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jolkaP

Joined: 17 Oct 2011
Posts: 12
Location: Marseille (France)

Posted: Thu Jan 12, 2012 11:57 pm Post subject: Surfaces from Alfred Grays "Modern Differential Geometr

Hello everybody,

There is a book called "Modern Differential Geometry of Curves and Surfaces with Mathematica" that has a lot of beautiful formulae inside. And the author put them on line, at first in his own website, now they are available on Wolfram (Mathematica's editor). The direct link to the file is http://library.wolfram.com/infocenter/Books/3759/Gray.zip?file_id=4520. Of course, they are in Mathematica syntaxe.

Here I share the isosurfaces from this source with the k3dsurf syntaxe. The morph parameter t in Scherk surfaces was added by myself.

Code:

Name: cossurface
/*(z - a*x*y)^2 - a^2(1 - x^2)(1 - y^2)=0 is the
nonparametric form of the surface (u,v)->(a*cos(u),a*cos(v),a*cos(u + v)).*/
F(): (z - x*y) ^2-(1 - x^2)*(1 - y^2)
[x]: -1 , 1
[y]: -1 , 1
[z]: -1 , 1
;
Name: crosscap(1,1)
/*(a*x^2 + b*y^2)(x^2 + y^2 + z^2) - 2*z*(x^2 + y^2) =0 is the nonparametric form of a crosscap*/
F(): (x^2+y^2)*(x^2+y^2+z^2)-2*z*(x^2+y^2)
[x]: -1 , 1
[y]: -1 , 1
[z]: -2 , 2
;
Name: ellipsoid(1,2,3)
/*(x/a)^2 + (y/b)^2 + (z/c)^2 - 1=0 is the nonparametric form of an ellipsoid.*/
F(): (x/1)^2 + (y/2)^2 + (z/3)^2 - 1
[x]: -1 , 1
[y]: -2 , 2
[z]: -3 , 3
;
Name: ellipticparaboloid(1,2,3)
/*(x/a)^2 + (y/b)^2 - c*z=0 is the nonparametric form of an elliptic paraboloid.*/
F(): x^2 + (y/2)^2 - 3*z
[x]: -6 , 6
[y]: -12 , 12
[z]: 0 , 12
;
Name: equihom1
/*xyz-1=0 is the nonparametric
form of the first equiaffinely homogeneous surface.*/
F(): x*y*z-1
[x]: -3 , 3
[y]: -3 , 3
[z]: -3 , 3
;
Name: equihom2
/*This is the nonparametric form of the second equiaffinely homogeneous surface.*/
F(): (x^2 + y^2)*z - 1
[x]: -3 , 3
[y]: -3 , 3
[z]: -3 , 3
;
Name: equihom3
/*This is the nonparametric form of the third equiaffinely homogeneous surface.*/
F(): x^2*(z - y^2)^3 - 1
[x]: -3 , 3
[y]: -2 , 2
[z]: 0 , 4
;
Name: equihom4
/*This is the nonparametric form of the fourth equiaffinely homogeneous surface.*/
F(): x^2*(z - y^2)^3 + 1
[x]: -3 , 3
[y]: -3 , 3
[z]: -5 , 3
;
Name: equihom5
/*This is the nonparametric form of the fifth equiaffinely homogeneous surface.*/
F(): z - x*y - x^3/3
[x]: -3 , 3
[y]: -3 , 3
[z]: -3 , 3
;
Name: equihom6
/*This is the nonparametric form of the sixth equiaffinely homogeneous surface.*/
F(): z - x*y + log(x)
[x]: 0.00001 , 5
[y]: -5 , 5
[z]: -5 , 5
;
Name: goursat(0.5,0.8,-1)
/* x^4 + y^4 + z^4 +a*(x^2 + y^2 + z^2)^2 + b*(x^2 + y^2 + z^2) + c=0. is the nonparametric form of Goursat's surface*/
F(): x^4 + y^4 + z^4 +0.5*(x^2 + y^2 + z^2)^2 + 0.8*(x^2 + y^2 + z^2) -1
[x]: -1 , 1
[y]: -1 , 1
[z]: -1 , 1
;
Name: hyperbolicparaboloid(1,2,3)
/*(x/a)^2 - (y/b)^2 - c*z=0 is the nonparametric form of a hyperbolic paraboloid.*/
F(): x^ 2-(y/2)^2-3*z
[x]: -3 , 3
[y]: -6 , 6
[z]: -3 , 3
;
Name: hyperboloid(1,2,3)
/*(x/a)^2 + (y/b)^2 - (z/c)^2 - 1=0 is the nonparametric form of a hyperboloid.*/
F(): x^ 2+(y/2)^2-(z/3)^2-1
[x]: -10 , 10
[y]: -10 , 10
[z]: -5 , 5
;
Name: hy2sheet(1,1,1)
/*(x/a)^2 - (y/b)^2 - (z/c)^2 - 1=0 is the nonparametric form of a hyperboloid of two sheets.*/
F(): x^2-y^2-z^2-1
[x]: -sqrt(10) , sqrt(10)
[y]: -3 , 3
[z]: -3 , 3
;
Name: hy2sheet(1,2,3)
/*(x/a)^2 - (y/b)^2 - (z/c)^2 - 1=0 is the nonparametric form of a hyperboloid of two sheets.*/
F(): x^2-(y/2)^2-(z/3)^2-1
[x]: -sqrt(10) , sqrt(10)
[y]: -6 , 6
[z]: -9 , 9
;
Name: kazoola(1,1,1,1)
/*d + c*x^2y^2z^2 - a*(x^2 + y^2 + z^2) - b*(x^4 + y^4 + z^4)=0 is the nonparametric form of a kazoola.*/
F(): 1 + x^2*y^2*z^2 - (x^2 + y^2 + z^2) - (x^4 + y^4 + z^4)
[x]: -sqrt(10) , sqrt(10)
[y]: -6 , 6
[z]: -9 , 9
;
Name: kummer
/*x^4 + y^4 + z^4 - (y^2*z^2 + z^2*x^2 + x^2*y^2) - (x^2 + y^2 + z^2) + 1=0 is the nonparametric form of Kummer's surface..*/
F(): x^4 + y^4 + z^4 - (y^2*z^2 + z^2*x^2 + x^2*y^2) - (x^2 + y^2 + z^2) + 1
[x]: -2 , 2
[y]: -2 , 2
[z]: -2 , 2
;
Name: scherk(1+t)
/*exp(a*z)*cos(a*x) - cos(a*y)==0 is the nonparametric form of Scherk's minimal surface.*/
F(): exp((1+t)*z)*cos((1+t)*x)-cos((1+t)*y)
[x]: -20 , 20
[y]: -20 , 20
[z]: -4 , 4
;
Name: scherk(1,pi-t)
/*exp(a/*tan(x/(a*cos(phi)))*tan(y/(a*sin(phi))) - tanh(z/a)==0 is the
nonparametric form of a twisted Scherk's minimal surface.*/
F(): tan(x/(cos(pi-t)))*tan(y/(sin(pi-t))) - tanh(z)
[x]: -pi , pi
[y]: -pi , pi
[z]: -pi/2 , pi/2
;
Name: scherk5
/*sinh(x)*sinh(y) - sin(z)=0 is the nonparametric form of a Scherk's fifth minimal surface.*/
F(): sinh(x)*sinh(y) - sin(z)
[x]: -3 , 3
[y]: -3 , 3
[z]: -5*pi/2 , 5*pi/2
;
Name: sinsurface
/*4*x^2*y^2*(a^2 - z^2) - a^2*(x^2 + y^2 - z^2)^2=0 is the nonparametric form of the surface (u,v)->a*sin(u),a*sin(v),a*sin(u + v))*/
F(): 4*x^2*y^2*(1 - z^2) - (x^2 + y^2 - z^2)^2
[x]: -1 , 1
[y]: -1 , 1
[z]: -1 , 1
;
Name: sphere(1)
/*x^2+y^2+z^2-a^2=0 is the nonparametric form of a sphere.*/
F(): x^2+y^2+z^2-1
[x]: -1 , 1
[y]: -1 , 1
[z]: -1 , 1
;
Name: sphere(2,1)
/*x^2n+y^2n+z^2n-a^2n=0 the nonparametric form of
the surface x^(2*n) + y^(2*n) +z^(2*n) = a^(2*n).*/
F(): x^4+y^4+z^4-1
[x]: -1 , 1
[y]: -1 , 1
[z]: -1 , 1
;
Name: torus(8,3)
/*z^2 + (sqrt(x^2 + y^2) - a)^2 - b^2=0 the nonparametric form of a torus.*/
F(): z^2 + (sqrt(x^2 + y^2) - 8)^2 - 3^2
[x]: -11 , 11
[y]: -11 , 11
[z]: -3 , 3
;
Name: twocusps
/*The nonparametric form of a surface with two cusps.*/
F(): (z - 1)^2*(x^2 - z^2) - (x^2 - z)^2 - y^4 - y^2*(2*x^2 + z^2 + 2*z - 1)
[x]: -1 , 1
[y]: -1.2 , 1.2
[z]: -0.5 , 0.9
;

I know some of them are already in the k3dsurf's default examples but there are some others that are not yet in the pre-loaded list of isosurfaces.

