3 \]

Properties: ¶

degree 4
two singular curves, both lines. one is a cusp, the other is a self-crossing. at least in terms of the real part…
two singular points, one at the intersection of the two singular lines – at the origin. the other is at \( (0,0,-1) \), and is a cusp singularity, lying only on the singular \(z\)-axis

Some story ¶

The support came off quite nicely, except where the two separate bodies join, at \( (0,0,-1) \), where the printed surface broke. Not surprising. PLA is a brittle plastic, so even a little extra force while removing support is likely to break these point singularities. So I hot-glued it back together.

Sphere of radius 5, random projection.

Bertini_real input files ¶

input

% surface 'geisha'
% http://homepage.univie.ac.at/herwig.hauser/bildergalerie/gallery.html
% http://homepage.univie.ac.at/herwig.hauser/bildergalerie/gallery/geisha.jpg
% a degree 4 surface in three variables.
CONFIG
tracktype: 1;
securitylevel:1
sharpendigits: 30;
condnumthreshold: 1e300;
odepredictor: 7;
finaltol: 1e-16;
numsamplepoints: 6;
endgamenum: 2;
maxstepsbeforenewton: 0;
maxnewtonits: 1;
END;
INPUT

variable_group x, y, z;
function f;
f = x^2*y*z + x^2*z^2 -(y^3*z + y^3);
END;