ribbon.spiralpolar module

Classes for spirals described by a polar equation r = f(theta).

class ribbon.spiralpolar.SpiralPolarBase(tincr, r0)

Bases: ABC

Abstract base class for spirals with polar equation r = f(theta).

The starting point is at r = r0 and theta = t0. This need not be the origin of the spiral. The arclength s is measured from (r0, t0) in the direction of increasing radius.

Attributes:

tincrbool

Whether radius increases with increasing theta.

r0, t0float

The spiral curve segment begins at (r = r0, theta = t0).

Methods

arclength_to_r(s)	Returns the radial coordinates corresponding to the arclengths.
arclength_to_theta(s[, s0, t0])	Returns the angular coordinates corresponding to the arclengths.
get_arclength(v[, var, v0])	Returns the arclength at the given values of polar coordinates.
get_cart_coords(v[, var])	Returns the cartesian coordinates at the given values of polar coordinates or arclengths.
get_curvature(v[, var])	Returns the curvature at the given values of polar coordinates or arclengths.
get_polar_coords(v[, var])	Returns the polar coordinates at the given values of theta, r, or arclengths.
get_tangent(v[, var])	Returns the unit tangent vector at the given values of polar or arclength coordinates.
r_to_theta(r)	Returns the angular coordinates corresponding to the given radial coordinates.
theta_to_r(theta)	Returns the radial coordinates corresponding to the given angular coordinates.

abstractmethod theta_to_r(theta)

Returns the radial coordinates corresponding to the given angular coordinates.

Parameters:

thetaarray_like

Angular coordinates in radians.

Returns:

float or ndarray

Radial coordinates (float if input is scalar, else ndarray)

abstractmethod r_to_theta(r)

Returns the angular coordinates corresponding to the given radial coordinates.

Parameters:

rarray_like

Radial coordinates.

Returns:

float or ndarray

Angular coordinates in radians (float if input is scalar, else ndarray).

abstractmethod get_tangent(v, var='s')

Returns the unit tangent vector at the given values of polar or arclength coordinates.

Parameters:

varray_like

Value of the polar coordinates or arclengths.

var{‘r’, ‘theta’, ‘s’}

Descriptor for v. Radial coordinates — 'r', angular coordinates — 'theta', and arclengths — 's'.

Returns:

ndarray

Unit tangent vectors. Shape is (2,) if v is scalar, else (n,2) where n is the length of v.

arclength_to_r(s)

Returns the radial coordinates corresponding to the arclengths.

Parameters:

sarray_like

Arclengths measured from the starting point (as set by the instance attribute r0) along the direction of increasing radius.

Returns:

float or ndarray

Radial coordinates (float if input is scalar, else ndarray).

arclength_to_theta(s, s0=0.0, t0=None)

Returns the angular coordinates corresponding to the arclengths.

Parameters:

sarray_like

Arclengths measured from the starting point (as set by the instance attribute r0) along the direction of increasing radius.

s0float

Value of the arclength at angle t0.

t0float | None

Measure arclength from this value of the angular coordinate. If None, set to t0, which corresponds to the radial coordinate r0.

Returns:

float or ndarray

Radial coordinates (float if input is scalar, else ndarray).

get_arclength(v, var='theta', v0=None)

Returns the arclength at the given values of polar coordinates.

Parameters:

varray_like

Value of the polar coordinates.

var{‘r’, ‘theta’}

Descriptor for v. Radial coordinates — 'r', angular coordinates — 'theta'.

v0float or None

Origin of the arclength. If None, will be set to self.t0 if var = 'theta' or to self.r0 if var = 'r'.

Returns:

float or ndarray

Arclength (float if input is scalar, else ndarray).

get_curvature(v, var='s')

Returns the curvature at the given values of polar coordinates or arclengths.

Parameters:

varray_like

Value of the polar coordinates or arclengths.

var{‘r’, ‘theta’, ‘s’}

Descriptor for v. Radial coordinates — 'r', angular coordinates — 'theta', and arclengths — 's'.

Returns:

float or ndarray

Curvatures (float if input is scalar, else ndarray).

get_cart_coords(v, var='s')

Returns the cartesian coordinates at the given values of polar coordinates or arclengths.

Parameters:

varray_like

Value of the polar coordinates or arclengths.

var{‘r’, theta’, ‘s’}

Descriptor for v. Radial coordinates — 'r', angular coordinates — 'theta', and arclengths — 's'.

Returns:

x, ytuple

Cartesian coordinates. If v is a scalar, x and y are floats, else they are 1D *ndarray*s.

get_polar_coords(v, var='s')

Returns the polar coordinates at the given values of theta, r, or arclengths.

Parameters:

varray_like

Value of the polar coordinates or arclengths.

var{‘r’, ‘theta’, ‘s’}

Descriptor for v. Radial coordinates — 'r', angular coordinates — 'theta', and arclengths — 's'.

