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Classification of isolated singularities |
| Here is a list of
canonical models from the classification of isolated singularities. |
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Classification of Steiner surfaces |
| Here is a list of
canonical forms of Steiner surfaces. |
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Barth Sextic |
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Equation:
67.77708776*x^2*y^2*z^2-27.41640789*x^4*y^2-27.41640789*x^2*z^4+10.47213596*x^4*z^2-27.41640789*y^4*z^2+10.47213596*y^4*x^2+10.47213596*y^2*z^4-4.236067978*x^4-8.472135956*x^2*y^2-8.472135956*x^2*z^2+8.472135956*x^2-4.236067978*y^4-8.472135956*y^2*z^2+8.472135956*y^2-4.236067978*z^4+8.472135956*z^2-4.236067978
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Bicorne |
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Equation:
y^2*x^2+y^2*z^2-x^4-2*x^2*z^2-z^4
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Bifolia |
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Equation:
x^4+y^4+z^4+2*x^2*y^2+2*x^2*z^2+2*y^2*z^2-3*y*x^2-3*y*z^2
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Bohemian Dome |
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Equation: -y^4-x^4-z^4+2*x^2*z^2+4*y^2-2*x^2*y^2-2*y^2*z^2
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Bohemian Star |
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Equation: y^4*x^4+2*y^6*x^2+2*y^4*x^2*z^2+y^8+2*y^6*z^2+y^4*z^4-4*y^4*x^2-4*y^4*z^2-16*x^2*z^2*y^2+16*x^2*z^2
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Boy |
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Equation: -2749.231165*x^3*z*y^2-1832.820776*y*z^2*x^2+648*z*x^2*y^2-1620*z^2*x^2*y^2+1832.820776*y*z^3*x^2-4123.846747*y^4*x*z+916.4103882*z^3*x*y^2-288*z^4+64*z^3+432*z^5-216*z^6-729*x^6-729*y^6-144*z^2*x^2-144*z^2*y^2+324*z^4*x^2+324*z^4*y^2+324*z*x^4+324*z*y^4+610.9402588*y^3*z^2-810*z^2*x^4-810*z^2*y^4-610.9402588*y^3*z^3-2187*x^4*y^2-2187*x^2*y^4+1374.615582*x^5*z-305.4701294*z^3*x^3
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C8 surface |
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Surface of degree 8, with 168 isolated singular points.
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Equation:
64*x^8-128*x^6+80*x^4-16*x^2+2+64*y^8-128*y^6+80*y^4-16*y^2+64*z^8-128*z^6+80*z^4-16*z^2
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Cayley 1 |
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Equation: 2*x^3-6*x*y^2-1.0*z*x^2-1.0*z*y^2+3*x^2+3*y^2+z^3-0.33333333*z^2+z-1
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Cayley 2 |
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Equation:
-1.6*x^3+4.8*x*y^2+z^3+.8*z^2
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Cayley 3 |
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Equation:
x^2+y^2+z^3+3.2*x^3-9.6*x*y^2
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Cayley 4 |
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Equation:
25*z^3+16*z*y^2+60*x^2*y+50*z^2
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Cayley Cubic |
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Equation:
4*x^2+4*y^2+4*z^2+16*x*y*z-1
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Cassini |
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Equation:
x^4+y^4-z^4-0.5*x^2+0.5*y^2+2*x^2*y^2+0.0625
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Chair |
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This surface looks like a chair, by a modern style designer.
