ddg.math.parametrizations.confocal2d module

Parametrizations for curves, surfaces, coordinate systems, …

ddg.math.parametrizations.confocal2d.confocal_conics_sqrt(u1, u2, a1, a2)[source]

2-dimensional coordinate system along confocal conics

x**2 / (a1 + u) + y**2 / (a2 + u) = 1    with some a1 > a2.

These are

hyperbolas for u = u1 with -a1 < u1 < -a2,
ellipses   for u = u2 with -a2 < u2.

The parametrization is given in terms of square roots:

x(u1, u2) = x_sgn * sqrt(u1 + a1) * sqrt(u2 + a1) / sqrt(a1 - a2)
y(u1, u2) = y_sgn * sqrt(-(u1 + a2)) * sqrt(u2 + a2) / sqrt(a1 - a2)

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:

u1float in [-a1, -a2]: First parameter.
u2float in [-a2, inf): Second parameter.
a1float with a1 > a2: Determines the first semi axis of the conics.
a2float with a1 > a2: Determines the second semi axis of the conics.

Returns:

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

ddg.math.parametrizations.confocal2d.confocal_conics_trigonometric(s1, s2, a1, a2)[source]

2-dimensional coordinate system along confocal conics

x**2 / (a1 + u) + y**2 / (a2 + u) = 1    with some a1 > a2.

The parametrization is given in terms of trigonometric functions:

x(s1, s2) = sqrt(a1 - a2) * cos(s1) * cosh(s2)
y(s1, s2) = sqrt(a1 - a2) * sin(s1) * sinh(s2)

The parametrization is 2*pi periodic in s1.

The relation to the parameters u = u1 (hyperbolas) and u = u2 (ellipses) of the confocal system are given by

u1(s1) = (a1-a2)*cos(s1)**2 - a1
u2(s2) = (a1-a2)*cosh(s2)**2 - a1

Parameters:

s1float: First parameter.
s2float: Second parameter.
a1float with a1 > a2: Determines the first semi axis of the conics.
a2float with a1 > a2: Determines the second semi axis of the conics.

Returns:

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

ddg.math.parametrizations.confocal2d.confocal_conics_concentric(s1, s2, a1, a2)[source]

2-dimensional coordinate system along confocal conics

x**2 / (a1 + u) + y**2 / (a2 + u) = 1    with some a1 > a2.

The parametrization is given by

x(s1, s2) = s1 * s2
y(s1, s2) = sqrt(1 - s1**2) * sqrt(s2**2 - 1)

The relation to the parameters u = u1 (hyperbolas) and u = u2 (ellipses) of the confocal system are given by

u1(s1) = (a1-a2)*s1**2 - a1
u2(s2) = (a1-a2)*s2**2 - a1

The parametrization covers the upper half-plane. It is diagonally related to two families of concentric circles with the two focii as centers. The diagonals s1 + s2 = xi = const lie on concentric circles with center (-sqrt(a1-a2),0) and radius sqrt(a1-a2)*xi. The diagonals s2 - s1 = eta = const lie on concentric circles with center (sqrt(a1-a2),0) and radius sqrt(a1-a2)*eta.

Parameters:

s1float in (-1, 1): First parameter.
s2float in (1, inf): Second parameter.
a1float with a1 > a2: Determines the first semi axis of the conics.
a2float with a1 > a2: Determines the second semi axis of the conics.

Returns:

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

ddg.math.parametrizations.confocal2d.confocal_conics_ic_ellipse(s1, s2, a1, a2, k)[source]

2-dimensional coordinate system along confocal conics (outside an ellipse)

x**2 / (a1 + u) + y**2 / (a2 + u) = 1    with some a1 > a2.

The parametrization is given in terms of elliptic functions:

x(s1, s2) = sqrt(a1 - a2) * (sn(s1, k) * dn(s2, k)) / (k * cn(s2, k))
y(s1, s2) = sqrt(a1 - a2) * (sqrt(1 - k**2) * cn(s1, k)) / (k * cn(s2, k))

The relation to the parameters u = u1 (hyperbolas) and u = u2 (ellipses) of the confocal system are given by

u1(s1) = (a1 - a2) * sn(s1)**2 - a1
u2(s2) = (a1 - a2) * (dn(s2) / (k * cn(s2)))**2 - a1

The diagonals s1 +- s2 = const lie on lines, which are tangent to an ellipse. Along the s2 direction the parametrization is bounded by this ellipse and geos to infinity in finite time, while it is periodic along the s1 direction.

The parametrization is closely related to incircular nets (IC-nets): The straight diagonals of a uniform samlping of this parametrization constitutes an IC-net.

Parameters:

s1float: First parameter (fundamental domain: [0, 2K)).
s2float: Second parameter (fundamental domain: (0, K)).
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).

Returns:

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

ddg.math.parametrizations.confocal2d.confocal_conics_ic_hyperbola(s1, s2, a1, a2, k)[source]

2-dimensional coordinate system along confocal conics (outside a hyperbola)

x**2 / (a1 + u) + y**2 / (a2 + u) = 1    with some a1 > a2.

The parametrization is given in terms of elliptic functions:

x(s1, s2) = k * sqrt(a1 - a2) * sn(s1, k) / sn(s2, k)
y(s1, s2) = sqrt(a1 - a2) * dn(s1, k) * cn(s2, k) / sn(s2, k)

The relation to the parameters u = u1 (hyperbolas) and u = u2 (ellipses) of the confocal system are given by

u1(s1) = k**2 * (a1 - a2) * sn(s1)**2 - a1
u2(s2) = (a1 - a2) * ns(s2)**2 - a1

The diagonals s1 +- s2 = const lie on lines, which are tangent to a hyperbola. Along the s2 direction the parametrization is bounded by this hyperbola, and geos to infinity in finite time, while it is periodic along the s1 direction.

The parametrization is closely related to incircular nets (IC-nets): The straight diagonals of a uniform samlping of this parametrization constitutes an IC-net.

Parameters:

s1float: First parameter (fundamental domain: [0, 2K)).
s2float: Second parameter (fundamental domain: (0, K)).
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).

Returns:

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

ddg.math.parametrizations.confocal2d.confocal_conics_hyperbolic_pencil(s1, s2, a1, a2)[source]

2-dimensional coordinate system along confocal conics (outside a hyperbola)

x**2 / (a1 + u) + y**2 / (a2 + u) = 1    with some a1 > a2.

The parametrization is given in terms of elliptic functions:

x(s1, s2) = sqrt(a1 - a2) * exp(s1 + s2)
y(s1, s2) = sqrt(a1 - a2) * sqrt((1 - exp(2 * s1)) * (exp(2 * s2) - 1))

The relation to the parameters u = u1 (hyperbolas) and u = u2 (ellipses) of the confocal system are given by

u1(s1) = (a1 - a2) * exp(2*s1) - a1
u2(s2) = (a1 - a2) * exp(2*s2) - a1

The parametrization covers the first quadrant, where the y-axis is approached in the limit s1 -> -inf. It is diagonally related to vertical lines and to a hyperbolic pencil of circles which has the two foci of the confocal conics as limiting points: The diagonals s1 + s2 = xi = const lie on vertical lines. The diagonals s2 - s1 = eta = const lie on circles with center (sqrt(a1-a2)*cosh(eta),0) and radius sqrt(a1-a2)*sinh(eta).

Parameters:

s1float in (-inf, 0): First parameter.
s2float in (0, inf): Second parameter.
a1float with a1 > a2: Determines the first semi axis of the conics.
a2float with a1 > a2: Determines the second semi axis of the conics.

Returns:

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