Returns:

r, thetatuple

Polar coordinates. If v is a scalar, r and theta are floats, else they are 1D *ndarray*s.

class ribbon.spiralpolar.SpiralCircleInvolute(b, r0)

Bases: SpiralPolarBase

Spirals with polar equation: r = b * (1+theta^2)^(1/2), where b >= 0 is the radius of the circle.

The starting point is at r = r0 and theta = t0, where r0 must be > b.

The arclength s is measured from (r0, t0) in the direction of increasing radius.

For all points on this spiral r must be >= r0, theta must be >= t0, and s must be >= 0.

Parameters:

bfloat

The radius of circle, must be >= 0.

r0

Radial coordinate of the starting point of the spiral (must be > b).

Methods

arclength_to_r(s)	Returns the radial coordinates corresponding to the arclengths.
arclength_to_theta(s[, s0, t0])	Returns the angular coordinates corresponding to the arclengths.
get_arclength(v[, var, v0])	Returns the arclength at the given values of polar coordinates.
get_cart_coords(v[, var])	Returns the cartesian coordinates at the given values of polar coordinates or arclengths.
get_curvature(v[, var])	Returns the curvature at the given values of polar coordinates or arclengths.
get_polar_coords(v[, var])	Returns the polar coordinates at the given values of theta, r, or arclengths.
get_tangent(v[, var])	Returns the unit tangent vector at the given values of polar or arclength coordinates.
r_to_theta(r)	Returns the angular coordinates corresponding to the given radial coordinates.
theta_to_r(theta)	Returns the radial coordinates corresponding to the given angular coordinates.

theta_to_r(theta)

Returns the radial coordinates corresponding to the given angular coordinates.

Parameters:

thetaarray_like

Angular coordinates in radians.

Returns:

float or ndarray

Radial coordinates (float if input is scalar, else ndarray)

r_to_theta(r)

Returns the angular coordinates corresponding to the given radial coordinates.

Parameters:

rarray_like

Radial coordinates.

Returns:

float or ndarray

Angular coordinates in radians (float if input is scalar, else ndarray).

get_tangent(v, var='s')

Returns the unit tangent vector at the given values of polar or arclength coordinates.

Parameters:

varray_like

Value of the polar coordinates or arclengths.

var{‘r’, ‘theta’, ‘s’}

Descriptor for v. Radial coordinates — 'r', angular coordinates — 'theta', and arclengths — 's'.

Returns:

ndarray

Unit tangent vectors. Shape is (2,) if v is scalar, else (n,2) where n is the length of v.

arclength_to_r(s)

Returns the radial coordinates corresponding to the arclengths.

Parameters:

sarray_like

Arclengths measured from the starting point (as set by the instance attribute r0) along the direction of increasing radius.

Returns:

float or ndarray

Radial coordinates (float if input is scalar, else ndarray).

arclength_to_theta(s, s0=0.0, t0=None)

Returns the angular coordinates corresponding to the arclengths.

Parameters:

sarray_like

Arclengths measured from the starting point (as set by the instance attribute r0) along the direction of increasing radius.

s0float

Value of the arclength at angle t0.

t0float | None

Measure arclength from this value of the angular coordinate. If None, set to t0, which corresponds to the radial coordinate r0.

Returns:

float or ndarray

Radial coordinates (float if input is scalar, else ndarray).

get_arclength(v, var='theta', v0=None)

Returns the arclength at the given values of polar coordinates.

Parameters:

varray_like

Value of the polar coordinates.

var{‘r’, ‘theta’}

Descriptor for v. Radial coordinates — 'r', angular coordinates — 'theta'.

v0float or None

Origin of the arclength. If None, will be set to self.t0 if var = 'theta' or to self.r0 if var = 'r'.

Returns:

float or ndarray

Arclength (float if input is scalar, else ndarray).

get_cart_coords(v, var='s')

Returns the cartesian coordinates at the given values of polar coordinates or arclengths.

Parameters:

varray_like

Value of the polar coordinates or arclengths.

var{‘r’, theta’, ‘s’}

Descriptor for v. Radial coordinates — 'r', angular coordinates — 'theta', and arclengths — 's'.

Returns:

x, ytuple

Cartesian coordinates. If v is a scalar, x and y are floats, else they are 1D *ndarray*s.

get_curvature(v, var='s')

Returns the curvature at the given values of polar coordinates or arclengths.

Parameters:

varray_like

Value of the polar coordinates or arclengths.

var{‘r’, ‘theta’, ‘s’}

Descriptor for v. Radial coordinates — 'r', angular coordinates — 'theta', and arclengths — 's'.

Returns:

float or ndarray

Curvatures (float if input is scalar, else ndarray).

get_polar_coords(v, var='s')

Returns the polar coordinates at the given values of theta, r, or arclengths.

Parameters:

varray_like

Value of the polar coordinates or arclengths.

var{‘r’, ‘theta’, ‘s’}

Descriptor for v. Radial coordinates — 'r', angular coordinates — 'theta', and arclengths — 's'.

Returns:

r, thetatuple

Polar coordinates. If v is a scalar, r and theta are floats, else they are 1D *ndarray*s.