To use these examples just save the code in a file with k3ds extension and then load it while working in the isosurfaces environement inside k3dsurf. They will be added to the list of surfaces directly available for this session.

jolkaP

Joined: 17 Oct 2011
Posts: 12
Location: Marseille (France)

Posted: Fri Jan 13, 2012 12:34 am Post subject: Parametric morphed surfaces

This source has also an interesting list of morphed parametric surfaces: like a catenoid becoming a plane. Here you are:

Code:

Name:richmondpolar
/*
richmondpolar(n)[t](u,v):=
{-cos(t + v)/(2*u) -
u^(2*n + 1)*cos(t - (2*n + 1)*v)/(4*n + 2),
-sin(t + v)/(2*u) +
u^(2*n + 1)*sin(t - (2*n + 1)*v)/(4*n + 2),
u^n*cos(t - n*v)/n}
is the polar parametrization of a 1-parameter family of minimal surfaces such that {r,theta}->richmondpolar[n][0][r,theta] is a minimal surface with one planar end of degree n.*/
X():-cos(t + v)/(2*u) - u^3*cos(t - 3*v)/6
Y():-sin(t + v)/(2*u) + u^3*sin(t - 3*v)/6
Z():u*cos(t - v)
[u]:0.3,1.3
[v]:0,2*pi

Code:

Name:planetocat
/*
{u,v}->planetocat[t](u,v) is a 1-parameter family of minimal surfaces containing a plane and a catenoid. The principal curves of each surface are planar.
*/
X():cos(t)*u + sin(u)*cosh(v)
Y():sin(t)*cos(u)*cosh(v)
Z():v + cos(t)*cos(u)*sinh(v)
[u]:0,2*pi
[v]:-2,2

Code:

Name:hypocycloidinvolutemin
/*
(u,v)->hypocycloidinvolutemin(a,b)[t](u,v)=
(cos(t)*((a - b)*cos(u)*cosh(v) +
b*cos((a - b)*u/b)*cosh((a - b)*v/b)) +
sin(t)*((-a + b)*sin(u)*sinh(v) -
b*sin((a - b)*u/b)*sinh((a - b)*v/b)),
cos(t)*((a - b)*cosh(v)*sin(u) -
b*cosh((a - b)*v/b)*sin((a - b)*u/b)) +
sin(t)*((a - b)*cos(u)*sinh(v) -
b*cos(((a - b)*u)/b)*sinh((a - b)*v/b)),
(-4*(a - b)*b*cos((a*u)/(2*b))*cosh((a*v)/(2*b))*sin(t))/a -
(4*(a - b)*b*cos(t)*sin((a*u)/(2*b))*sinh((a*v)/(2*b)))/a) is a 1-parameter family of minimal surfaces such that (u,v)->hypocycloidinvolutemin(a,b)[0](u,v) is a minimal surface containing the involute of a hypocycloid as a geodesic.
*/
X():cos(t)*(3*cos(u)*cosh(v) + cos(3*u)*cosh(3*v)) + sin(t)*(-3*sin(u)*sinh(v) - sin(3*u)*sinh(3*v))
Y():cos(t)*(3*cosh(v)*sin(u) - cosh(3*v)*sin(3*u)) + sin(t)*(3*cos(u)*sinh(v) - cos((3*u))*sinh(3*v))
Z():-3*cos(2*u)*cosh(2*v)*sin(t) - 3*cos(t)*sin(2*u)*sinh(2*v)
[u]:-pi,pi
[v]:-1,1

Code:

Name:heltocat
/*
(u,v)->heltocat[t](u,v)=(
cos(u)*cosh(v)*sin(t) + cos(t)*sin(u)*sinh(v),
cosh(v)*sin(t)*sin(u) - cos(t)*cos(u)*sinh(v),
u*cos(t) + v*sin(t))
is a 1-parameter family of minimal surfaces connecting a helicoid to a catenoid.
*/
X():cos(u)*cosh(v)*sin((1-t)*pi/2) + cos((1-t)*pi/2)*sin(u)*sinh(v)
Y():cosh(v)*sin((1-t)*pi/2)*sin(u) - cos((1-t)*pi/2)*cos(u)*sinh(v)
Z():u*cos((1-t)*pi/2) + v*sin((1-t)*pi/2)
[u]:0,4*pi
[v]:-1.5,1.5

Code:

Name:enneperpolar(5)
/*
enneperpolar(n)[t](u_,v_):=
{u*cos(t - v) - u^(2*n + 1)*cos(t - (2*n + 1)*v)/(2*n + 1),
u*sin(t - v) + u^(2*n + 1)*sin(t - (2*n + 1)*v)/(2*n + 1),
2*u^(n + 1)*cos(t - (n + 1)*v)/(n + 1)}
is the polar parametrization of a 1-parameter family of minimal surfaces such that {r,theta}->enneperpolar[n][0][r,theta] is Enneper's minimal surface of degree n.
*/
X():u*cos(t - v) - u^11*cos(t - 11*v)/11
Y():u*sin(t - v) + u^11*sin(t - 11*v)/11
Z():2*u^6*cos(t - 6*v)/6
[u]:0, 1.57
[v]:-pi, pi

Code:

Name:ennepercatenoidpolar(1,1,1)
/*
ennepercatenoidpolar(n)[t](u,v):=(
(n*(2*n + 1)*u^2*cos(t - v) + n*(2*n + 1)*cos(t + v) - u^(n + 1)*(n*u^(n + 1)*cos(t - v - 2*n*v) + 2*cos(t - n*v) + 4*n*cos(t - n*v)))/(n*(2*n + 1)*u),
(n*(2*n + 1)*u^2*sin(t - v) - n*(2*n + 1)*sin(t + v) + u^(n + 1)*(2*sin(t - n*v) + 4*n*sin(t - n*v) + n*u^(n + 1)*sin(t + v - 2*(n + 1)*v)))/(n*(2*n + 1)*u),
2*(u^(n + 1)*cos(t - (n + 1)*v) + (n + 1)*(cos(t)*log(u) + v*sin(t)))/(n + 1))
ennepercatenoidpolar(n,a,b)[t](u,v):=
((n*(2*n + 1)*u^2*cos(t - v) + a^2*n*(2*n + 1)*cos(t + v) - b*u^(n + 1)*(2*a*cos(t - n*v) + 4*a*n*cos(t - n*v) + b*n*u^(n + 1)*cos(t - (2*n + 1)*v)))/(n*(2*n + 1)*u),
(n*(2*n + 1)*u^2*sin(t - v) - a^2*n*(2*n + 1)*sin(t + v) + b*u^n*(b*n*u^(n + 2)*sin(t - v - 2*n*v) + 2*a*u*sin(t + v - (n + 1)*v) + 4*a*n*u*sin(t + v - (n + 1)*v)))/(n*(2*n + 1)*u),
(2*b*u^(n + 1)*cos(t - (n + 1)*v) + 2*a*(n + 1)*cos(t)*log(u) + 2*a*(n + 1)*v*sin(t))/(n + 1))
ennepercatenoidpolar(n,a,b)[t](r,theta) is the polar parametrization of a 1-parameter family of minimal surfaces constructed from the minimal curve whose Weierstrass representation is given by f[z_]:=2 and g[z_]:=a/z + b*z^n. For t=0 and a=b=1 the surface has a catenoid-like bottom and an enneper-like top with n + 1 lobes.
*/
X():(3*u^2*cos(t - v) + 3*cos(t + v) - u^2*(2*cos(t - v) + 4*cos(t - v) + u^2*cos(t - 3*v)))/(3*u)
Y():(3*u^2*sin(t - v) - 3*sin(t + v) + u*(u^3*sin(t - 4*v) + 2*u*sin(t + v - 2*v) + 4*u*sin(t - v)))/(3*u)
Z():(2*u^2*cos(t - 2*v) + 4*cos(t)*log(u) + 4*v*sin(t))/2
[u]:0.40, 1.08
[v]:-pi, pi

Code:

Name:deltoidinvolutemin(1)
/*
deltoidinvolutemin(a)[t](u,v) = a*deltoidinvolutemin(1)[t](u,v) is a 1-parameter family of minimal surfaces such that (u,v)->deltoidinvolutemin(a)[0](u,v) is a minimal surface containing the involute of a deltoid as a geodesic.
*/
X():(1/3)*cos(t)*(8*cos(u/2)*cosh(v/2) + 2*cos(u)*cosh(v) - cos(2*u)*cosh(2*v)) + (1/3)*sin(t)*(-8*sin(u/2)*sinh(v/2) - 2*sin(u)*sinh(v) + sin(2*u)*sinh(2*v))
Y():(cos(t)*(-8*cosh(v/2)*sin(u/2) + 2*cosh(v)*sin(u) + cosh(2*v)*sin(2*u)))/3 + (sin(t)*(-8*cos(u/2)*sinh(v/2) + 2*cos(u)*sinh(v) + cos(2*u)*sinh(2*v)))/3
Z():(4*sin(t)*(3*u - 2*cosh(3*v/2)*sin(3*u/2)))/9 - (4*cos(t)*(3*v - 2*cos(3*u/2)*sinh(3*v/2)))/9
[u]:-2*pi, 2*pi
[v]:-0.3, 0.3

Code:

Name:deltoidmin(1)
/*deltoidmin(a)(t)(u,v):=
{a*cos(t)*(2*cos(u)*cosh(v) + cos(2*u)*cosh(2*v)) +
2*a*sin(t)*sin(u)*(-sinh(v) - cos(u)*sinh(2*v)),
-2*a*(cos(u)*sinh(v)*((-1 + cos(u)*cosh(v))*sin(t) +
2*a*cos(t)*sin(u)*sinh(v))) +
cosh(v)*sin(u)*(2*a*cos(t)*(1 - cos(u)*cosh(v)) +
2*a*sin(t)*sin(u)*sinh(v)),
-(8/3)*a*cos(3*u/2)*cosh(3*v/2)*sin(t) -
(8/3)*a*cos(t)*sin(3*u/2)*sinh(3*v/2)} is a 1-parameter family of minimal surfaces such that (u,v)->deltoidmin(a)(0)(u,v) is a minimal surface containing a deltoid as a geodesic.
*/
X():cos(t)*(2*cos(u)*cosh(v) + cos(2*u)*cosh(2*v)) +
2*sin(t)*sin(u)*(-sinh(v) - cos(u)*sinh(2*v))
Y():-2*(cos(u)*sinh(v)*((-1 + cos(u)*cosh(v))*sin(t) +
2*cos(t)*sin(u)*sinh(v))) +
cosh(v)*sin(u)*(2*cos(t)*(1 - cos(u)*cosh(v)) +
2*sin(t)*sin(u)*sinh(v))
Z():-(8/3)*cos(3*u/2)*cosh(3*v/2)*sin(t) -
(8/3)*cos(t)*sin(3*u/2)*sinh(3*v/2)
[u]:-pi, pi
[v]:-0.3, 0.3

Code:

Name:circleinvolutemin(1,1)
/*
circleinvolutemin(1,a)[t]=a*circleinvolutemin(1,1)[t] is a 1-parameter family of minimal surfaces such that (u,v)->circleinvolutemin(1,a)[0](u,v) is a minimal surface containing the first involute of a circle as a geodesic.
*/
X():cos(t)*(cos(u)*cosh(v) + u*cosh(v)*sin(u) - v*cos(u)*sinh(v)) + sin(t)*(v*cosh(v)*sin(u) + u*cos(u)*sinh(v) - sin(u)*sinh(v))
Y():sin(t)*(-v*cos(u)*cosh(v) + cos(u)*sinh(v) + u*sin(u)*sinh(v)) + cos(t)*(-u*cos(u)*cosh(v) + cosh(v)*sin(u) - v*sin(u)*sinh(v))
Z():-u*v*cos(t) + (u^2 - v^2)*sin(t)/2
[u]:0, pi
[v]:-1, 1

