Algebraic cylinders
Equations in three space (x,y,z) which are independent of one dimension (here z).
x^3 - x^2 +y^2 = 0
2*x^4 -3*x^2*y +y^2 -2*y^3 +y^4 = 0
x^4 +x^2*y^2 -2*x^2*y -x*y^2 +y^2 = 0
x^4 +y^4 +2*x^2*y^2 +3*x^2*y -y^3 = 0Astrodal Ellipsoid
This surface goes in where the ellipsoid goes out. Which gives the surface an overall hyperbolic look.
Parametric form:
x= pow(a*cos(u)*cos(v),3)
y= pow(b*sin(u)*cos(v),3)
z= pow(c*sin(v),3)
Example:
Barth sextic
This is (the real points of a part of) the Barth-sextic, an algebraic surface in complex three-dimensional projective space with 65 double points, given by the equation:
4(t^2 x^2 - y^2)(t^2 y^2 - z^2)(t^2 z^2 - x^2) -
(1+2t)(x^2 + y^2 + z^2 - 1)^2 = 0
t = 0.5*(1 + sqrt(5))Barth decic
t= (1+Sqrt[5])/2
w=1
8 (x^2-t^4 y^2) (y^2-t^4 z^2) (z^2-t^4 x^2)*
(x^4+y^4+z^4-2 x^2 y^2-2 x^2 z^2-2 y^2 z^2)+
+(3+5t) (x^2+y^2+z^2-w^2)^2 (x^2+y^2+z^2-(2-t) w^2)^2 w^2=0
Bicorn:
This curve looks like the top part of a paraboloid, bounded from below by another paraboloid. The basic equation is:
y^2*(a^2 - (x^2 + z^2)) - (x^2 + z^2 + 2*a*y - a^2)^2 =0 example a=0
Bifolia
(x^2 + y^2 + z^2)^2 - a*(x^2 + z^2)*y =0example a=3
Bohemian Dome
Okay - take a deep breath.
If you rotate a circle which is parallell to a plane in a circle that is
perpendicular to the same plane, the resulting envelope is a Bohemian dome.
Parametric form:
x= a*cos(u)
y= b*cos(v) + a*sin(u)
z= c*sin(v)
Example:
Boy surface
Model of the projective plane without singularities. Found by Werner Boy on assignment from David Hilbert. Polynomial by Francois Apery.
Parametric equation:
x =(2/3)*(cos(u)*cos(2*v)+sqrt(2)*sin(u)*cos(v))*cos(u) /(sqrt(2) - sin(2*u)*sin(3*v))
y =(2/3)*(cos(u)*sin(2*v)-sqrt(2)*sin(u)*sin(v))*cos(u) /(sqrt(2)-sin(2*u)*sin(3*v))
z =sqrt(2)*cos(u)^2 / (sqrt(2) - sin(2*u)*sin(2*v))Polynomial:
64*(1-z)^3*z^3-48*(1-z)^2*z^2*(3*x^2+3*y^2+2*z^2)+
12*(1-z)*z*(27*(x^2+y^2)^2-24*z^2*(x^2+y^2)+
36*sqrt(2)*y*z*(y^2-3*x^2)+4*z^4)+
(9*x^2+9*y^2-2*z^2)*(-81*(x^2+y^2)^2-72*z^2*(x^2+y^2)+
108*sqrt(2)*x*z*(x^2-3*y^2)+4*z^4)=0Cassini ovals
Locus of points whose products of distances to two fixed points is a constant. Also cross-sections of circular torus. Giovanni Domenico Cassini.
Rotated around the y axis
(x^2 + y^2 + z^2 + a^2)^2 - c*a^2*(x^2 + z^2) - b^2 )=0
Example: a=0.45 , b=0.5 , c=16Cayley cubic
-5(x^2*y+x^2*z+y^2*x+y^2*z+z^2*y+z^2*x)+
2*(x*y+x*z+y*z)=0Chair
A tetrahedral surface looking like an inflateable chair from the 70's.
Implicit form:
(x^2+y^2+z^2-a*k^2)^2-b*((z-k)^2-2*x^2)*((z+k)^2-2*y^2)=0
with k=5, a=0.95 and b=0.8.
