def Dini(a=1, b=1, name="Dini's surface"):
r"""
Return Dini's surface, with parametrization
.. MATH::
\begin{aligned}
x(u, v) & = a \cos(u)\sin(v); \\
y(u, v) & = a \sin(u)\sin(v); \\
z(u, v) & = u + \log(\tan(v/2)) + \cos(v).
\end{aligned}
INPUT:
- ``a, b`` -- surface parameters.
- ``name`` -- string. Name of the surface.
For more information, see :wikipedia:`Dini%27s_surface`.
EXAMPLES::
sage: dini = surfaces.Dini(a=3, b=4); dini
Parametrized surface ('Dini's surface') with equation (3*cos(u)*sin(v), 3*sin(u)*sin(v), 4*u + 3*cos(v) + 3*log(tan(1/2*v)))
sage: dini.plot()
Graphics3d Object
"""
u, v = var('u, v')
dini_eq = [
a * cos(u) * sin(v), a * sin(u) * sin(v),
a * (cos(v) + log(tan(v / 2))) + b * u
]
coords = ((u, 0, 2 * pi), (v, 0, 2 * pi))
return ParametrizedSurface3D(dini_eq, coords, name)
def WhitneyUmbrella(name="Whitney's umbrella"):
r"""
Return Whitney's umbrella, with parametric representation
.. MATH::
x(u, v) = uv, \quad y(u, v) = u, \quad z(u, v) = v^2.
INPUT:
- ``name`` -- string. Name of the surface.
For more information, see :wikipedia:`Whitney_umbrella`.
EXAMPLES::
sage: whitney = surfaces.WhitneyUmbrella(); whitney
Parametrized surface ('Whitney's umbrella') with equation (u*v, u, v^2)
sage: whitney.plot()
Graphics3d Object
"""
u, v = var('u, v')
whitney_eq = [u * v, u, v**2]
coords = ((u, -1, 1), (v, -1, 1))
return ParametrizedSurface3D(whitney_eq, coords, name)
def Crosscap(r=1, name="Crosscap"):
r"""
Return a crosscap surface, with parametrization
.. MATH::
\begin{aligned}
x(u, v) & = r(1 + \cos(v)) \cos(u); \\
y(u, v) & = r(1 + \cos(v)) \sin(u); \\
z(u, v) & = - r\tanh(u - \pi) \sin(v).
\end{aligned}
INPUT:
- ``r`` -- surface parameter.
- ``name`` -- string. Name of the surface.
For more information, see :wikipedia:`Cross-cap`.
EXAMPLES::
sage: crosscap = surfaces.Crosscap(); crosscap
Parametrized surface ('Crosscap') with equation ((cos(v) + 1)*cos(u), (cos(v) + 1)*sin(u), -sin(v)*tanh(-pi + u))
sage: crosscap.plot()
Graphics3d Object
"""
u, v = var('u, v')
crosscap_eq = [
r * (1 + cos(v)) * cos(u), r * (1 + cos(v)) * sin(u),
-tanh(u - pi) * r * sin(v)
]
coords = ((u, 0, 2 * pi), (v, 0, 2 * pi))
return ParametrizedSurface3D(crosscap_eq, coords, name)
def MonkeySaddle(name="Monkey saddle"):
r"""
Return a monkey saddle surface, with equation
.. MATH::
z = x^3 - 3xy^2.
INPUT:
- ``name`` -- string. Name of the surface.
For more information, see :wikipedia:`Monkey_saddle`.
EXAMPLES::
sage: saddle = surfaces.MonkeySaddle(); saddle
Parametrized surface ('Monkey saddle') with equation (u, v, u^3 - 3*u*v^2)
sage: saddle.plot()
Graphics3d Object
"""
u, v = var('u, v')
monkey_eq = [u, v, u**3 - 3 * u * v**2]
coords = ((u, -2, 2), (v, -2, 2))
return ParametrizedSurface3D(monkey_eq, coords, name)
def Torus(r=2, R=3, name="Torus"):
r"""
Return a torus obtained by revolving a circle of radius ``r`` around
a coplanar axis ``R`` units away from the center of the circle. The
parametrization used is
.. MATH::
\begin{aligned}
x(u, v) & = (R + r \cos(v)) \cos(u); \\
y(u, v) & = (R + r \cos(v)) \sin(u); \\
z(u, v) & = r \sin(v).
\end{aligned}
INPUT:
- ``r``, ``R`` -- Minor and major radius of the torus.
- ``name`` -- string. Name of the surface.
For more information, see :wikipedia:`Torus`.