Code:

Name:catalandef(1)
/*
(u,v)->catalandef(a)[t](u,v)={a*cos(t)*(u - cosh(v)*sin(u)) +
a*sin(t)*(v - cos(u)*sinh(v)),
a*cos(t)*(1 - cos(u)*cosh(v)) +
a*sin(t)*sin(u)*sinh(v),
4*a*(1 - cos(u/2)*cosh(v/2))*sin(t) -
4*a*cos(t)*sin(u/2)*sinh(v/2))
is a 1-parameter family of minimal surfaces connecting Catalan's minimal surface to its conjugate.
*/
X():cos(t)*(u - cosh(v)*sin(u)) + sin(t)*(v - cos(u)*sinh(v))
Y():cos(t)*(1 - cos(u)*cosh(v)) + sin(t)*sin(u)*sinh(v)
Z():4*(1 - cos(u/2)*cosh(v/2))*sin(t) - 4*cos(t)*sin(u/2)*sinh(v/2)
[u]:-pi*2, pi*2
[v]:-pi/2, pi/2

Code:

Name:bourpolar(1,1)
/*bourpolar(m)[t](u,v):=
(u^(m - 1)*cos(t - (m - 1)*v)/(m - 1) -
u^(m + 1)*cos(t - (m + 1)*v)/(m + 1),
u^(m - 1)*sin(t - (m - 1)*v)/(m - 1) +
u^(m + 1)*sin(t - (m + 1)*v)/(m + 1),
2*u^m*cos(t - m*v)/m)
bourpolar(m,n)[t](r,theta):=
(r^(m - 1)*cos(t - (m - 1)*theta)/(m - 1) -
r^(m + 2*n - 1)*cos(t - (m + 2*n - 1)*theta)/(m + 2*n - 1),
r^(m - 1)*sin[t - (m - 1)*theta]/(m - 1) +
r^(m + 2*n - 1)*sin(t - (m + 2*n - 1)*theta)/(m + 2*n - 1),
2*r^(m + n - 1)*cos(t - (m + n - 1)*theta)/(m + n - 1))
{r,theta}->bourpolar[m,n][t][r,theta] is the polar parametrization of a 1-parameter family of minimal surfaces such that {r,theta}->bourpolar[m,n][0][r,theta] is Bour's minimal surface of degree {m,n}.
*/
X():cos(t) -u^2*cos(t - 2*v)/2
Y():cos(t) -u^2*sin(t - 2*v)/2
Z():2*u*cos(t-v)
[u]:0, pi/2
[v]:-pi, pi

Please, let me know if you find any formula wrong.

To use these examples of parametric surfaces just save any code block in a file with k3ds extension and then load it while working in the parametric environement inside k3dsurf.

Phil999

Joined: 08 Feb 2007
Posts: 24

Posted: Thu Jan 19, 2012 9:38 pm Post subject:

thank you

jolkaP

Joined: 17 Oct 2011
Posts: 12
Location: Marseille (France)

Posted: Tue Jan 31, 2012 11:06 pm Post subject: Almost all parametric surfaces from this source

You welcome,

Here go all the parametric surfaces described in this source that can be expressed in k3dsurf. I tried to categorize them but of course some of them fall into multiples categories

FAMILIES OF MINIMAL SURFACES ( those where already in my post above, but here they are written in a clearer way; activate morph to see what happens)

In the next one, for t=0 you get Bour's minimal surface of degree (m,n)

Code:

X():u^(m - 1)*cos(t - (m - 1)*v)/(m - 1) -
u^(m + 2*n - 1)*cos(t - (m + 2*n - 1)*v)/(m + 2*n - 1)
Y():u^(m - 1)*sin(t - (m - 1)*v)/(m - 1) +
u^(m + 2*n - 1)*sin(t - (m + 2*n - 1)*v)/(m + 2*n - 1)
Z():2*u^(m + n - 1)*cos(t - (m + n - 1)*v)/(m + n - 1)
[u]:0, 1.7
[v]:-pi, pi

You have to replace m and n with integers. For example, for m=2 and n=1 you get

Code:

X():u*cos(t - v) -u^3*cos(t - 3*v)/3
Y():u*cos(t - v) -u^3*sin(t - 3*v)/3
Z():2*u^2*cos(t - 2*v)/2
[u]:0, 1.7
[v]:-pi, pi

See Catalan's minimal surface become its conjugate and viceversa.

Code:

X():cos(t)*(u - cosh(v)*sin(u)) + sin(t)*(v - cos(u)*sinh(v))
Y():cos(t)*(1 - cos(u)*cosh(v)) + sin(t)*sin(u)*sinh(v)
Z():4*(1 - cos(u/2)*cosh(v/2))*sin(t) - 4*cos(t)*sin(u/2)*sinh(v/2)
[u]:-pi*2, pi*2
[v]:-pi/2, pi/2

In the next one, for t=0 you get a minimal surface containing first involute of a circle as a geodesic.

Code:

X():cos(t)*(cos(u)*cosh(v) + u*cosh(v)*sin(u) - v*cos(u)*sinh(v)) + sin(t)*(v*cosh(v)*sin(u) + u*cos(u)*sinh(v) - sin(u)*sinh(v))
Y():sin(t)*(-v*cos(u)*cosh(v) + cos(u)*sinh(v) + u*sin(u)*sinh(v)) + cos(t)*(-u*cos(u)*cosh(v) + cosh(v)*sin(u) - v*sin(u)*sinh(v))
Z():-u*v*cos(t) + (u^2 - v^2)*sin(t)/2
[u]:0, pi
[v]:-1, 1

In the next one, for t=0 you get a minimal surface containing the involute of a deltoid as a geodesic.

Code:

X():(1/3)*cos(t)*(8*cos(u/2)*cosh(v/2) + 2*cos(u)*cosh(v) - cos(2*u)*cosh(2*v)) + (1/3)*sin(t)*(-8*sin(u/2)*sinh(v/2) - 2*sin(u)*sinh(v) + sin(2*u)*sinh(2*v))
Y():(cos(t)*(-8*cosh(v/2)*sin(u/2) + 2*cosh(v)*sin(u) + cosh(2*v)*sin(2*u)))/3 + (sin(t)*(-8*cos(u/2)*sinh(v/2) + 2*cos(u)*sinh(v) + cos(2*u)*sinh(2*v)))/3
Z():(4*sin(t)*(3*u - 2*cosh(3*v/2)*sin(3*u/2)))/9 - (4*cos(t)*(3*v - 2*cos(3*u/2)*sinh(3*v/2)))/9
[u]:-2*pi, 2*pi
[v]:-0.3, 0.3

In the next one, for t=0 you get a minimal surface containing a deltoid as a geodesic.

Code:

X():a*cos(t)*(2*cos(u)*cosh(v) + cos(2*u)*cosh(2*v)) + 2*a*sin(t)*sin(u)*(-sinh(v) - cos(u)*sinh(2*v))
Y():-2*a*(cos(u)*sinh(v)*((-1 + cos(u)*cosh(v))*sin(t) + 2*a*cos(t)*sin(u)*sinh(v))) + cosh(v)*sin(u)*(2*a*cos(t)*(1 - cos(u)*cosh(v)) + 2*a*sin(t)*sin(u)*sinh(v))
Z():-(8/3)*a*cos(3*u/2)*cosh(3*v/2)*sin(t) - (8/3)*a*cos(t)*sin(3*u/2)*sinh(3*v/2)
[u]:-pi, pi
[v]:-0.3, 0.3

You have to replace a with any positive number. For example, for a=1, you get.

Code:

X():cos(t)*(2*cos(u)*cosh(v) + cos(2*u)*cosh(2*v)) + 2*sin(t)*sin(u)*(-sinh(v) - cos(u)*sinh(2*v))
Y():-2*(cos(u)*sinh(v)*((-1 + cos(u)*cosh(v))*sin(t) +
2*cos(t)*sin(u)*sinh(v))) + cosh(v)*sin(u)*(2*cos(t)*(1 - cos(u)*cosh(v)) + 2*sin(t)*sin(u)*sinh(v))
Z():-(8/3)*cos(3*u/2)*cosh(3*v/2)*sin(t) - (8/3)*cos(t)*sin(3*u/2)*sinh(3*v/2)
[u]:-pi, pi
[v]:-0.3, 0.3

The following describes a family of minimal surfaces constructed from the minimal curve whose Weierstrass representation is given by f[z_]:=2 and g[z_]:=a/z + b*z^n.

Code:

X():(n*(2*n + 1)*u^2*cos(t - v) + a^2*n*(2*n + 1)*cos(t + v) - b*u^(n + 1)*(2*a*cos(t - n*v) + 4*a*n*cos(t - n*v) + b*n*u^(n + 1)*cos(t - (2*n + 1)*v)))/(n*(2*n + 1)*u)
Y():(n*(2*n + 1)*u^2*sin(t - v) - a^2*n*(2*n + 1)*sin(t + v) + b*u^n*(b*n*u^(n + 2)*sin(t - v - 2*n*v) + 2*a*u*sin(t + v - (n + 1)*v) + 4*a*n*u*sin(t + v - (n + 1)*v)))/(n*(2*n + 1)*u)
Z():(2*b*u^(n + 1)*cos(t - (n + 1)*v) + 2*a*(n + 1)*cos(t)*log(u) + 2*a*(n + 1)*v*sin(t))/(n + 1)
[u]:0.4,1.08
[v]:-pi,pi

Replace n with any integer, a and b with real numbers. For instance, for n=1, a=b=1 you get

Code:

X():(3*u^2*cos(t - v) + 3*cos(t + v) - u^2*(2*cos(t - v) + 4*cos(t - v) + u^2*cos(t - 3*v)))/(3*u)
Y():(3*u^2*sin(t - v) - 3*sin(t + v) + u*(u^3*sin(t - 4*v) +
2*u*sin(t + v - 2*v) + 4*u*sin(t - v)))/(3*u)
Z():(2*u^2*cos(t - 2*v) + 4*cos(t)*log(u) + 4*v*sin(t))/2
[u]:0.40, 1.08
[v]:-pi, pi

In the next, for t=0, you get Enneper's minimal surface of degree n.

