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 cmp_ir(self,z):
"""
returns -1 for left, 0 for in, and 1 for right from initial region
cut line is on the north ray from L.
"""
L = self.L
x0 = self.x0
if x0 > 0.5:
if real(z) > real(L) and abs(z) < abs(L):
return 0
if real(z) < real(L):
return -1
if real(z) > real(L):
return 1
else:
if imag(z) > imag(L):
if real(z) > real(L):
return 1
if real(z) < real(L):
return -1
if real(z) < real(L) and real(z) > log(real(L)) + log(sqrt(1+tan(imag(z))**2)):
return 0
if real(z) > real(L):
return 1
if real(z) < real(L):
return -1
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 cmp_ir(self,z):
"""
returns -1 for left, 0 for in, and 1 for right from initial region
cut line is on the north ray from pfp.
Works only for real x0.
"""
pfp = self.pfp
x0 = self.x0
if x0 > 0.5:
print z,abs(z)
if real(z) >= real(pfp) and abs(z) < abs(pfp):
return 0
if real(z) < real(pfp):
return -1
if real(z) > real(pfp):
return 1
else:
if imag(z) > imag(pfp):
if real(z) > real(pfp):
return 1
if real(z) < real(pfp):
return -1
if real(z) < real(pfp) and real(z) > log(real(pfp)) + log(sqrt(1+tan(imag(z))**2)):
return 0
if real(z) > real(pfp):
return 1
if real(z) < real(pfp):
return -1
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)