EXAMPLES::
sage: torus = surfaces.Torus(); torus
Parametrized surface ('Torus') with equation ((2*cos(v) + 3)*cos(u), (2*cos(v) + 3)*sin(u), 2*sin(v))
sage: torus.plot()
Graphics3d Object
"""
u, v = var('u, v')
torus_eq = [(R + r * cos(v)) * cos(u), (R + r * cos(v)) * sin(u),
r * sin(v)]
coords = ((u, 0, 2 * pi), (v, 0, 2 * pi))
return ParametrizedSurface3D(torus_eq, coords, name)
def Catenoid(c=1, name="Catenoid"):
r"""
Return a catenoid surface, with parametric representation
.. MATH::
\begin{aligned}
x(u, v) & = c \cosh(v/c) \cos(u); \\
y(u, v) & = c \cosh(v/c) \sin(u); \\
z(u, v) & = v.
\end{aligned}
INPUT:
- ``c`` -- surface parameter.
- ``name`` -- string. Name of the surface.
For more information, see :wikipedia:`Catenoid`.
EXAMPLES::
sage: cat = surfaces.Catenoid(); cat
Parametrized surface ('Catenoid') with equation (cos(u)*cosh(v), cosh(v)*sin(u), v)
sage: cat.plot()
Graphics3d Object
"""
u, v = var('u, v')
catenoid_eq = [c * cosh(v / c) * cos(u), c * cosh(v / c) * sin(u), v]
coords = ((u, 0, 2 * pi), (v, -1, 1))
return ParametrizedSurface3D(catenoid_eq, coords, name)
def Klein(r=1, name="Klein bottle"):
r"""
Return the Klein bottle, in the figure-8 parametrization given by
.. MATH::
\begin{aligned}
x(u, v) & = (r + \cos(u/2)\cos(v) - \sin(u/2)\sin(2v)) \cos(u); \\
y(u, v) & = (r + \cos(u/2)\cos(v) - \sin(u/2)\sin(2v)) \sin(u); \\
z(u, v) & = \sin(u/2)\cos(v) + \cos(u/2)\sin(2v).
\end{aligned}
INPUT:
- ``r`` -- radius of the "figure-8" circle.
- ``name`` -- string. Name of the surface.
For more information, see :wikipedia:`Klein_bottle`.
EXAMPLES::
sage: klein = surfaces.Klein(); klein
Parametrized surface ('Klein bottle') with equation (-(sin(1/2*u)*sin(2*v) - cos(1/2*u)*sin(v) - 1)*cos(u), -(sin(1/2*u)*sin(2*v) - cos(1/2*u)*sin(v) - 1)*sin(u), cos(1/2*u)*sin(2*v) + sin(1/2*u)*sin(v))
sage: klein.plot()
Graphics3d Object
"""
u, v = var('u, v')
x = (r + cos(u / 2) * sin(v) - sin(u / 2) * sin(2 * v)) * cos(u)
y = (r + 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)
klein_eq = [x, y, z]
coords = ((u, 0, 2 * pi), (v, 0, 2 * pi))
return ParametrizedSurface3D(klein_eq, coords, name)
def Helicoid(h=1, name="Helicoid"):
r"""
Return a helicoid surface, with parametrization
.. MATH::
\begin{aligned}
x(\rho, \theta) & = \rho \cos(\theta); \\
y(\rho, \theta) & = \rho \sin(\theta); \\
z(\rho, \theta) & = h\theta/(2\pi).
\end{aligned}
INPUT:
- ``h`` -- distance along the z-axis between two
successive turns of the helicoid.
- ``name`` -- string. Name of the surface.
For more information, see :wikipedia:`Helicoid`.
EXAMPLES::
sage: helicoid = surfaces.Helicoid(h=2); helicoid
Parametrized surface ('Helicoid') with equation (rho*cos(theta), rho*sin(theta), theta/pi)
sage: helicoid.plot()
Graphics3d Object
"""
rho, theta = var('rho, theta')
helicoid_eq = [
rho * cos(theta), rho * sin(theta), h * theta / (2 * pi)
]
coords = ((rho, -2, 2), (theta, 0, 2 * pi))
return ParametrizedSurface3D(helicoid_eq, coords, name)
def Enneper(name="Enneper's surface"):
r"""
Return Enneper's surface, with parametrization
.. MATH::
\begin{aligned}
x(u, v) & = u(1 - u^2/3 + v^2)/3; \\
y(u, v) & = -v(1 - v^2/3 + u^2)/3; \\
z(u, v) & = (u^2 - v^2)/3.
\end{aligned}
INPUT:
- ``name`` -- string. Name of the surface.
For more information, see :wikipedia:`Enneper_surface`.