Code:

X():u*cos(t - v) - u^(2*n + 1)*cos(t - (2*n + 1)*v)/(2*n + 1)
Y():u*sin(t - v) + u^(2*n + 1)*sin(t - (2*n + 1)*v)/(2*n + 1)
Z():2*u^(n + 1)*cos(t - (n + 1)*v)/(n + 1)
[u]:0,1.58
[v]:-pi,pi

Replace n with any integer. For instance, for n=2 you get

Code:

X():u*cos(t - v) - u^5*cos(t - 5*v)/5
Y():u*sin(t - v) + u^5*sin(t - 5*v)/5
Z():2*u^3*cos(t - 3*v)/3
[u]:0, 1.58
[v]:-pi, pi

See an helicoid become a catenoid and vice-versa.

Code:

X():cos(u)*cosh(v)*sin((1-t)*pi/2) + cos((1-t)*pi/2)*sin(u)*sinh(v)
Y():cosh(v)*sin((1-t)*pi/2)*sin(u) - cos((1-t)*pi/2)*cos(u)*sinh(v)
Z():u*cos((1-t)*pi/2) + v*sin((1-t)*pi/2)
[u]:0,4*pi
[v]:-1.5,1.5

In the next, for t=0, you get a minimal surface containing the involute of a hypocycloid as a geodesic

Code:

X():cos(t)*((a - b)*cos(u)*cosh(v) + b*cos((a - b)*u/b)*cosh((a - b)*v/b)) + sin(t)*((-a + b)*sin(u)*sinh(v) - b*sin((a - b)*u/b)*sinh((a - b)*v/b))
Y():cos(t)*((a - b)*cosh(v)*sin(u) - b*cosh((a - b)*v/b)*sin((a - b)*u/b)) + sin(t)*((a - b)*cos(u)*sinh(v) - b*cos(((a - b)*u)/b)*sinh((a - b)*v/b))
Z():(-4*(a - b)*b*cos((a*u)/(2*b))*cosh((a*v)/(2*b))*sin(t))/a - (4*(a - b)*b*cos(t)*sin((a*u)/(2*b))*sinh((a*v)/(2*b)))/a
[u]:-pi,pi
[v]:-1,1

You have to replace parameters a and b. For a=4 and b=1 you get

Code:

X():cos(t)*(3*cos(u)*cosh(v) + cos(3*u)*cosh(3*v)) + sin(t)*(-3*sin(u)*sinh(v) - sin(3*u)*sinh(3*v))
Y():cos(t)*(3*cosh(v)*sin(u) - cosh(3*v)*sin(3*u)) + sin(t)*(3*cos(u)*sinh(v) - cos((3*u))*sinh(3*v))
Z():-3*cos(2*u)*cosh(2*v)*sin(t) - 3*cos(t)*sin(2*u)*sinh(2*v)
[u]:-pi,pi
[v]:-1,1

In the next, for t=0, you get a minimal surface containing a parabola as a geodesic

Code:

X():2*(u*cos(t) + v*sin(t))
Y():((u^2 - v^2)*cos(t) + 2*u*v*sin(t))
Z():((-2*(u*v + atan(2*v/(u + sqrt(1 + u^2))))*cos(t) +
(-2*v^2 + log((u + sqrt(1 + u^2))^2 + 4*v^2))*sin(t) +
2*sqrt(1 + u^2)*(-(v*cos(t)) + u*sin(t))))/2
[u]:-2,2
[v]:-0.3,0.3

See a plane become a catenoid and vice-versa. The principal curves of each surface in this family are planar.

Code:

X():cos(t)*u + sin(u)*cosh(v)
Y():sin(t)*cos(u)*cosh(v)
Z():v + cos(t)*cos(u)*sinh(v)
[u]:0,2*pi
[v]:-2,2

In the next, for t=0, you get Richmond's surface with one planar end of degree n.

Code:

X():-cos(t + v)/(2*u) - u^3*cos(t - 3*v)/6
Y():-sin(t + v)/(2*u) + u^(2*n + 1)*sin(t - (2*n + 1)*v)/(4*n + 2)
Z():u^n*cos(t - n*v)/n
[u]:0.3,1.3
[v]:0,2*pi

Replace n with any integer. For instance, for n=1 you get

Code:

X():-cos(t + v)/(2*u) - u^3*cos(t - 3*v)/6
Y():-sin(t + v)/(2*u) + u^3*sin(t - 3*v)/6
Z():u*cos(t - v)
[u]:0.3,1.3
[v]:0,2*pi

MINIMAL SURFACES

Bour's minimal surface

Code:

X():(2*u^2 - u^4 - 2*v^2 + 6*u^2*v^2 - v^4)/4
Y():u*v*(-1 - u^2 + v^2)
Z():(2/3)*(u^3 - 3*u*v^2)
[u]:-1, 1
[v]:-1, 1

Catalan's minimal surface

Code:

X():(u - cosh(v)*sin(u))
Y():(1 - cos(u)*cosh(v))
Z():- 4*sin(u/2)*sinh(v/2)
[u]:-pi*2, pi*2
[v]:-pi/2, pi/2

Enneper's minimal surface

Code:

X():u -u^3/3 + u*v^2
Y():-v +v^3/3 - v*u^2
Z():u^2 - v^2
[u]:-1.75, 1.75
[v]:-1.75, 1.75

Henneberg's minimal surface

Code:

X():2*sinh(u)*cos(v) - (2/3)*sinh(3*u)*cos(3*v)
Y():2*sinh(u)*sin(v) + (2/3)*sinh(3*u)*sin(3*v)
Z():2*cosh(2*u)*cos(2*v)
[u]:0.3,1
[v]:-pi,pi

Richmond's minimal surface is a minimal surface with one planar end

Code:

X():(-3*u - u^5 + 2*u^3*v^2 + 3*u*v^4)/(6*(u^2 + v^2))
Y():(-3*v - 3*u^4*v - 2*u^2*v^3 + v^5)/(6*(u^2 + v^2))
Z():u
[u]:-1,1
[v]:-1,1

Scherk's minimal surface

Code:

X():u
Y():v
Z():(1/a)*log(cos(a*v)/cos(a*u))
[u]:0.01,3*pi/2 + 0.01
[v]:-pi/2 + 0.01,3*pi/2 + 0.01

You have to replace parameter a. For a=1 you get

Code:

X():u
Y():v
Z():log(cos(v)/cos(u))
[u]:0.01,3*pi/2 + 0.01
[v]:-pi/2 + 0.01,3*pi/2 + 0.01

Thomsen's minimal surface

Code:

X():b*u/a + sqrt(1+b^2)*sinh(a*u)*cos(a*v)/a^2
Y():sqrt(1 + b^2)*v/a + b*cosh(a*u)*sin(a*v)/a^2
Z():sinh(a*u)*sin(a*v)/a^2
[u]:-pi/2,pi/2
[v]:0,2*pi

You have to replace parameters a and b. For a=1 and b=2 you get:

Code:

X():-2*u+sqrt(5)*sinh(u)*cos(v)
Y():sqrt(5)*v-2*cosh(u)*sin(v)
Z():sinh(u)*sin(v)
[u]:-pi/2,pi/2
[v]:0,2*pi

COMPACT SURFACES

Astroidial ellipsoid with axes of lengths a, b and c

Code:

X():(a*cos(u)*cos(v))^3
Y():(b*sin(u)*cos(v))^3
Z():(c*sin(v))^3
[u]:0, 2*pi
[v]:-pi/2, pi/2

You have to replace a,b and c. For example, for a=b=c=1 you get

Code:

X():(cos(u)*cos(v))^3
Y():(sin(u)*cos(v))^3
Z():sin(v)^3
[u]:0, 2*pi
[v]:-pi/2, pi/2

Bohemian dome formed by moving an ellipse along a circle in a perpendicular plane so that the ellipse remains parallel to a plane

Code:

X():a*cos(u)
Y():a*sin(u)+b*cos(v)
Z():c*sin(v)
[u]:0, 2*pi
[v]:-pi, pi

You have to replace a,b and c. For example, for a=1, b=2 and c=1 you get

Code:

X():cos(u)
Y():sin(u)+2*cos(v)
Z():sin(v)
[u]:0, 2*pi
[v]:-pi, pi

Boy's surface

Code:

X():(a/2)*(-cos(v)^2 + 2*cos(u)^2*sin(v)^2 + cos(v)*cos(u)*sin(v)*(-cos(v)^2 + cos(u)^2*sin(v)^2) - sin(v)^2*sin(u)^2 + 2*cos(v)*sin(v)*sin(u)*(-cos(v)^2 + sin(v)^2*sin(u)^2) - sin(v)^4*sin(4*u)/4)
Y():(sqrt(3)/2)*b*(-cos(v)^2 + cos(v)*cos(u)*sin(v)*(cos(v)^2 - cos(u)^2*sin(v)^2) + sin(v)^2*sin(u)^2 - (sin(v)^4*sin(4*u))/4)
Z():c*(cos(v) + cos(u)*sin(v) + sin(u)*sin(v))*(-4*sin(v)*(cos(u) - sin(u))*(-cos(v) + cos(u)*sin(v))* (cos(v) - sin(u)*sin(v)) + (cos(v) + cos(u)*sin(v) + sin(u)*sin(v)
[u]:-pi/2, pi/2
[v]:-pi/2, pi/2

You have to replace a,b and c. For example, for a=b=4 and c=1 you get

Code:

X():2*(-cos(v)^2 + 2*cos(u)^2*sin(v)^2 + cos(v)*cos(u)*sin(v)*(-cos(v)^2 + cos(u)^2*sin(v)^2) - sin(v)^2*sin(u)^2 + 2*cos(v)*sin(v)*sin(u)*
(-cos(v)^2 + sin(v)^2*sin(u)^2) - sin(v)^4*sin(4*u)/4)
Y():(sqrt(3)/2)*4*(-cos(v)^2 + cos(v)*cos(u)*sin(v)*(cos(v)^2 - cos(u)^2*sin(v)^2) +
sin(v)^2*sin(u)^2 - (sin(v)^4*sin(4*u))/4)
Z():(cos(v) + cos(u)*sin(v) + sin(u)*sin(v))*
(-4*sin(v)*(cos(u) - sin(u))*(-cos(v) + cos(u)*sin(v))*
(cos(v) - sin(u)*sin(v)) +(cos(v) + cos(u)*sin(v) + sin(u)*sin(v))^3)
[u]:-pi/2, pi/2
[v]:-pi/2, pi/2