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Equation: 10*x^4-12*x^2*y^2+36*x^2*z^2-75*x^2+y^4+36*y^2*z^2-75*y^2+2*z^4-75*z^2+640.625-16*y^2*z+16*x^2*z
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Chubs |
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Equation:
x^4+y^4+z^4-x^2-y^2-z^2+0.5
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Clebsh cubic |
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Equation:
81*x^3+81*y^3+81*z^3-189*x^2*y-189*x^2*z-189*x*y^2-189*x*z^2-189*y^2*z-189*y*z^2+54*x*y*z+126*x*y+126*x*z+126*y*z-9*x^2-9*y^2-9*z^2-9*x-9*y-9*z+1
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Diagonal surface of Clebsh |
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Equation:
16*x^3+16*y^3-31*z^3+24*x^2*z-48*x^2*y-48*x*y^2+24*y^2*z-93.5307*z^2-72*z
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Cross cap propeller |
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Equation:
x*y*z+0.1*x^2+0.1*y^2
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Cubic27L |
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Equation:
9*y^2*z-10*y*x*z-2*y*z^2+3*x^2*z-25*y^2+30*y*x-9*x^2+2*x*z+10*y-6*x
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Cushion |
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Equation:
z^2*x^2-z^4+2*z^3+x^2-2*z^2-x^4-y^4-2*x^2*y^2-y^2*z^2+2*y^2*z+y^2
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Dervish |
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Equation:
a*F+q=0
F=h1*h2*h3*h4*h5
h1=x-z
h2=cos(2*Pi/5)*x-sin(2*Pi/5)*y-z
h3=cos(4*Pi/5)*x-sin(4*Pi/5)*y-z
h4=cos(6*Pi/5)*x-sin(6*Pi/5)*y-z
h5=cos(8*Pi/5)*x-sin(8*Pi/5)*y-z
q=(1-c*z)*(x^2+y^2-1+r*z^2)^2
r=(1+3*sqrt(5))/4
a=-(8/5)*(1+1/sqrt(5))*sqrt(5-sqrt(5))
c=sqrt(5-sqrt(5))/2
Pi= 3.141592653589793
sqrt(5)= (2.23606797749979)
sqrt(5-sqrt(5))=1.662507751109814
-------------------------
a = ( -(8/5)*(1+1/(2.23606797749979)) * 1.662507751109814 )
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Devil |
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Equation:
x^4+2*x^2*z^2-0.36*x^2-y^4+0.25*y^2+z^4
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DevTgt |
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Equation:
3*x^2*y^2-4*y^3-4*x^3*z+6*x*y*z-z^2
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Ding dong |
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Equation:
x^2+y^2-(1-z)z^2
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Duplin Cycles |
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Equation:
2+128*x+290.24*y^2+8*z^2-51.1119*x^4+8*x^2-51.1119*y^4-51.1119*z^4
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Endrass Octic |
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Equation:
64(x^2-w^2)(y^2-w^2)((x+y)^2-2 w^2)((x-y)^2-2 w^2)
-( a (x^2+y^2)^2
+(b z^2+c w^2)(x^2+y^2)
-16 z^4
+d z^2 w^2
+f w^4
)^2 = 0 w = 1
a = -4(1+sqrt(2)) b = 8(2+sqrt(2))
c = 2(2+7sqrt(2)) d = 8(1-2sqrt(2))
f = -(1+12sqrt(2))
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Folium surface |
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Equation:
y^2-3*x*y^2+z^2-3*x*z^2+2*x^2
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Glob teardrop |
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Equation:
0.5*x^5+0.5*x^4-y^2-z^2
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Gumdrop torus |
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Equation:
4*x^4+4*y^4+8*y^2*z^2+4*z^4+17*x^2*y^2+17*x^2*z^2-20*x^2-20*y^2-20*z^2+17
-3.2 -3.2 -3.2 3.1 3.1 3.1 0.2 0.1 0.9 1 0.003 0.003 0.02
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Heart |
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Equation:
-1-12*z^2*x^2-6*y^2*z^2-12*x^2*y^2+12*x^2*y^2*z^2-3*z^4+6*x^2+3*z^2-3*y^4+3*y^2-12*x^4+12*x^4*y^2+12*x^4*z^2+6*x^2*y^4+6*x^2*z^4+3*y^4*z^2+3*y^2*z^4+8*x^6+y^6+z^6-1/10*x^2*z^3-y^2*z^3
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Horned Cube |
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Equation:
-3*x^8-3*y^8-2*z^8+5*x^4*y^2*z^2+3*x^2*y^4*z^2+1
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Hunt |
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Equation:
1380*y^2+4620*z^2+87*x^4+84*x^2-129*y^4+276*z^4-150*x^2*y^2-960*x^2*z^2-528*y^2*z^2+12*x^4*y^2+12*x^2*z^4+12*x^4*z^2+12*y^4*z^2+12*y^4*x^2+12*y^2*z^4+24*x^2*y^2*z^2+4*x^6+4*y^6+4*z^6-4900
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Hyberboloid one sheet |
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Equation:
x^2-y^2-z^2+0.1
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Hyberboloid two sheet |
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Equation:
x^2-y^2-z^2-0.1
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Indented Octohedron |
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Equation:
a*(x^8+y^8+z^8)+b*(x^6+y^6+z^6)
+c*(x^4+y^4+z^4)+d*(x^2+y^2+z^2)
+e*(x^4*y^2+x^2*y^4+x^4*z^2+x^2*z^4+y^4*z^2+y^2*z^4)
+g(x^2 y^2 + x^2 z^2 + y^2 z^2)
+h+i(x^2 y^2 z^2)
b = -183/16*a h = 54*a e = -297/16*a c = 375/8*a
d = -82*a g = 297/4*a
a = 1 i = -400
#h = -6*a-2*b c = -6*a-3*b d = 8*a+3*b e = 0 g = 0
#a = 1 b = -1 i = 0
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Intersection style cross cap |
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Equation:
x^2*y-z^5-z^2
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KampyleOfEudoxus |
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Equation:
y^2+z^2-x^4+0.04*x^2
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Kiss |
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Equation:
0.5*x^5+0.5*x^4-y^2-z^2
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Klein bottle |
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Equation:
-1-4*y*z^2*x^2-2*y+6*z^2*x^2*y^2-16*x*z+16*x*z*y^2+3*x^2+7*y^2+11*z^2-11*z^4+z^6-3*x^4-7*y^4+x^6+y^6-14*z^2*x^2-18*z^2*y^2+3*z^4*x^2+3*z^4*y^2-10*x^2*y^2-4*y^3*z^2+3*z^2*x^4+3*z^2*y^4+16*x^3*z+3*x^4*y^2+3*x^2*y^4+4*x^2*y-12*z^2*y-2*x^4*y-4*x^2*y^3-2*z^4*y+16*x*z^3+12*y^3-2*y^5-32*x*z*y
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Kusner-Schmitt |
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Equation:
x^2*y^2*z^2+3*x^2*y^2+3*x^2*z^2+9*x^2+3*y^2*z^2+9*y^2+9*z^2-32*x*y*z-5
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Kusner Schmitt rate |
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Equation:
x^2*y^2*z^2+3*x^2*y^2+3*x^2*z^2+9*x^2+3*y^2*z^2+9*y^2-32*x*y*z-5
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Lemniscate of Gerono |
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Equation:
x^4-x^2+y^2+z^2
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Miter |
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Equation:
4*x^4+y^4+4*x^2*y^2+4*x^2*z^2+y^2*z^2-y^2
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Odd |
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Equation:
x^2*z^2-z^4+2*z^3+x^2-2*z^2-x^4-y^4-2*x^2*y^2-y^2*z^2+2*y^2*z+y^2
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ouaw |
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Equation:
x^2*y^2-x*z^5-z^2
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Parabolic Torus |
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Equation:
x^4+2*x^2*y^2-2*x^2*z-.61*x^2+y^4-2*y^2*z+.11*y^2+z^2+.61*z+.0121
-3.1 -3.1 -3.1 3.2 3.2 3.2 0.1 0.2 0.9 1 0.003 0.003 0.02
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Piriform teardrop |
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Equation:
x^4-x^3+y^2+z^2
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Q1 |
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Equation:
x^4+y^4+z^4-4*x^2-4*y^2*z^2-4*y^2-4*z^2*x^2-4*z^2-4*x^2*y^2+20.7846*x*y*z+1
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Q2 |
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Equation:
-x^4+y^4-z^4+4.6666666666*x^2+4.6666666666*y^2*z^2-8.3366*y^2+8.3366*z^2*x^2+4.6666666666*z^2+4.6666666666*x^2*y^2-25.2673*x*y*z+1
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Q3 |
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Equation:
5.229914547374508*y^2*z^2+3.597883597883598*x^2*y^2+y^4+z^4-x^4-19.49816368932737*x*y*z+5.229914547374508*x^2-7.43880040039534*y^2-3.597883597883598*z^2+7.43880040039534*z^2*x^2-1
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Quadrifolia |
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Equation:
x^4+y^4+z^4+16*x^2*y^2+16*x^2*z^2+16*y^2*z^2-20*y*x^2-20*y*z^2
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Quartic Cylinder |
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Equation:
y^2*x^2+y^2*z^2+0.01*x^2+0.01*z^2-0.01
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S20 |
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Equation:
y^3+y^2*z+y*x^2-2*x*y-z*x^2
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S21 |
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Equation:
y^3+y^2*z+y*x^2-2*x*y+z*x^2+x*y*z
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S22 |