Clebsch diagonal cubic
with 10 eckhard points (3 lines in a tritangent meeting)
81*(x^3+y^3+z^3)-189*(x^2*y+x^2*z+y^2*x+y^2*z+z^2*x+z^2*y)+ 54*(x*y*z)+126*(x*y+x*z+y*z)-9*(x^2+y^2+z^2)-9*(x+y+z)+1=0
Costa Minimal Surface
The famous Costa-Hoffman-Meeks minimal surface. I will only include the Mathematica parametrization due to Alfred Gray et al. Look elsewhere on my homepage for a PoV parametrization with only "elementary" functions
c=189.07272;
e1=6.87519;
costa1[u_,v_]:= (1/2) Re[-WeierstrassZeta[u+I v,{c,0}] +Pi u +
Pi^2/(4 e1)+(Pi/(2 e1)) (WeierstrassZeta[u+I v-1/2,{c,0}]-
WeierstrassZeta[u+I v-I/2,{c,0}])]
costa2[u_,v_]:= (1/2) Re[-I*WeierstrassZeta[u+I v,{c,0}] +Pi v +
Pi^2/(4 e1)-(Pi/(2 e1))*(I*WeierstrassZeta[u+I v-1/2,{c,0}]-
I*WeierstrassZeta[u+I v-I/2,{c,0}])]
costa3[u_,v_]:=(Sqrt[2 Pi]/4) Log[Abs[(WeierstrassP[u+I v,{c,0}]-e1)/
(WeierstrassP[u+I v,{c,0}]+e1)]]
costa[u_,v_]:={costa1[u,v],costa2[u,v],costa3[u,v]}
costaplot80=ParametricPlot3D[costa[u,v],{u,0.001,1.001},
{v,0.001,1.001},PlotPoints->80]
selectgraphics3d[graphics3dobj_,bound_,opts___]:=
Show[Graphics3D[Select[graphics3dobj,
(Abs[#[[1,1,1]]] < bound && Abs[#[[1,1,2]]] < bound &&
Abs[#[[1,1,3]]] < bound && Abs[#[[1,2,1]]] < bound &&
Abs[#[[1,2,2]]] < bound && Abs[#[[1,2,3]]] < bound &&
Abs[#[[1,3,1]]] < bound && Abs[#[[1,3,2]]] < bound &&
Abs[#[[1,3,3]]] < bound && Abs[#[[1,4,1]]] < bound &&
Abs[#[[1,4,2]]] < bound && Abs[#[[1,4,2]]] < bound
)&]],opts]
dip[ins_][g_]:=$DisplayFunction[Insert[g,ins,{1,1}]]
selectgraphics3d[costaplot80[[1]],8,
Boxed->False,ViewPoint->{2.9,-1.4,1.2},
PlotRange->{{-4,4},{-4,4},{-2,2}},
DisplayFunction->dip[EdgeForm[]]]
Crossed Trough
This is a surface with four pieces that sweep up from the x-z plane.
x^2*z^2 - y = 0 Cubic saddle.
x^3 - y^3 - z = 0Cushion
From siggraph
z^2*x^2 - z^4 - 2*z*x^2 + 2*z^3 + x^2 - z^2
-(x^2 - z)^2 - y^4 - 2*x^2*y^2 - y^2*z^2 + 2*y^2*z + y^2 =0 Dervish.
This surface reminded me of the Whirling Dervishes. Hands out, and skirt billowing.
Implicit form:
a*F+q=0
with:
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
Devil's curve in 3-space variant.
x^4 + 2*x^2*z^2 - 0.36*x^2 - y^4 + 0.25*y^2 + z^4 = 02D :
x^2 * (x^2 - a^2) - y^2 * (y^2 - b^2)=0Dini's surface
Dini's surface of constant negative curvature. Looks like a flower
.
x=a*cos(u)*sin(v)
y=a*sin(u)*sin(v)
z=a*(cos(v)+log(tan((v/2))))+b*uExample:
a=1,b=0.2,u={ 0,4*pi},v={0.001,2}
Dupin Cyclid
Envelope of spheres kissing three other spheres. Also envelope of spheres with centres on a conic and touching a sphere. Every Dupin Cyclid is the inverse of a Torus. This formula is the inversion of a torus in the x-z plane, with the origin as inversion center.
(r1^2 - dy^2 - (dx + r0)^2)*
(r1^2 - dy^2 - (dx - r0)^2)*
(x^4+y^4+z^4)+
2*((r1^2 - dy^2 - (dx + r0)^2 )*
(r1^2 - dy^2 - (dx - r0)^2)*
(x^2*y^2+x^2*z^2+y^2*z^2))+
2*ri^2*((-dy^2-dx^2+r1^2+r0^2)*
(2*x*dx+2*y*dy-ri^2)-4*dy*r0^2*y)*
(x^2+y^2+z^2)+
4*ri^4(dx*x+dy*y)*(-ri^2+dy*y+dx*x)+
4*ri^4*r0^2*y^2+ri^8=0
r0=Major radius of torus.
r1=Minor radius of torus.
dx,dy=Torus displacement.
ri=Inversion radius.