Cossurface

Code:

X():cos(u)
Y():cos(v)
Z():cos(u+v)
[u]:-pi,pi
[v]:-pi,pi

A cross cap

Code:

X():sin(u) * sin(2 * v) / 2
Y():sin(2 * u) * cos(v) * cos(v)
Z():cos(2 * u) * cos(v) * cos(v)
[u]:-pi/2, pi/2
[v]:-pi/2, pi/2

Eight turning on another eight

Code:

X():sin(u)*sin(v)
Y():cos(u)*sin(u)*sin(v)
Z():cos(v)*sin(v)
[u]:-pi, pi
[v]:-pi/2, pi/2

A cyclide of Dupin of radius k whose focal sets are the ellipse u->(a*cos(u),sqrt(a^2 - c^2)*Sin(u),0} and the hyperbola v->(c*sec(v),0,sqrt(a^2 - c^2)*tan(v))

Code:

X():(c*(k - c*cos(u)) + a*cos(u)*(a - k*cos(v)))/
(a - c*cos(u)*cos(v))
Y():
(sqrt(a^2 - c^2)*(a - k*cos(v))*sin(u))/
(a - c*cos(u)*cos(v))
Z():(sqrt(a^2 - c^2)*(k - c*cos(u))*sin(v))/(a - c*cos(u)*cos(v))
[u]:-pi, pi
[v]:-pi, pi

You have to replace a,c and k. For example for a=8, c=3 and k=5 you get

Code:

X():(3*(5 - 3*cos(u)) + 8*cos(u)*(8 - 5*cos(v)))/(8 - 3*cos(u)*cos(v))
Y():(sqrt(8^2 - 3^2)*(8 - 5*cos(v))*sin(u))/(8 - 3*cos(u)*cos(v))
Z():(sqrt(8^2 - 3^2)*(5 - 3*cos(u))*sin(v))/(8 - 3*cos(u)*cos(v))
[u]:-pi, pi
[v]:-pi, pi

An ellipsoid with axes of lengths a, b and c

Code:

X():a*cos(v)*cos(u)
Y():b*cos(v)*sin(u)
Z():c*sin(v)
[u]:-pi, pi
[v]:-pi/2, pi/2

You have to replace a,b and c with non-zero real numbers. For a=1, b=2, c=3 you get

Code:

X():cos(v)*cos(u)
Y():2*cos(v)*sin(u)
Z():3*sin(v)
[u]:-pi, pi
[v]:-pi/2, pi/2

A generalized cube

Code:

X():a*(cos(u)*sqrt(cos(u)^2)^(n - 1) + sin(u)*sqrt(sin(u)^2)^(n - 1))*(cos(v)*sqrt(cos(v)^2)^(n - 1) + sin(v)*sqrt(sin(v)^2)^(n - 1))
Y():b*(-cos(u)*sqrt(cos(u)^2)^(n - 1) + sin(u)*sqrt(sin(u)^2)^(n - 1))*(cos(v)*sqrt(cos(v)^2)^(n - 1) + sin(v)*sqrt(sin(v)^2)^(n - 1))
Z():c*(-cos(v)*sqrt(cos(v)^2)^(n - 1) + sin(v)*sqrt(sin(v)^2)^(n - 1))
[u]:-pi/2, pi/2
[v]:-pi, pi

You have to replace n with a natural number and a,b,c with positive real numbers. For example, for n=4, a=b=c=1 you get

Code:

X():cos(u)*abs(cos(u))^3+sin(u)*abs(sin(u))^3*(cos(v)*abs(cos(v))^3+sin(v)*abs(sin(v))^3)
Y():-cos(u)*abs(cos(u))^3+sin(u)*abs(sin(u))^3*(cos(v)*abs(cos(v))^3+sin(v)*abs(sin(v))^3)
Z():-cos(v)*abs(cos(v))^3+sin(v)*abs(sin(v))^3
[u]:-pi/2, pi/2
[v]:-pi, pi

A generalized octahedron.

Code:

X():a*cos(u)*cos(v)*sqrt(cos(u)^2*cos(v)^2)^(n - 1)
Y():b*sin(u)*cos(v)*sqrt(sin(u)^2*cos(v)^2)^(n - 1)
Z():c*sin(v)*sqrt(sin(v)^2)^(n - 1)
[u]:-pi/2, pi/2
[v]:-pi, pi

You have to replace n with a natural number and a,b,c with positive real numbers. For n=1 you get an ellipsoid, for n=2 and a=b=c you get an ordinary octahedron and for n=3 you get an astroidial ellipsoid. For example, for n=2, a=b=c=1 you get

Code:

X():cos(u)*cos(v)*abs(cos(u)*cos(v))
Y():sin(u)*cos(v)*abs(sin(u)*cos(v))
Z():sin(v)*abs(sin(v))
[u]:-pi/2, pi/2
[v]:-pi, pi

A Klein bottle formed by moving and twisting a figure eight along a circle of radius a. The surface is nonorientable and a neighborhood of the self-intersection curve is nonorientable.

Code:

X():(a + cos(u/2)*sin(v) - sin(u/2)*sin(2*v))*cos(u)
Y():(a + cos(u/2)*sin(v) - sin(u/2)*sin(2*v))*sin(u)
Z():sin(u/2)*sin(v) + cos(u/2)*sin(2*v)
[u]:-pi/4,3*pi/2
[v]:-pi,pi

You have to replace a. For a=2, you get

Code:

X():(2 + cos(u/2)*sin(v) - sin(u/2)*sin(2*v))*cos(u)
Y():(2 + cos(u/2)*sin(v) - sin(u/2)*sin(2*v))*sin(u)
Z():sin(u/2)*sin(v) + cos(u/2)*sin(2*v)
[u]:-pi/4,3*pi/2
[v]:-pi,pi

A Klein bottle in which a neighbourhood of self-intersection curve is orientable

Code:

X():((u-abs(u))/(2*u))*sin(u)+
((u+abs(u))/(2*u))*(10*sin(0.105*u) + 2*(1 + 0.066*u)*
(cos(0.105*u) + 4*cos(0.21*u))*cos(v)*
(sin(0.105*u) + 2*sin(0.21*u))/
sqrt(100*cos(0.105*u)^2 + 4*(cos(0.105*u) +
4*cos(0.21*u))^2*(sin(0.105*u) + 2*sin(0.21*u))^2))
Y():((u-abs(u))/(2*u))*(- 2 + cos(u))*cos(v)+
((u+abs(u))/(2*u))*((sin(0.105*u) + 2*sin(0.21*u))^2 -
(10*(1 + 0.033*2*u)*cos(0.105*u)*cos(v))/
sqrt(100*cos(0.105*u)^2 + 4*(cos(0.105*u) +
4*cos(0.21*u))^2*(sin(0.105*u) + 2*sin(0.21*u))^2))
Z():((u-abs(u))/(2*u))*(2 - cos(u))*sin(v)+
((u+abs(u))/(2*u))*((1 + 0.066*u)*sin(v))
[u]:-pi,29.9517
[v]:-pi,pi

WRI's version of a Klein bottle. The self-intersection curve has an orientable neighbourhood.

Code:

X():if(sin(u)<0,(6*cos(u)*(1 + sin(u)) + 4*(1 - cos(u)/2)*cos(v + pi)),
(6*cos(u)*(1 + sin(u)) + 4*(1 - cos(u)/2)*cos(u)*cos(v)))
Y():if(sin(u)<0,(16*sin(u)),(16*sin(u) + 4*(1 - cos(u)/2)*sin(u)*cos(v)))
Z():4*(1 - cos(u)/2)*sin(v)
[u]:0,2*pi
[v]:0,2*pi

A pillow

Code:

X():cos(u)
Y():cos(v)
Z():sin(u)*sin(v)
[u]:-pi,pi
[v]:-pi,pi

A pseudocrosscap in R^3

Code:

X():(1 - u^2)*sin(v)
Y():(1 - u^2)*sin(2*v)
Z():u
[u]:-1,1
[v]:0,2*pi

Steiner's Roman surface

Code:

X():(1/2)*sin(2*u)*cos(v)^2
Y():(1/2)*sin(u)*sin(2*v)
Z():(1/2)*cos(u)*sin(2*v)
[u]:0,pi
[v]:-pi/2,pi/2

n-th Roman surface

Code:

X():(1/2)*(cos(u)*cos(v)^2*sin(u))^n
Y():(1/2)*(cos(v)*sin(u)*sin(v))^n
Z():(1/2)*(cos(u)*cos(v)*sin(v))^n
[u]:0,pi
[v]:-pi,pi

You have to replace n with any positive integer. For n=3 you get.

Code:

X():(1/2)*(cos(v)^2*sin(u))^3
Y():(1/2)*(cos(v)*sin(u)*sin(v))^3
Z():(1/2)*(cos(u)*cos(v)*sin(v))^3
[u]:0,pi
[v]:-pi,pi

Sinsurface

Code:

X():sin(u)
Y():sin(v)
Z():sin(u+v)
[u]:-pi,pi
[v]:-pi,pi

A surface resembling a snail

Code:

X():u*cos(v)*sin(u)
Y():u*cos(u)*cos(v)
Z():-u*sin(v)
[u]:-pi,pi
[v]:-pi,pi

COMPACT REVOLUTION SURFACES

A parametrization of the Earth (with equatorial diameter 12756.4 kilometers and polar diameter 12713.2 kilometers)

Code:

X():6378.2*cos(u)*cos(v)
Y():6378.2*sin(u)*cos(v)
Z():6356.6*sin(v)
[u]:-pi, pi
[v]:-pi/2, pi/2

Eight surface is a surface of revolution generated by an eight-shaped curve.

Code:

X():cos(u)*sin(2*v)
Y():sin(u)*sin(2*v)
Z():sin(v)
[u]:-pi, pi
[v]:-pi/2, pi/2

Heart is a surface of revolution generated by a cardioid with the axis of revolution passing through the cusp

Code:

X():2*cos(u)*(1 + cos(v))*sin(v)
Y():2*sin(u)*(1 + cos(v))*sin(v)
Z():-2*cos(v)*(1 + cos(v))
[u]:-pi/2,pi/2
[v]:-pi,pi

A circular cylinder of radius a inverted with respect to a sphere of radius rho. It is a cyclide of Dupin.