EXAMPLES::
sage: enn = surfaces.Enneper(); enn
Parametrized surface ('Enneper's surface') with equation (-1/9*(u^2 - 3*v^2 - 3)*u, -1/9*(3*u^2 - v^2 + 3)*v, 1/3*u^2 - 1/3*v^2)
sage: enn.plot()
Graphics3d Object
"""
u, v = var('u, v')
enneper_eq = [
u * (1 - u**2 / 3 + v**2) / 3, -v * (1 - v**2 / 3 + u**2) / 3,
(u**2 - v**2) / 3
]
coords = ((u, -3, 3), (v, -3, 3))
return ParametrizedSurface3D(enneper_eq, coords, name)
def Dini(a=1, b=1, name="Dini's surface"):
r"""
Returns Dini's surface, with parametrization
.. MATH::
\begin{aligned}
x(u, v) & = a \cos(u)\sin(v); \\
y(u, v) & = a \sin(u)\sin(v); \\
z(u, v) & = u + \log(\tan(v/2)) + \cos(v).
\end{aligned}
INPUT:
- ``a, b`` -- surface parameters.
- ``name`` -- string. Name of the surface.
EXAMPLES::
sage: dini = surfaces.Dini(a=3, b=4); dini
Parametrized surface ('Dini's surface') with equation (3*cos(u)*sin(v), 3*sin(u)*sin(v), 4*u + 3*cos(v) + 3*log(tan(1/2*v)))
sage: dini.plot() # not tested -- known bug (see #10132)
"""
u, v = var('u, v')
dini_eq = [a*cos(u)*sin(v), a*sin(u)*sin(v),
a*(cos(v) + log(tan(v/2))) + b*u]
coords = ((u, 0, 2*pi), (v, 0, 2*pi))
return ParametrizedSurface3D(dini_eq, coords, name)
def Paraboloid(a=1, b=1, c=1, elliptic=True, name=None):
r"""
Return a paraboloid with equation
.. MATH::
\frac{z}{c} = \pm \frac{x^2}{a^2} + \frac{y^2}{b^2}
When the plus sign is selected, the paraboloid is elliptic. Otherwise
the surface is a hyperbolic paraboloid.
INPUT:
- ``a``, ``b``, ``c`` -- Surface parameters.
- ``elliptic`` (default: True) -- whether to create an elliptic or
hyperbolic paraboloid.
- ``name`` -- string. Name of the surface.
For more information, see :wikipedia:`Paraboloid`.
EXAMPLES::
sage: epar = surfaces.Paraboloid(1, 3, 2); epar
Parametrized surface ('Elliptic paraboloid') with equation (u, v, 2*u^2 + 2/9*v^2)
sage: epar.plot()
Graphics3d Object
sage: hpar = surfaces.Paraboloid(2, 3, 1, elliptic=False); hpar
Parametrized surface ('Hyperbolic paraboloid') with equation (u, v, -1/4*u^2 + 1/9*v^2)
sage: hpar.plot()
Graphics3d Object
"""
u, v = var('u, v')
x = u
y = v
if elliptic:
z = c * (v**2 / b**2 + u**2 / a**2)
else:
z = c * (v**2 / b**2 - u**2 / a**2)
paraboloid_eq = [x, y, z]
coords = ((u, -3, 3), (v, -3, 3))
if name is None:
if elliptic:
name = "Elliptic paraboloid"
else:
name = "Hyperbolic paraboloid"
return ParametrizedSurface3D(paraboloid_eq, coords, name)
def Ellipsoid(center=(0,0,0), axes=(1,1,1), name="Ellipsoid"):
r"""
Returns an ellipsoid centered at ``center`` whose semi-principal axes
have lengths given by the components of ``axes``. The
parametrization of the ellipsoid is given by
.. MATH::
\begin{aligned}
x(u, v) & = x_0 + a \cos(u) \cos(v); \\
y(u, v) & = y_0 + b \sin(u) \cos(v); \\
z(u, v) & = z_0 + c \sin(v).
\end{aligned}
INPUT:
- ``center`` -- 3-tuple. Coordinates of the center of the ellipsoid.
- ``axes`` -- 3-tuple. Lengths of the semi-principal axes.
- ``name`` -- string. Name of the ellipsoid.
EXAMPLES::
sage: ell = surfaces.Ellipsoid(axes=(1, 2, 3)); ell
Parametrized surface ('Ellipsoid') with equation (cos(u)*cos(v), 2*cos(v)*sin(u), 3*sin(v))
sage: ell.plot()
Graphics3d Object
"""
u, v = var ('u, v')
x, y, z = center
a, b, c = axes
ellipsoid_parametric_eq = [x + a*cos(u)*cos(v),
y + b*sin(u)*cos(v),
z + c*sin(v)]
coords = ((u, 0, 2*pi), (v, -pi/2, pi/2))
return ParametrizedSurface3D(ellipsoid_parametric_eq, coords, name)