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Equation:
y^3+4*x^2*z+4*y*x
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S23 |
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Equation:
y^3+x^2+x*y*z
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S26 |
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Equation:
x^4+3*x^2*y^2+x^2*z^2-4*x^2*z+2y^4+2*y^2*z^2-4*y^2*z
-2.1 -2.1 -2.1 5.3 5.3 5.3
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Sarti octic 144 |
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Equation:
-1728*x^4*y^2*z^2+928.0*z^4*x^4+9024.0*z^2*x^4+928.0*z^4*y^4+9024.0*z^2*y^4+9024.0*x^2*y^2+2720.0*x^4*y^4-1728*x^2*z^2-1728*y^2*z^2-1728*x^4*y^2-1728*x^2*y^4-1728*x^2*z^4-1728*y^2*z^4-576*x^6*y^2-576*x^6*z^2-576*x^2*y^6-576*x^2*z^6-576*y^6*z^2+9024.0*z^4*x^2*y^2-24960.0*z^2*x^2*y^2-1728*x^2*y^4*z^2+2720.0*z^4+112.0*z^8-576*z^2+928.0*x^4+112.0*x^8+928.0*y^4+112.0*y^8-576*x^2-576*y^2-576*x^6-576*y^6-576*z^6+112.0-576*y^2*z^6
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Sarti octic 72 |
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Equation:
3584.0*z^4+256.0*z^8+1792.0*z^4*x^4+10752.0*z^2*x^4+1792.0*x^4+256.0*x^8+256.0+1792.0*z^4*y^4+10752.0*z^2*y^4+1792.0*y^4+256.0*y^8+10752.0*z^4*x^2*y^2-21504.0*z^2*x^2*y^2+10752.0*x^2*y^2+3584.0*x^4*y^4-192-192*12*x^4*y^2*z^2-192*24*x^2*y^2*z^2-192*12*x^2*y^2-192*12*x^2*z^2-192*12*y^2*z^2-192*12*x^4*y^2-192*12*x^4*z^2-192*12*x^2*y^4-192*12*x^2*z^4-192*12*y^4*z^2-192*12*y^2*z^4-192*4*x^6*y^2-192*4*x^6*z^2-192*6*x^4*y^4-192*6*x^4*z^4-192*4*x^2*y^6-192*4*x^2*z^6-192*4*y^6*z^2-192*6*y^4*z^4-192*4*y^2*z^6-192*12*x^2*y^4*z^2-192*12*x^2*y^2*z^4-192*4*x^2-192*4*y^2-192*4*z^2-192*6*x^4-192*6*y^4-192*6*z^4-192*4*x^6-192*4*y^6-192*4*z^6-192*x^8-192*y^8-192*z^8
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Shallowtail |
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Equation:
-4*z^3*y^2-27*y^4+16*x*z^4-128*x^2*z^2+144*x*y^2*z+256*x^3
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Pillow Tooth |
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Equation:
x^4+y^4+z^4-x^2-y^2-z^2
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Sphere |
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Equation:
x^2+y^2+z^2-1
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Classification of Steiner surfaces |
| Here is a list of
canonical forms of Steiner surfaces. |
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Strophoid |
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Equation:
.1*y^2+.1*z^2-x*y^2-x*z^2-.16*x^2-.16*x^3.1*y^2+.1*z^2-x*y^2-x*z^2-.16*x^2-.16*x^3.1*y^2+.1*z^2-x*y^2-x*z^2-.16*x^2-.16*x^3
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TangleCube |
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Equation:
x^4-5*x^2+y^4-5*y^2+z^4-5*z^2+11.8
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Tetrahedral |
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Equation:
x^4+2*x^2*y^2+2*x^2*z^2+y^4+2*y^2*z^2+z^4+8*x*y*z-10*x^2-10*y^2-10*z^2+25
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Torus |
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Equation:
x^4+y^4+z^4+2*x^2*y^2+2*x^2*z^2+2*y^2*z^2-1.06*x^2+0.9*y^2-1.06*z^2+0.2025
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Trifolia |
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Equation:
x^3+y^3+z^3+13*x^3*y^3+13*x^3*z^3+13*y^3*z^3-20*y*x^2-20*y*z^2-20*x*z^2
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Triple plan |
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Equation:
x*y*z
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Tritrumpet |
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Equation:
8*z^2+6*x*y^2-2*x^3+3*x^2+3*y^2-1
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Tubey |
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Equation:
-2+4*y+4*x+12*x*y^2*z+12*x*y*z^2+5*x^4*y^2*z^2+12*x*y+12*x*z+6*y^2+12*y*z+6*z^2+12*x^2*y+12*x^2*z+12*x*y^2+12*x*z^2+12*y^2*z+12*y*z^2+3*x^2*y^4*z2+4*z+24*x*y*z-3*x^8-3*y^8-2*z^8+x^4+6*x^2+y^4+z^4+12*x^2*y*z+4*x^3*y+4*x^3*z+6*x^2*y^2+6*x^2*z^2+4*x*y^3+4*x*z^3+4*y^3*z+6*y^2*z^2+4*y*z^3
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Whitney umbrella |
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Surface corresponding to a so-called Whitney singularity, which contains the
line (x=0,y=0) as singular locus.
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Equation: x^2*z-y^2
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