Code:

X():a*rho^2*cos(u)/(a^2 + v^2)
Y():a*rho^2*sin(u)/(a^2 + v^2)
Z():rho^2*v/(a^2 + v^2)
[u]:-pi,pi
[v]:-rho,rho

You have to replace rho and a. For example, for rho=40, a=1 you get

Code:

X():40^2*cos(u)/(1+v^2)
Y():40^2*sin(u)/(1+v^2)
Z():40^2*v/(1+v^2)
[u]:-pi,pi
[v]:-40,40

A torus inverted with respect to a sphere of radius rho. It is a cyclide of Dupin.

Code:

X():rho^2*cos(u)*(a + b*cos(v))/(a^2 + b^2 + 2*a*b*cos(v))
Y():rho^2*(a + b*cos(v))*sin(u)/(a^2 + b^2 + 2*a*b*cos(v))
Z():b*rho^2*sin(v)/(a^2 + b^2 + 2*a*b*cos(v))
[u]:-pi,pi
[v]:-pi,pi

You have to replace rho, a and b. For example, for rho=40, a=8 and b=3 you get

Code:

X():40^2*cos(u)*(8+3*cos(v))/(90+48*cos(v))
Y():40^2*(8+3*cos(v))*sin(u)/(90+48*cos(v))
Z():8*40^2*sin(v)/(90+48*cos(v))
[u]:-pi,pi
[v]:-pi,pi

Kidney is a surface of revolution generated by a nephroid

Code:

X():cos(u)*(3*cos(v) - cos(3*v))
Y():sin(u)*(3*cos(v) - cos(3*v))
Z():(3*sin(v) - sin(3*v))
[u]:0,2*pi
[v]:-pi/2,pi/2

Generalized paraboloid's polar parametrization

Code:

X():a*u*cos(v)
Y():b*u*sin(v)
Z():u^n
[u]:0,1
[v]:0,2*pi

You have to replace n, a and b. For example, for a=n=2 and c=1 you get

Code:

X():2*u*cos(v)
Y():u*sin(v)
Z():u^2
[u]:0,1
[v]:0,2*pi

The sphere's standard parametrization

Code:

X():cos(v)*cos(u)
Y():cos(v)*sin(u)
Z():sin(v)
[u]:-pi,pi
[v]:-pi/2,pi/2

The torus formed by revolving a circle of radius b in the xz-plane about the z-axis along a circle of radius a in the xy-plane.

Code:

X():(a + b*cos(v))*cos(u)
Y():(a + b*cos(v))*sin(u)
Z():b*sin(v)
[u]:0,2*pi
[v]:-pi,pi

You have to replace a and b. For example, for a=8 and b=3 you get

Code:

X():(8+3*cos(v))*cos(u)
Y():(8+3*cos(v))*sin(u)
Z():3*sin(v)
[u]:0,2*pi
[v]:-pi,pi

FRUITS REVOLUTION SURFACES

Apple

Code:

X():cos(u)*(4+3.8*cos(v))
Y():sin(u)*(4+3.8*cos(v))
Z():(cos(v)+sin(v)-1)*(1+sin(v))*log(1 - pi*v/10) + 7.5*sin(v)
[u]:0, 2*pi
[v]:-pi, pi

Pear

Code:

X():cos(u)*(4 + 3.8*cos(v))
Y():sin(u)*(4 + 3.8*cos(v))
Z():((cos(v) - 3.5)*(1 + sin(v))*log(1 - pi*v/10) + 10*sin(v))*(1 - 0.4*cos(v/2)*(1 + 0.2*sin(v^2)))
[u]:0,2*pi
[v]:-30*pi/31,30*pi/31

NON-COMPACT REVOLUTION SURFACES

Rotated witch of Agnesi

Code:

X():-cos(u)*cos(v)^2
Y():-cos(v)^2*sin(u)
Z():tan(v)
[u]:0, 2*pi
[v]:-1, 1

A catenoid is the minimal surface of revolution generated by the catenary

Code:

X():a*cosh(v/a)*cos(u)
Y():a*cosh(v/a)*sin(u)
Z():v
[u]:-pi, pi
[v]:-pi, pi

You have to replace a with a positive real number. For example, for a=1 you get.

Code:

X():2*cosh(v/2)*cos(u)
Y():2*cosh(v/2)*sin(u)
Z():v
[u]:-pi, pi
[v]:-pi, pi

Circular cone of radius a and slope b/a

Code:

X():a*v*cos(u)
Y():a*v*sin(u)
Z():b*v
[u]:-pi, pi
[v]:-1, 1

You have to replace a and b. For example for a=1 and b=0.5 you get

Code:

X():v*cos(u)
Y():v*sin(u)
Z():v/2
[u]:-pi, pi
[v]:-1, 1

Circular cylinder

Code:

X():cos(u)
Y():sin(u)
Z():v
[u]:-pi, pi
[v]:-4, 4

A circular cone inverted with respect to a sphere of radius rho. It is a cyclide of Dupin

Code:

X():a*rho^2*cos(u)/(a^2*v + b^2*v)
Y():a*rho^2*sin(u)/(a^2*v + b^2*v)
Z():b*rho^2/(a^2*v + b^2*v)
[u]:-pi,pi
[v]:0.1,rho

You have to replace rho, a and b. For example, for rho=40, a=b=1 you get

Code:

X():40^2*cos(u)/(2*v)
Y():40^2*sin(u)/(2*v)
Z():40^2/(2*v)
[u]:-pi,pi
[v]:0.1,40

Sphere's Mercator parametrization

Code:

X():cos(u)/cosh(v)
Y():sin(u)/cosh(v)
Z():tanh(v)
[u]:0,3*pi/2
[v]:-0.49*pi,0.49*pi

Polar parametrization of a plane

Code:

X():u*cos(v)
Y():u*sin(v)
Z():0
[u]:0.01,pi
[v]:-pi,pi

Pseudosphere: surface of revolution of constant negative curvature.

Code:

X():cos(u)*sin(v)
Y():sin(u)*sin(v)
Z():(cos(v) + log(tan(v/2)))
[u]:0,2*pi
[v]:0.001,pi-0.001

Pseudosphere's another parametrization

Code:

X():cos(u)*tanh(v)
Y():sin(u)*tanh(v)
Z():(1/cosh(v) + log(tanh(v/2)))
[u]:0,2*pi
[v]:0.01,2

Sphere's polar stereographic parametrization

Code:

X():2*u*cos(v)/(1 + u^2)
Y():2*u*sin(v)/(1 + u^2)
Z():(-1 + u^2)/(1 + u^2)
[u]:0,3
[v]:0,2*pi

Torus parametrization

Code:

X():u*cos(v)
Y():u*sin(v)
Z():pm*sqrt(b^2 - (r - a)^2)
[u]:a-b,a+b
[v]:0,2*pi

You have to replace pm with +1 or -1 and a and b with positive real numbers such as a>b. For example for pm=1, a=8 and b=3 you get

Code:

X():u*cos(v)
Y():u*sin(v)
Z():sqrt(9-(u-8)^2)
[u]:5,11
[v]:0,2*pi

PARAMETRIZATIONS BY PRINCIPAL CURVES

Dini's surface of constant negative curvature

Code:

X():cos(b)*cos(v)/cosh((u - v*sin(b))/cos(b))
Y():cos(b)*sin(v)/cosh((u - v*sin(b))/cos(b))
Z():(u - cos(b)*tanh((u - v*sin(b))/cos(b)))
[u]:-3, 3
[v]:0, 6*pi

You have to replace b. For example, for b=0.2 you get

Code:

X():cos(0.2)*cos(v)/cosh((u - v*sin(0.2))/cos(0.2))
Y():cos(0.2)*sin(v)/cosh((u - v*sin(0.2))/cos(0.2))
Z():(u - cos(0.2)*tanh((u - v*sin(0.2))/cos(0.2)))
[u]:-3, 3
[v]:0, 6*pi

Circular helicoid

Code:

X():cos(u + v)*sinh(-u + v)
Y():sin(u + v)*sinh(-u + v)
Z():u + v
[u]:-pi/2,pi/2
[v]:-pi/2,pi/2

Kuen's surface of constant negative curvature

Code:

X():2*cosh(u)*(cos(v) + v*sin(v))/(v^2 + cosh(u)^2)
Y():2*cosh(u)*(sin(v) - v*cos(v))/(v^2 + cosh(u)^2)
Z():(u - sinh(2*u)/(v^2 + cosh(u)^2))
[u]:-1.91, 1.91
[v]:-1.13, 1.13

Pseudosphere

Code:

X():cos(v)/cosh(u)
Y():sin(v)/cosh(u)
Z():u-tanh(u)
[u]:-2.8,2.8
[v]:0,2*pi

PARAMETRISATIONS BY ASYMPTOTIC CURVES

Catenoid

Code:

X():cos((u + v)/2)*cosh((u - v)/2)
Y():sin((u + v)/2)*cosh((u - v)/2)
Z():(u-v)/2
[u]:-pi, pi
[v]:-pi, pi

Dini's surface of constant negative curvature

Code:

X():cos(b)*cos(u-v)/cosh((u+v - (u-v)*sin(b))/cos(b))
Y():cos(b)*sin(u-v)/cosh((u+v - (u-v)*sin(b))/cos(b))
Z():(u+v - cos(b)*tanh((u+v - (u-v)*sin(b))/cos(b)))
[u]:-3, 3
[v]:0, 6*pi

You have to replace b. For example, for b=0.2 you get

Code:

X():cos(0.2)*cos(u-v)/cosh((u+v - (u-v)*sin(0.2))/cos(0.2))
Y():cos(0.2)*sin(u-v)/cosh((u+v - (u-v)*sin(0.2))/cos(0.2))
Z():(u+v - cos(0.2)*tanh((u+v - (u-v)*sin(0.2))/cos(0.2)))
[u]:-3, 3
[v]:0, 6*pi

Exponential twist (v cos(u),v sin(u), exp(c u))

Code:

X():exp(c*(u - v)/2)*cos(u)
Y():exp(c*(u - v)/2)*sin(u)
Z():exp(c*u)
[u]:-pi, 2*pi
[v]:-pi, 2*pi

You have to replace c. For example, for c=0.3 you get

Code:

X():exp(0.3*(u - v)/2)*cos(u)
Y():exp(0.3*(u - v)/2)*sin(u)
Z():exp(0.3*u)
[u]:-pi, 2*pi
[v]:-pi, 2*pi

Funnel (v cos(u), v sin(u), log(v))

Code:

