ddg.math.parametrizations.discrete_confocal2d module

Discrete parametrized surfaces and coordinate systems.

ddg.math.parametrizations.discrete_confocal2d.discrete_confocal_conics_gamma(n1, n2, b1, b2)[source]

2-dimensional discrete confocal coordinate system defined on 1/2 Z^2.

The parametrization is given in terms of gamma functions

x(n1, n2) = x_sgn * D * (n1 + b1)_1/2 * (n2 + b1 - 0.5)_1/2
y(n1, n2) = y_sgn * D * (-n1 - b2)_1/2 * (n2 + b2)_1/2

where

D = 1 / sqrt(b1 - b2 - 0.5)

and (.)_1/2 denotes the Pochhammer symbol

(n)_1/2 = gamma(n + 1/2) / gamma(n).

Incident points from dual sublattices, e.g. Z^2 and (Z+1/2)^2 are related by polarity

x * x* / (a1 + u) + y * y* / (a2 + u) = 1

where

a1 = b1 + 0.5
a2 = b2 + 1.
(d1, d2) in {-1,1} x {-1, 1}
x = x(n1, n2)
x* = x(n1 + 0.5*d1, n2 + 0.5*d2)
y = y(n1, n2)
y* = y(n1 + 0.5*d1, n2 + 0.5*d2)
u = u1(n1 + 0.25*d1) or u = u2(n2 + 0.25*d2)

The relation to the corresponding confocal conics that constitute the polarity is given by

u1(n1 + 0.25) = n1 + b1 - a1 = n1 - 0.5
u2(n2 + 0.25) = n2 + b2 - a2 = n2 - 1.0

as computed in discrete_confocal_conics_gamma_u1 and discrete_confocal_conics_gamma_u2.

With x_sgn = y_sgn = 1 this defines a parametrization of the first quadrant. Here the parametrization is extended to the entire plane by reflection along the coordinate axes. This is done by computing x_sgn and y_sgn accordingly.

Parameters:

n1float in [-b1, -b2]: First parameter (should be integer and half-integer values).
n2float in [-b2, inf): Second parameter (should be integer and half-integer values).
b1float with b1 > b2: Determines the first semi axis of the conics.
b2float with b1 > b2: Determines the second semi axis of the conics.

Returns:

np.ndarray of shape (2,): Point in R^2.

ddg.math.parametrizations.discrete_confocal2d.discrete_confocal_conics_gamma_u1(n1)[source]

Auxiliary function for discrete confocal coordinates discrete_confocal_conics_gamma to compute the cooresponding continuous confocal conics

x**2 / (a1 + u) + y**2 / (a2 + u) = 1

that establish the polarity relation.

It computes the u = u1 value for the conic adjacent to points with parameter n1:

u1(n1 + 0.25) = n1 - 0.5

Note that 0.25 is implicitely added to the input parameter.

Parameters:

n1float: Should be integer or half-integer value.

Returns:

float: The corresponding u1 value.

See also

discrete_confocal_conics_gamma

ddg.math.parametrizations.discrete_confocal2d.discrete_confocal_conics_gamma_u2(n2)[source]

Auxiliary function for discrete confocal coordinates discrete_confocal_conics_gamma to compute the cooresponding continuous confocal conics

x**2 / (a1 + u) + y**2 / (a2 + u) = 1

that establish the polarity relation.

It computes the u = u2 value for the conic adjacent to points with parameter n2:

u2(n2 + 0.25) = n2 - 1.0

Note that 0.25 is implicitely added to the input parameter.

Parameters:

n2float: Should be integer or half-integer value.

Returns:

float: The corresponding u2 value.

See also

discrete_confocal_conics_gamma

ddg.math.parametrizations.discrete_confocal2d.discrete_confocal_conics_trigonometric(n1, n2, a1, a2, h1, h2, c1=0.0, c2=0.0)[source]

2-dimensional discrete confocal coordinate system defined on 1/2 Z^2.

