Complex Plots

AUTHORS:

  • Robert Bradshaw (2009): initial version

  • David Lowry-Duda (2022): incorporate matplotlib colormaps

class sage.plot.complex_plot.ComplexPlot(rgb_data, x_range, y_range, options)

Bases: sage.plot.primitive.GraphicPrimitive

The GraphicsPrimitive to display complex functions in using the domain coloring method

INPUT:

  • rgb_data – An array of colored points to be plotted.

  • x_range – A minimum and maximum x value for the plot.

  • y_range – A minimum and maximum y value for the plot.

get_minmax_data()

Return a dictionary with the bounding box data.

EXAMPLES:

sage: p = complex_plot(lambda z: z, (-1, 2), (-3, 4))
sage: sorted(p.get_minmax_data().items())
[('xmax', 2.0), ('xmin', -1.0), ('ymax', 4.0), ('ymin', -3.0)]
sage: p = complex_plot(lambda z: z, (1, 2), (3, 4))
sage: sorted(p.get_minmax_data().items())
[('xmax', 2.0), ('xmin', 1.0), ('ymax', 4.0), ('ymin', 3.0)]
sage.plot.complex_plot.add_contours_to_rgb(rgb, delta, dark_rate=0.5)

Return an rgb array from given array of \((r, g, b)\) and \((delta)\).

Each input \((r, g, b)\) is modified by delta to be lighter or darker depending on the size of delta. Negative delta values darken the color, while positive delta values lighten the pixel.

We assume that the delta values come from a function like sage.plot.complex_plot.mag_to_lightness(), which maps magnitudes to the range \([-1, +1]\).

INPUT:

  • rgb – a grid of length 3 tuples \((r, g, b)\), as an \(N \times M \times 3\) numpy array.

  • delta – a grid of values as an \(N \times M\) numpy array; these represent how much to change the lightness of each \((r, g, b)\). Values should be in \([-1, 1]\).

  • dark_rate – a positive number (default: \(0.5\)); affects how strongly visible the contours appear.

OUTPUT:

An \(N \times M \times 3\) floating point Numpy array X, where X[i,j] is an (r, g, b) tuple.

ALGORITHM:

Each pixel and lightness-delta is mapped from \((r, g, b, delta) \mapsto (h, l, s, delta)\) using the standard RGB-to-HLS formula.

Then the lightness is adjusted via \(l \mapsto l' = l + 0.5 \cdot delta\).

Finally map \((h, l', s) \mapsto (r, g, b)\) using the standard HLS-to-RGB formula.

EXAMPLES:

sage: import numpy as np
sage: from sage.plot.complex_plot import add_contours_to_rgb
sage: add_contours_to_rgb(np.array([[[0, 0.25, 0.5]]]), np.array([[0.75]]))  # abs tol 1e-4
array([[[0.25 , 0.625, 1.   ]]])
sage: add_contours_to_rgb(np.array([[[0, 0, 0]]]), np.array([[1]]))  # abs tol 1e-4
array([[[0.5, 0.5, 0.5]]])
sage: add_contours_to_rgb(np.array([[[1, 1, 1]]]), np.array([[-0.5]])) # abs tol 1e-4
array([[[0.75, 0.75, 0.75]]])

Raising dark_rate leads to bigger adjustments:

sage: add_contours_to_rgb(np.array([[[0.5, 0.5, 0.5]]]),  # abs tol 1e-4
....:                     np.array([[0.5]]), dark_rate=0.1)
array([[[0.55, 0.55, 0.55]]])
sage: add_contours_to_rgb(np.array([[[0.5, 0.5, 0.5]]]),  # abs tol 1e-4
....:                     np.array([[0.5]]), dark_rate=0.5)
array([[[0.75, 0.75, 0.75]]])
sage.plot.complex_plot.add_lightness_smoothing_to_rgb(rgb, delta)

Return an rgb array from given array of colors and lightness adjustments.