X():exp(u - v)*cos(u + v)
Y():-exp(u - v)*sin(u + v)
Z():(u - v)
[u]:-pi/2, pi/2
[v]:0.0, pi

Kuen's surface of constant negative curvature

Code:

X():2*cosh(u+v)*(cos(u-v) + (u-v)*sin(u-v))/((u-v)^2 + cosh(u+v)^2)
Y():2*cosh(u+v)*(sin(u-v) - (u-v)*cos(u-v))/((u-v)^2 + cosh(u+v)^2)
Z():(u+v - sinh(2*(u+v))/((u-v)^2 + cosh(u+v)^2))
[u]:-2,2
[v]:-2,2

Pseudosphere

Code:

X():cos(u-v)/cosh(u+v)
Y():sin(u-v)/cosh(u+v)
Z():u+v-tanh(u+v)
[u]:-1,1
[v]:0,pi

Shoe

Code:

X():(-3*b/(4*a))^(1/3)*(u-v)^(2/3)
Y():-u-v
Z():(b/4)*(u^2+14*u*v+v^2)
[u]:-1,1.2
[v]:-1,1.2

You have to replace a and b. For example for a=-1 and b=1 you get

Code:

X():(3/4)^(1/3)*(u-v)^(2/3)
Y():-u-v
Z():(-1/4)*(u^2+14*u*v+v^2)
[u]:-1,1.2
[v]:-1,1.2

Torus

Code:

X():(cos((u + v)/2)/(1 + cosh((u - v)/sqrt(8))^2))
Y():(sin((u + v)/2)/(1 + cosh((u - v)/sqrt(8))^2))
Z():pm*(cosh((u - v)/sqrt(8))/(1 + cosh((u - v)/sqrt(8))^2))
[u]:-pi,pi
[v]:-pi,pi

You have to replace pm with +1 or -1. For example for pm=1 you get

Code:

X():(cos((u + v)/2)/(1 + cosh((u - v)/sqrt(8))^2))
Y():(sin((u + v)/2)/(1 + cosh((u - v)/sqrt(8))^2))
Z():(cosh((u - v)/sqrt(8))/(1 + cosh((u - v)/sqrt(8))^2))
[u]:-pi,pi
[v]:-pi,pi

OTHER SURFACES

A modified catenoid

Code:

X():a*cosh(v)*cos(u)
Y():b*cosh(v)*sin(u)
Z():c*v
[u]:-pi, pi
[v]:-pi, pi

You have to replace a,b and c. For example for a=3, b=2, c=1 you get

Code:

X():2*cosh(v)*cos(u)
Y():3*cosh(v)*sin(u)
Z():v
[u]:-pi, pi
[v]:-pi, pi

An isothermal parametrization of a catenoid

Code:

X():a*cosh(v/a)*cos(u/a)
Y():a*cosh(v/a)*sin(u/a)
Z():v
[u]:-pi, pi
[v]:-pi, pi

You have to replace a, for example for a=2 you get

Code:

X():2*cosh(v/2)*cos(u/2)
Y():2*cosh(v/2)*sin(u/2)
Z():v
[u]:-pi, pi
[v]:-pi, pi

Dini's surface of constant negative curvature -1/a^2. It is the generalized helicoid of slant b generated by a tractrix. The case b=0 is the standard parametrization of a pseudosphere.

Code:

X():a*cos(u)*sin(v)
Y():a*sin(u)*sin(v)
Z():a*(cos(v) + log(tan(v/2))) + b*u
[u]:0, 4*pi
[v]:0.05, 1

You have to replace a and b, for example for a=1 and b=0.2 you get

Code:

X():cos(u)*sin(v)
Y():sin(u)*sin(v)
Z():(cos(v) + log(tan(v/2))) + 0.2*u
[u]:0, 4*pi
[v]:0.05, 1

Elliptic paraboloid

Code:

X():u
Y():v
Z():u^2/a^2 + v^2/b^2
[u]:-1, 1
[v]:-1, 1

You have to replace a and b. For example for a=1 and b=2 you get

Code:

X():u
Y():v
Z():u^2 + v^2/2^2
[u]:-1, 1
[v]:-1, 1

Helicoid-like surface whose twisting varies exponentially

Code:

X():v*cos(u)
Y():v*sin(u)
Z():exp(c*u)
[u]:0, 4*pi
[v]:-1, 1

You have to replace c. For example for c=0.2 you get

Code:

X():v*cos(u)
Y():v*sin(u)
Z():exp(0.2*u)
[u]:0, 4*pi
[v]:-1, 1

Funnel

Code:

X():a*v*cos(u)
Y():b*v*sin(u)
Z():c*log(v)
[u]:-pi, pi
[v]:0.01, 1

You have to replace a, b and c. For example for a=5, b=1, c=2 you get

Code:

X():5*v*cos(u)
Y():v*sin(u)
Z():2*log(v)
[u]:-pi, pi
[v]:0.01, 1

Handkerchief shaped surface

Code:

X():u
Y():u
Z():(1/3)*u^3 + u*v^2 +a*(u^2 - v^2)
[u]:-1 , 1
[v]:0 , 1

You have to replace a. For example, for a=1 you get

Code:

X():u
Y():u
Z():(1/3)*u^3 + u*v^2 +(u^2 - v^2)
[u]:-1 , 1
[v]:0 , 1

Elliptical helicoid of slant c.

Code:

X():a*v*cos(u)
Y():b*v*sin(u)
Z():c*u
[u]:0,4*pi
[v]:-1,1

You have to replace a,b and c. For example for a=1, b=1, c=0.3

Code:

X():v*cos(u)
Y():v*sin(u)
Z():0.3*u
[u]:0,4*pi
[v]:-1,1

Elliptical hyperboloid of two sheets

Code:

X():a*cosh(u)*cosh(v)
Y():b*sinh(u)*cosh(v)
Z():c*sinh(v)
[u]:-1,1
[v]:-1,1

You have to replace a, b and c. For example, for a=b=c=1 you get

Code:

X():cosh(u)*cosh(v)
Y():sinh(u)*cosh(v)
Z():sinh(v)
[u]:-1,1
[v]:-1,1

Hyperbolic paraboloid

Code:

X():u
Y():v
Z():u*v
[u]:-1,1
[v]:-1,1

Elliptical hyperboloid of one sheet

Code:

X():a*cosh(v)*cos(u)
Y():b*cosh(v)*sin(u)
Z():c*sinh(v)
[u]:0,2*pi
[v]:-2,2

You have to replace a, b and c. For example for a=1, b=2 and c=3 you get

Code:

X():cosh(v)*cos(u)
Y():2*cosh(v)*sin(u)
Z():3*sinh(v)
[u]:0,2*pi
[v]:-2,2

Another parametrization of an hyperboloid

Code:

X():a*sec(v)*cos(u)
Y():b*sec(v)*sin(u)
Z():c*tan(v)
[u]:0,3*pi/2
[v]:-pi/2,pi/2

You have to replace a, b and c. For example for a=1, b=2 and c=3 you get

Code:

X():sec(v)*cos(u)
Y():2*sec(v)*sin(u)
Z():3*tan(v)
[u]:0,3*pi/2
[v]:-pi/2,pi/2

Polar parametrization of Jorge-Meeks 2-oid

Code:

X():(1/8)*log((1 + u^2 + 2*u*cos(v))/(1 + u^2 - 2*u*cos(v)))
Y():-u*(1 + u^2)*sin(v)/(2*(1 + u^4 - 2*u^2*cos(2*v)))
Z():(1 - u^4)/(4*(1 + u^4 - 2*u^2*cos(2*v)))
[u]:-1.9,1.9
[v]:0.1,pi-0.1

Kuen's surface of constant negative curvature

Code:

X():2*(cos(u) + u*sin(u))*sin(v)/(1 + u^2*sin(v)^2)
Y():2*(sin(u) - u*cos(u))*sin(v)/(1 + u^2*sin(v)^2)
Z():log(tan(v/2)) + 2*cos(v)/(1 + u^2*sin(v)^2)
[u]:-4,4
[v]:0.01, pi
-0.01

Menn's surface

Code:

X():u
Y():v
Z():a*u^4 + u^2*v - v^2
[u]:-1.5,1.5
[v]:-1.5,1.5

You have to replace a. For example, for a=1, you get

Code:

X():u
Y():v
Z():u^4 + u^2*v - v^2
[u]:-1.5,1.5
[v]:-1.5,1.5

Mercator injection of an ellipsoid with axes of lengths a, b and c

Code:

X():cos(u)/cosh(v)
Y():2*sin(u)/cosh(v)
Z():3*tanh(v)
[u]:0,3*pi/2
[v]:-0.49*pi,0.49*pi

You have to replace a,b and c. For example for a=1, b=2 and c=3 you get

Code:

X():cos(u)/cosh(v)
Y():2*sin(u)/cosh(v)
Z():3*tanh(v)
[u]:0,3*pi/2
[v]:-0.49*pi,0.49*pi

Parametrization of a Moebius strip that has a circle as boundary

Code:

X():(-2*cos(2*v)*sin(u))/
(-2 + sqrt(2)*cos(u)*sin(v) + sqrt(2)*sin(u)*sin(2*v))
Y():(sqrt(2)*(cos(u)*sin(v) - sin(u)*sin(2*v)))/
(-2 + sqrt(2)*cos(u)*sin(v) + sqrt(2)*sin(u)*sin(2*v))
Z():(-2*cos(u)*cos(v))/
(-2 + sqrt(2)*cos(u)*sin(v) + sqrt(2)*sin(u)*sin(2*v))
[u]:0,pi
[v]:0,pi

Moebius strip's standard parametrization

Code:

X():(cos(u) + v*cos(u/2)*cos(u))
Y():(sin(u) + v*cos(u/2)*sin(u))
Z():v*sin(u/2)
[u]:0,2*pi
[v]:-0.3,0.3

Monkey saddle

Code:

X():u
Y():v
Z():u^3 - 3*u*v^2
[u]:-1.1,1.1
[v]:-1.1,1.1

Polar parametrization of a monkey saddle with n - 2 tails.

Code:

X():u*cos(v)
Y():u*sin(v)
Z():u^n*cos(n*v)
[u]:0, 1
[v]:0, 2*pi

You have to replace n. For example, for n=3 you get.