The parametrization is given in terms of trigonometric functions

x(n1, n2) = A * cos(h1 * n1 + c1) * cosh(h2 * n2 + c2)
y(n1, n2) = A * sin(h1 * n1 + c1) * sinh(h2 * n2 + c2)

where

A = sqrt((a1 - a2) / (cos(h1 / 2) * cosh(h2 / 2))).

Incident points from dual sublattices are related by polarity. The parameters of the corresponding confocal conics that constitute the polarity are computed in discrete_confocal_conics_trigonometric_u1 and discrete_confocal_conics_trigonometric_u2.

The parametrization becomes periodic in n1 for h1 = 2*pi / m with some integer m.

Parameters:

n1float: First parameter (should be integer and half-integer values).
n2float: Second parameter (should be integer and half-integer values).
a1float with a1 > a2: Determines the first semi axis of the conics.
a2float with a1 > a2: Determines the second semi axis of the conics.
h1float: Frequency in first parameter direction.
h2float: Frequency in second parameter direction.
c1float: Shift in first parameter direction.
c2float: Shift in second parameter direction.

Returns:

np.ndarray of shape (2,): Point in R^2.

ddg.math.parametrizations.discrete_confocal2d.discrete_confocal_conics_trigonometric_u1(n1, a1, a2, h1, c1=0.0)[source]

Auxiliary function for discrete confocal coordinates discrete_confocal_conics_trigonometric to compute the cooresponding continuous confocal conics that establish the polarity relation.

It computes the u = u1(n1 + 0.25) value for the conic adjacent to points with parameter n1.

Parameters:

n1float: Should be integer or half-integer value.
a1float with a1 > a2: Determines the first semi axis of the conics.
a2float with a1 > a2: Determines the second semi axis of the conics.
h1float: Frequency in first parameter direction.
c1float: Shift in first parameter direction.

Returns:

float: The corresponding u1 value.

See also

discrete_confocal_conics_trigonomitric

ddg.math.parametrizations.discrete_confocal2d.discrete_confocal_conics_trigonometric_u2(n2, a1, a2, h2, c2=0.0)[source]

Auxiliary function for discrete confocal coordinates discrete_confocal_conics_trigonometric to compute the cooresponding continuous confocal conics that establish the polarity relation.

It computes the u = u2(n2 + 0.25) value for the conic adjacent to points with parameter n2.

Parameters:

n2float: Should be integer or half-integer value.
a1float with a1 > a2: Determines the first semi axis of the conics.
a2float with a1 > a2: Determines the second semi axis of the conics.
h2float: Frequency in second parameter direction.
c2float: Shift in second parameter direction.

Returns:

float: The corresponding u2 value.

See also

discrete_confocal_conics_trigonomitric

ddg.math.parametrizations.discrete_confocal2d.discrete_confocal_conics_concentric(n1, n2, a1, a2, d, c1=0.0, c2=0.0)[source]

2-dimensional discrete confocal coordinate system defined on 1/2 Z^2.

The parametrization is given by

x(n1, n2) = sqrt(a1 - a2) * d**2 * (n1 + c1) * (n2 + c2)
y(n1, n2) = sqrt(a1 - a2) * (n1 + p(c1))_1/2 * (-(n1 + m(c1)) + 0.5)_1/2
                          * (n2 + p(c2))_1/2 * (-(n2 + m(c2)))_1/2

where (.)_1/2 denotes the Pochhammer symbol

(n)_1/2 = gamma(n + 1/2) / gamma(n),

and

p(c) = c + 0.25 + sqrt(16 + d**2) / (4*d),
m(c) = c + 0.25 - sqrt(16 + d**2) / (4*d).

Incident points from dual sublattices are related by polarity. The parameters of the corresponding confocal conics that constitute the polarity are computed in discrete_confocal_conics_concentric_u1 and discrete_confocal_conics_concentric_u2.

With c1 = c2 = 0 and

d = 4 / sqrt( (2*l + 1)**2 - 1 )  for some positive integer l

one can achieve boundary conditions for n1 in [-(l+1)/2, (l+1)/2] and n2 in [l/2, inf].