Example r0=4.9,r1=5,dx=2,dy=0,ri=3 (double crescent)
or r0=3,r1=5,dx=3,dy=0,ri=9 (degenerate w. arch)
or r0=6,r1=0.5,dx=3,dy=0,ri=12 (plain)
Ennepers surface.
This one is a classic. In POV format it's a 9. degree polynomial.
6*((y^2-x^2)/(4*z)-(1/4)*(x^2+y^
((y^2-x^2)/(2*z)+2*z^2/9+2/3)^3-2+(8/9)*z^2)+2/9)^2=0Parametric
x=u-(u*u*u/3)+u*v*v
y=v-(v*v*v/3)+u*u*v
z=u*u-v*v Try u={-2,2},v={-2,2}
Folium surface
(y^2 + z^2) * (1 + (b - 4*a)*x) + x^2*(1 + b) = 0
Example a=1,b=1
2D :
y^2 * (1 + (b - 4*a)*x) + x^2*(1 + b) = 0
Glob
Sort of like basic teardrop shape.
0.5*x^5 + 0.5*x^4 - (y^2 + z^2) = 0
Heart
Looks like a heart
(2*x^2+y^2+z^2-1)^3-(1/10)*x^2*z^3-y^2*z^3 = 0
Hunt surface
4*(x^2+y^2+z^2-13)^3 + 27*(3*x^2+y^2-4*z^2-12)^2 = 0
Hyperbolic torus
x^4 + 2*x^2*y^2 - 2*x^2*z^2 - 2*(r0^2+r1^2)*x^2 + y^4 -
2*y^2*z^2 + 2*(r0^2-r1^2)*y^2 + z^4 + 2*(r0^2+r1^2)*z^2 +
(r0^2-r1^2)^2 = 0Example r0= 0.6, r1= 0.4
Kampyle of Eudoxus
(y^2 + z^2) - c^2 * x^4 + c^2 * a^2 * x^2 =0
Example a=0.2, c=1
2D.
y^2 - c^2 * x^4 + c^2 * a^2 * x^2 = 0
Klein Bottle
Think of it as a rectangle where one pair of opposite sides are
joined directly, and the other pair are joined with a half twist.
One sided surface.
Felix Klein.
Parametric:
x = cos(u)*(cos(u/2)*(sqrt_2+cos(v))+(sin(u/2)*sin(v)*cos(v)))
y = sin(u)*(cos(u/2)*(sqrt_2+cos(v))+(sin(u/2)*sin(v)*cos(v)))
z = -1*sin(u/2)*(sqrt_2+cos(v))+cos(u/2)*sin(v)*cos(v)Polynomial; Looks odd, from Ian Stewart.
(x^2+y^2+z^2+2*y-1)*((x^2+y^2+z^2-2*y-1)^2-8*z^2)+
16*x*z*(x^2+y^2+z^2-2*y-1)=0Knot or Threefoil knot
This is just clipped from a POV file I did to draw a knot
//various constants
uumin = 0,uumax = 4*pi
vvmin = 0,vvmax = 2*pi
a=1, b=0.3, c=0.5, d=0.3
//preliminary calculations
r=a+b*cos(1.5*uu)
xx=r*cos(uu)
yy=r*sin(uu)
zz=c*sin(1.5*uu)
dx=-1.5*b*sin(1.5*uu)*cos(uu)-(a+b*cos(1.5*uu))*sin(uu)
dy=-1.5*b*sin(1.5*uu)*sin(uu)+(a+b*cos(1.5*uu))*cos(uu)
dz=1.5*c*cos(1.5*uu) //Derivatives
qn=vnormalize() //Vector operatons
qvn=vnormalize()
ww=vcross(qn,qvn)
//points and normals
x1=xx+d*(qvn.x*cos(vv)+ww.x*sin(vv)) //Calculate the
y1=yy+d*(qvn.y*cos(vv)+ww.y*sin(vv)) //points. ww.x is the
z1=zz+d*ww.z*sin(vv) //x value of ww vector
nx1=qvn.x*cos(vv)+ww.x*sin(vv) //Normals needed to
ny1=qvn.y*cos(vv)+ww.y*sin(vv) //make smooth triangles
nz1=ww.z*sin(vv) Kuen's surface
Kuens's surface of constant negative curvature. Curvature is here the product of the extrema of the directional curvatures.
x=2*(cos(u)+u*sin(u))*sin(v)/(1+u*u*sin(v)*sin(v))
y=2*(sin(u)-u*cos(u))*sin(v)/(1+u*u*sin(v)*sin(v))
z=log(tan(v/2))+2*cos(v)/(1+u*u*sin(v)*sin(v)) Try these values.