Code:

X():u*cos(v)
Y():u*sin(v)
Z():u^3*cos(3*v)
[u]:0, 1
[v]:0, 2*pi

Cyclide of Dupin of radius k whose focal sets are the parabolas u->{u,0,-u^2/(8*a) + a} and v->{0,v,v^2/(8*a) - a}

Code:

X():u*(8*a^2 + k + v^2)/(16*a^2 + u^2 + v^2)
Y():v*(8*a^2 - k + u^2)/(16*a^2 + u^2 + v^2)
Z():(16*a^2*(k - u^2 + v^2) - k*(u^2 + v^2))/(8*a*(16*a^2 + u^2 + v^2))
[u]:-40,40
[v]:-40,40

You have to replace a and k. For example for a=2 and k=1 you get

Code:

X():u*(33 + v^2)/(64 + u^2 + v^2)
Y():v*(31 + u^2)/(64 + u^2 + v^2)
Z():(64*(1 - u^2 + v^2) - (u^2 + v^2))/(16*(64 + u^2 + v^2))
[u]:-40,40
[v]:-40,40

Elliptic or hyperbolic paraboloid

Code:

X():u
Y():v
Z():a*u^2 + b*v^2+c*u*v
[u]:-1,1
[v]:-1,1

You have to replace a, b and c. For example for a=b=1 and c=0 you get

Code:

X():u
Y():v
Z():u^2 + v^2
[u]:-1,1
[v]:-1,1

Polar parametrization of a generalized paraboloid

Code:

X():a*u*cos(v)
Y():b*u*sin(v)
Z():b*u^n
[u]:0,1
[v]:0,2*pi

You have to replace n, a and b. For example for n=2, a=2 and b=1 you get

Code:

X():2*u*cos(v)
Y():u*sin(v)
Z():u^2
[u]:0,1
[v]:0,2*pi

Monkey saddle perturbed by a circular paraboloid

Code:

X():u
Y():v
Z():u^3 - 3*u*v^2 + a*(u^2 + v^2)
[u]:-1,1
[v]:-1,1

You have to replace a. For example for a=-1 you get

Code:

X():u
Y():v
Z():u^3 - 3*u*v^2 - (u^2 + v^2)
[u]:-1,1
[v]:-1,1

Polar parametrization of a monkey saddle of order n perturbed by a circular paraboloid.

Code:

X():u*cos(v)
Y():u*sin(v)
Z():u^n*cos(n*v) + a*u^2
[u]:0,1
[v]:0,2*pi

You have to replace n and a. For example for n=3 and a=-1 you get

Code:

X():u*cos(v)
Y():u*sin(v)
Z():u^3*cos(3*v) - u^2
[u]:0,1
[v]:0,2*pi

A plane

Code:

X():a1*u + a2*v
Y():b1*u + b2*v
Z():c1*u + c2*v
[u]:-pi,pi
[v]:-pi,pi

You have to replace a1, b1, c1, a2, b2 and c2. For example

Code:

X():u+v
Y():u-v
Z():2*u-3*v
[u]:-pi,pi
[v]:-pi,pi

Generalization of the monkey saddle and Plucker's surface

Code:

X():u
Y():v
Z():(u+v)^(m/2)*sin(4*atan(v/u))
[u]:-1,1
[v]:-1,1

You have to replace m and n. For example for m=2 and n=4 you get

Code:

X():u
Y():v
Z():(u+v)*sin(4*atan(v/u))
[u]:-1,1
[v]:-1,1

Plucker's surface with n folds

Code:

X():u
Y():v
Z():sin(n*atan(v/u))
[u]:-1,1
[v]:-1,1

You have to replace n. For example, for n=6 you get

Code:

X():u
Y():v
Z():sin(6*atan(v/u))
[u]:-1,1
[v]:-1,1

Plucker's surface

Code:

X():u
Y():v
Z():2*u*v/(u^2 + v^2)
[u]:-1,1
[v]:-1,1

Generalization of the polar parametrizations of the monkey saddle and Plucker's surface.

Code:

X():u*cos(v)
Y():u*sin(v)
Z():u^m*sin(n*v)
[u]:0,1
[v]:-pi,pi

You have to replace m and n. For example for m=2 and n=3 you get

Code:

X():u*cos(v)
Y():u*sin(v)
Z():u^2*sin(3*v)
[u]:0,1
[v]:-pi,pi

Polar parametrization of Plucker's surface with n folds

Code:

X():u*cos(v)
Y():u*sin(v)
Z():sin(n*v)
[u]:0,1
[v]:-pi,pi

You have to replace n. For example for n=5 you get

Code:

X():u*cos(v)
Y():u*sin(v)
Z():sin(5*v)
[u]:0,1
[v]:-pi,pi

Elliptical "pseudosphere"

Code:

X():a*cos(u)*sin(v)
Y():b*sin(u)*sin(v)
Z():c*(cos(v) + log(tan(v/2)))
[u]:0,2*pi
[v]:0.001,pi-0.001

You have to replace a, b and c. For example for a=1, b=2 and c=3 you get

Code:

X():cos(u)*sin(v)
Y():2*sin(u)*sin(v)
Z():3*(cos(v) + log(tan(v/2)))
[u]:0,2*pi
[v]:0.001,pi-0.001

Right conoid

Code:

X():v*cos(u)
Y():v*sin(u)
Z():2*sin(u)
[u]:-pi,pi
[v]:-2,2

Shoe resembles the instep of a shoe.

Code:

X():u
Y():v
Z():a*u^3+b*v^2
[u]:-1.5,1.5
[v]:-1.5,1.5

You have to replace a and b. For example for a=1 and b=-1 you get

Code:

X():u
Y():v
Z():u^3-v^2
[u]:-1.5,1.5
[v]:-1.5,1.5

Sievert's surface of constant positive curvature a^2.

Code:

X():(2/(a + 1 - a*sin(v)^2*cos(u)^2))*(sqrt((a + 1)*(1 + a*sin(u)^2))*sin(v)/sqrt(a))*cos(-u/sqrt(a + 1) + atan(sqrt(a + 1)*tan(u)))
Y():(2/(a + 1 - a*sin(v)^2*cos(u)^2))*(sqrt((a + 1)*(1 + a*sin(u)^2))*sin(v)/sqrt(a))*sin(-u/sqrt(a + 1) + atan(sqrt(a + 1)*tan(u)))
Z():log(tan(v/2))/sqrt(a) + 2*(a + 1)*cos(v)/((a + 1 - a*sin(v)^2*cos(u)^2)*sqrt(a))
[u]:-pi/2,pi/2
[v]:0.1,pi-0.1

You have to replace a. For example for a=1 you get

Code:

X():(2/(2 - sin(v)^2*cos(u)^2))*(sqrt(2*(1 + sin(u)^2))*sin(v))*cos(-u/sqrt(2) + atan(sqrt(2)*tan(u)))
Y():(2/(2 - sin(v)^2*cos(u)^2))*(sqrt(2*(1 + sin(u)^2))*sin(v))*sin(-u/sqrt(2) + atan(sqrt(2)*tan(u)))
Z():log(tan(v/2)) + 4*cos(v)/((2 - sin(v)^2*cos(u)^2))
[u]:-pi/2,pi/2
[v]:0.1,pi-0.1

Stereographic parametrization of an ellipsoid with axes of lengths a, b and c

Code:

X():2*a*u/(u^2 + v^2 + 1)
Y():2*b*v/(u^2 + v^2 + 1)
Z():c*(u^2 + v^2 - 1)/(u^2 + v^2 + 1)
[u]:-2.8,2.8
[v]:-2.8,2.8

You have to replace a, b and c. For example for a=5, b=3 and c=1 you get

Code:

X():10*u/(u^2 + v^2 + 1)
Y():6*v/(u^2 + v^2 + 1)
Z():(u^2 + v^2 - 1)/(u^2 + v^2 + 1)
[u]:-2.8,2.8
[v]:-2.8,2.8

Stereographic parametrization of a sphere

Code:

X():2*u/(u^2 + v^2 + 1)
Y():2*v/(u^2 + v^2 + 1)
Z():(u^2 + v^2 - 1)/(u^2 + v^2 + 1)
[u]:-2.8,2.8
[v]:-2.8,2.8

Polar stereographic parametrization of a sphere

Code:

X():2*exp(b*v)*u*cos(v)/(1 + exp(2*b*v)*u^2)
Y():2*exp(b*v)*u*sin(v)/(1 + exp(2*b*v)*u^2)
Z():(-1 + exp(2*b*v)*u^2)/(1 + exp(2*b*v)*u^2)
[u]:0,3
[v]:0,2*pi

You have to replace b. For example for b=0.05 you get

Code:

X():2*exp(0.05*v)*u*cos(v)/(1 + exp(0.1*v)*u^2)
Y():2*exp(0.05*v)*u*sin(v)/(1 + exp(0.1*v)*u^2)
Z():(-1 + exp(0.1*v)*u^2)/(1 + exp(0.1*v)*u^2)
[u]:0,3
[v]:0,2*pi

Swallow tail shaped surface

Code:

X():3*v^4 + u*v^2
Y():-4*v^3 - 2*u*v
Z():u
[u]:-3,2
[v]:-0.8,0.8

Tetrahedral surface

Code:

X():A*(u - a)^m*(v - a)^n
Y():B*(u - b)^m*(v - b)^n
Z():C*(u - c)^m*(v - c)^n
[u]:0,1
[v]:0,1

You have to replace A, a, B, b, C and c. For example

Code:

X():(u-1)*(v-1)^2
Y():2*(u-1/2)*(v-1/2)^2
Z():1/3*(u-3)*(v-3)^2
[u]:0,1
[v]:0,1

Generalized helicoid of zero Gaussian curvature of slant sl

Code:

X():v*cos(u)
Y():v*sin(u)
Z():sl*u+asin(c/(v*a))*c+sqrt(v^2*a^2-c^2)
[u]:0,7*pi/2
[v]:1,5

You have to replace a, c and sl. For example, for a=c=sl=1 you get[code:1:89cab81d3d]X():v*cos(u)
Y():v*sin(u)
Z():u+asin(1/v)+sqrt(v^2-1)
[u]:0,7*pi/2

abdelhamid belaid

Joined: 13 Aug 2009
Posts: 170

Posted: Wed Feb 01, 2012 4:27 pm Post subject:

Very nice works, thank you much Jolkap _________________ My YouTube channel

nextstep
Site Admin

Joined: 06 Jan 2007
Posts: 539

Posted: Sun Feb 05, 2012 11:39 pm Post subject:

Hi, Great list indeed... Thank you for sharing _________________ Cheers, Abderrahman

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