Parameters:

n1float: First parameter (should be integer and half-integer values).
n2float: Second parameter (should be integer and half-integer values).
a1float with a1 > a2: Determines the first semi axis of the conics.
a2float with a1 > a2: Determines the second semi axis of the conics.
dfloat: Scaling factor.
c1float: Shift in first parameter direction.
c2float: Shift in second parameter direction.

Returns:

np.ndarray of shape (2,): Point in R^2.

Notes

Not tested yet.

ddg.math.parametrizations.discrete_confocal2d.discrete_confocal_conics_concentric_u1(n1, a1, a2, d, c1=0)[source]

Auxiliary function for discrete confocal coordinates discrete_confocal_conics_concentric to compute the cooresponding continuous confocal conics that establish the polarity relation.

Parameters:

n1float
a1float
a2float
dfloat
c1float

Returns:

float: The corresponding u1 value.

See also

discrete_confocal_conics_concentric

ddg.math.parametrizations.discrete_confocal2d.discrete_confocal_conics_concentric_u2(n2, a1, a2, d, c2=0)[source]

Auxiliary function for discrete confocal coordinates discrete_confocal_conics_concentric to compute the cooresponding continuous confocal conics that establish the polarity relation.

Parameters:

n1float
a1float
a2float
dfloat
c2float

Returns:

float: The corresponding u2 value.

See also

discrete_confocal_conics_concentric

ddg.math.parametrizations.discrete_confocal2d.discrete_confocal_conics_ic_ellipse(n1, n2, a1, a2, k, h, c1=0.0, c2=0.0)[source]

2-dimensional discrete confocal coordinate system defined on 1/2 Z^2.

The parametrization is given by

x(n1, n2) = A1 * (sn(h * n1 + c1, k) * dn(h * n2 + c2, k)) / cn(h * n2 + c2, k)
y(n1, n2) = A2 * cn(h * n1 + c1, k) / cn(h * n2 + c2, k)

where

A1 = sqrt(a1 - a2) * dn(h / 2, k) / (k * cn(h / 2, k))
A2 = sqrt(a1 - a2) * sqrt(1 - k**2) / (k * cn(h / 2, k))

Incident points from dual sublattices are related by polarity. The parameters of the corresponding confocal conics that constitute the polarity are computed in discrete_confocal_conics_ic_ellipse_u1 and discrete_confocal_conics_ic_ellipse_u2.

The constants k and h should satisfy 0 < k**2 < 1 and cn(h/2,k) > 0. Symmetry is achieved by c1 = c1 = 0 and periodicity by

h = K(k) / m    for some positive integer or half-integer m.

Then the parameters may be restricted to

n1 in [-2*m, 2*m],    n2 in [0, m - 0.5].

Parameters:

n1float: First parameter (should be integer and half-integer values).
n2float: Second parameter (should be integer and half-integer values).
a1float with a1 > a2: Determines the first semi axis of the conics.
a2float with a1 > a2: Determines the second semi axis of the conics.
kfloat: Modulus of the elliptic functions (0 < k**2 < 1).
hfloat: Scaling / frequency factor
c1float: Shift in first parameter direction.
c2float: Shift in second parameter direction.

Returns:

np.ndarray of shape (2,): Point in R^2.

Notes

Not tested yet.

ddg.math.parametrizations.discrete_confocal2d.discrete_confocal_conics_ic_ellipse_u1(n1, a1, a2, k, h, c1)[source]

Auxiliary function for discrete confocal coordinates discrete_confocal_conics_ic_ellipse to compute the cooresponding continuous confocal conics that establish the polarity relation.

Parameters:

n1float
a1float
a2float
kfloat
hfloat
c1float

Returns:

float: The corresponding u1 value.

See also

discrete_confocal_conics_ic_ellipse

ddg.math.parametrizations.discrete_confocal2d.discrete_confocal_conics_ic_ellipse_u2(n2, a1, a2, k, h, c2)[source]

Auxiliary function for discrete confocal coordinates discrete_confocal_conics_ic_ellipse to compute the cooresponding continuous confocal conics that establish the polarity relation.