u = {-4, 4}
v = {0.05, pi-0.05}
Kummer Surface
Radiating Rods. I haven't tested this extensively.
v1: x^4+y^4+z^4-x^2-y^2-z^2-x^2*y^2-x^2*z^2-y^2*z^2+1 = 0
v2: x^4+y^4+z^4+a*(x^2+y^2+z^2)+b*(x^2*y^2+x^2*z^2+y^2*z^2)+
c*x*y*z-1=0 Lemniscate of Gerono, or Eight Curve
This figure looks like two teardrops with their pointed ends connected. It is formed by rotating the Lemniscate of Gerono about the x-axis.
x^4 - x^2 + y^2 + z^2 = 0
2D
y^2 - c^2 * a^2 * x^2 + c^2*x^4 =0
Mitre surface
4*x^2*(x^2 + y^2 + z^2) - y^2*(1 - y^2 - z^2) = 0
Moebius strip
x=cos(u)+v*cos(u/2)*cos(u)
y=sin(u)+v*cos(u/2)*sin(u)
z=v*sin(u/2) u={0,2pi} v={-0.3,0.3}
Nodal_cubic
y^3 + z^3 - 6*y*z=0
Odd surface
z^2*x^2-z^4-2*z*x^2+2*z^3+x^2-z^2-(x^2-z)^2-y^4 -
2*y^2*x^2-y^2*z^2+2*y^2*z+y^2=0 Paraboloid
x^2 - y + z^2=0
Parabolic Torus
x^4 + 2*x^2*y^2-2*x^2*z-(r0^2+r1^2)*x^2+y^4-2*y^2*z+
(r0^2-r1^2)*y^2+z^2+(r0^2+r1^2)*z+(r0^2-r1^2)^2 = 0 Example r0 = 0.6, r1=0.5
Pillow/Tooth object
From the back cover of the 1992 Siggraph proceedings
x^4 + y^4 + z^4 - (x^2 + y^2 + z^2) = 0
Piriform
Very nice teardrop shape
(x^4 - x^3) + y^2 + z^2 = 0
2D:
y^2 - a * c^2 * x^3 - b * c^2 * x^4 = 0 b>0
Quartic paraboloid.
Looks like the quadric paraboloid, but is squared off on the bottom and sides.
x^4 + z^4 - y = 0
Quartic saddle
Looks like the quadric saddle, but is squared off in the middle. The equation is:
x^4 - z^4 - y = 0
Quartic Cylinder
(x^2 + z^2) * y^2 + c^2 * (x^2 + z^2) - c^2 * a^2 = 0
Example a=1, c=0.1
Steiners Roman surface.
Model of the projective plane Jacob Steiner with algebra by Karl Weierstrass
x^2*y^2 + x^2*z^2 + y^2*z^2 + x*y*z = 0
There are several surfaces related to this. Here are a few:
x^2*y^2-x^2*z^2+y^2*z^2-x*y*z=0
y^2-2*x*y^2-x*z^2+x^2*y^2+x^2*z^2-z^4=0
Strophoid
(b - x)*(y^2 + z^2) - c^2*a*x^2 - c^2*x^3 = 0
Example a=1, b=-0.1, c=0.4 from A. Enzmann
2D:
(b - x)*y^2 - c^2*a*x^2 - c^2*x^3 = 0
a=b gives Right Strophoid
a=3*b gives trisectrix of Maclaurin
Swallowtail
A classic curve from Catastrophy theory. Parametric form:
x= u*pow(v,2) + 3*pow(v,4)
y= -2*u*v - 4*pow(v,3)
z= u
Example:
Tangle
A curious blobby cube.
Polynomial:
x^4 - 5*x^2 + y^4 - 5*y^2 + z^4 - 5*z^2 + 11.8 = 0
Torus
x^4 + y^4 + z^4 + 2 x^2 y^2 + 2 x^2 z^2 + 2 y^2 z^2
-2 (r0^2 + r1^2) x^2 + 2 (r0^2 - r1^2) y^2
-2 (r0^2 + r1^2) z^2 + (r0^2 - r1^2)^2 = 0
r0 is the "major" radius of the torus
r1 is the "minor"
Torus - sorta, but with two gumdrop shapes near the hole
4*(x^4 + (y^2 + z^2)^2) + 17 * x^2 * (y^2 + z^2) -
20 * (x^2 + y^2 + z^2) + 17 = 0Umbrella
x^2 - y*z^2=0
Witch of Agnesi
a * (y - 1) + (x^2 + z^2) * y =0
Example a=0.04 from A. Enzmann
2D:
a^2 * y + x^2 * y - c = 0
c=a^3 gives Wich of Agnesi