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jolkaP
Joined: 17 Oct 2011 Posts: 12 Location: Marseille (France)
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Posted: Thu Jan 12, 2012 11:57 pm Post subject: Surfaces from Alfred Grays "Modern Differential Geometr |
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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. |
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jolkaP
Joined: 17 Oct 2011 Posts: 12 Location: Marseille (France)
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Posted: Fri Jan 13, 2012 12:34 am Post subject: Parametric morphed surfaces |
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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
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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
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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
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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
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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. |
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Phil999
Joined: 08 Feb 2007 Posts: 24
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Posted: Thu Jan 19, 2012 9:38 pm Post subject: |
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thank you |
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jolkaP
Joined: 17 Oct 2011 Posts: 12 Location: Marseille (France)
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Posted: Tue Jan 31, 2012 11:06 pm Post subject: Almost all parametric surfaces from this source |
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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
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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
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abdelhamid belaid
Joined: 13 Aug 2009 Posts: 170
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Posted: Wed Feb 01, 2012 4:27 pm Post subject: |
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Very nice works, thank you much Jolkap _________________ My YouTube channel |
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nextstep Site Admin
Joined: 06 Jan 2007 Posts: 539
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Posted: Sun Feb 05, 2012 11:39 pm Post subject: |
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Hi,
Great list indeed... Thank you for sharing _________________ Cheers,
Abderrahman |
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