Parameters:

n1float
a1float
a2float
kfloat
hfloat
c2float

Returns:

float: The corresponding u2 value.

See also

discrete_confocal_conics_ic_ellipse

ddg.math.parametrizations.discrete_confocal2d.discrete_confocal_conics_ic_hyperbola(n1, n2, a1, a2, k, h, c1, c2)[source]

2-dimensional discrete confocal coordinate system defined on 1/2 Z^2.

The parametrization is given by

x(n1, n2) = A1 * sn(h * n1 + c1, k) / sn(h * n2 + c2, k)
y(n1, n2) = A2 * dn(h * n1 + c1, k) * cn(h * n2 + c2, k) / sn(h * n2 + c2, k)

where

A1 = k * sqrt(a1 - a2) * cn(h / 2, k) / dn(h / 2, k)
A2 = sqrt(a1 - a2) / dn(h / 2, k)

Incident points from dual sublattices are related by polarity.

Parameters:

n1float: First parameter (should be integer and half-integer values).
n2float: Second parameter (should be integer and half-integer values).
a1float with a1 > a2: Determines the first semi axis of the conics.
a2float with a1 > a2: Determines the second semi axis of the conics.
kfloat: Modulus of the elliptic functions (0 < k**2 < 1).
hfloat: Scaling / frequency factor
c1float: Shift in first parameter direction.
c2float: Shift in second parameter direction.

Returns:

np.ndarray of shape (2,): Point in R^2.

Notes

Not tested yet.

ddg.math.parametrizations.discrete_confocal2d.discrete_confocal_conics_hyperbolic_pencil(n1, n2, d, c1=0.0, c2=0.0)[source]

2-dimensional discrete confocal coordinate system defined on 1/2 Z^2.

The parametrization is given by

x(n1, n2) = exp(d * (n1 + n2 + c1 + c2))
y(n1, n2) = sqrt(1 - 1/q) * (qgamma(-n1 + 0.5 - c1, 1/q) / qgamma(-n1 -c1, 1/q))
            * sqrt(q - 1) * qgamma(n2 + 0.5 + c2, q) / qgamma(n2 + c2, q)

where

q = exp(2*d)

and qgamma denotes the q-gamma function, whiche is characerized by

qgamme(z+1, q) = qgamma(z, q) * (1 - q**z) / (1 - q)

Incident points from dual sublattices are related by polarity. The parameters of the corresponding confocal conics that constitute the polarity are computed in discrete_confocal_conics_hyperbolic_pencil_u1 and discrete_confocal_conics_ic_hyperbolic_pencil_u2.

Boundary conditions may be achieved by c1 = c2 = 0, and then the parameters may be restricted to n1 <= 0, n2 >=0.

Parameters:

n1float: First parameter (should be integer and half-integer values).
n2float: Second parameter (should be integer and half-integer values).
dfloat: Scaling / frequency factor
c1float: Shift in first parameter direction.
c2float: Shift in second parameter direction.

Returns:

np.ndarray of shape (2,): Point in R^2.

Notes

Not tested yet.

ddg.math.parametrizations.discrete_confocal2d.discrete_confocal_conics_hyperbolic_pencil_u1(n1, d, c1)[source]

Auxiliary function for discrete confocal coordinates discrete_confocal_conics_hyperbolic_pencil to compute the cooresponding continuous confocal conics that establish the polarity relation.

Parameters:

n1float
dfloat
c1float

Returns:

float: The corresponding u1 value.

See also

discrete_confocal_conics_ic_hyperbolic_pencil

ddg.math.parametrizations.discrete_confocal2d.discrete_confocal_conics_hyperbolic_pencil_u2(n2, d, c2)[source]

Auxiliary function for discrete confocal coordinates discrete_confocal_conics_hyperbolic_pencil to compute the cooresponding continuous confocal conics that establish the polarity relation.

Parameters:

n2float
dfloat
c2float

Returns:

float: The corresponding u2 value.

See also

discrete_confocal_conics_ic_hyperbolic_pencil