This smoothly adds lightness from black (when delta is \(-1\)) to white (when delta is \(1\)).

Each input \((r, g, b)\) is modified by delta to be lighter or darker depending on the size of delta. When delta is \(-1\), the output is black. When delta is \(+1\), the output is white. Colors piecewise-linearly vary from black to the initial \((r, g, b)\) to white.

We assume that the delta values come from a function like sage.plot.complex_plot.mag_to_lightness(), which maps magnitudes to the range \([-1, +1]\).

INPUT:

  • rgb – a grid of length 3 tuples \((r, g, b)\), as an \(N \times M \times 3\) numpy array.

  • delta – a grid of values as an \(N \times M\) numpy array; these represent how much to change the lightness of each \((r, g, b)\). Values should be in \([-1, 1]\).

OUTPUT:

An \(N \times M \times 3\) floating point Numpy array X, where X[i,j] is an (r, g, b) tuple.

EXAMPLES:

We can call this on grids of values:

sage: import numpy as np
sage: from sage.plot.complex_plot import add_lightness_smoothing_to_rgb
sage: add_lightness_smoothing_to_rgb(np.array([[[0, 0.25, 0.5]]]), np.array([[0.75]]))  # abs tol 1e-4
array([[[0.75  , 0.8125, 0.875 ]]])
sage: add_lightness_smoothing_to_rgb(np.array([[[0, 0.25, 0.5]]]), np.array([[0.75]]))  # abs tol 1e-4
array([[[0.75  , 0.8125, 0.875 ]]])
sage.plot.complex_plot.complex_plot(f, x_range, y_range, contoured=False, tiled=False, cmap=None, contour_type='logarithmic', contour_base=None, dark_rate=0.5, nphases=10, plot_points=100, interpolation='catrom', **options)

complex_plot takes a complex function of one variable, \(f(z)\) and plots output of the function over the specified x_range and y_range as demonstrated below. The magnitude of the output is indicated by the brightness and the argument is represented by the hue.

By default, zero magnitude corresponds to black output, infinite magnitude corresponds to white output. The options contoured, tiled, and cmap affect the output.

complex_plot(f, (xmin, xmax), (ymin, ymax), contoured, tiled, cmap, ...)

INPUT:

  • f – a function of a single complex value \(x + iy\)

  • (xmin, xmax) – 2-tuple, the range of x values

  • (ymin, ymax) – 2-tuple, the range of y values

  • cmapNone, or the string name of a matplotlib colormap, or an instance of a matplotlib Colormap, or the special string 'matplotlib' (default: None); If None, then hues are chosen from a standard color wheel, cycling from red to yellow to blue. If matplotlib, then hues are chosen from a preset matplotlib colormap.

The following named parameter inputs can be used to add contours and adjust their distribution:

  • contoured – boolean (default: False); causes the magnitude to be indicated by logarithmically spaced ‘contours’. The magnitude along one contour is either twice or half the magnitude along adjacent contours.

  • dark_rate – a positive number (default: \(0.5\)); affects how quickly magnitudes affect how light/dark the image is. When there are contours, this affects how visible each contour is. Large values (near \(1.0\)) have very strong, immediate effects, while small values (near \(0.0\)) have gradual effects.

  • tiled – boolean (default: False); causes the magnitude to be indicated by logarithmically spaced ‘contours’ as in contoured, and in addition for there to be \(10\) evenly spaced phase contours.

  • nphases – a positive integer (default: \(10\)); when tiled=True, this is the number of divisions the phase is divided into.

  • contour_type – either 'logarithmic', or 'linear' (default: 'logarithmic'); causes added contours to be of given type when contoured=True.

  • contour_base – a positive integer; when contour_type is 'logarithmic', this sets logarithmic contours at multiples of contour_base apart. When contour_type is 'linear', this sets contours at distances of contour_base apart. If None, then a default is chosen depending on contour_type.

The following inputs may also be passed in as named parameters:

  • plot_points – integer (default: 100); number of points to plot in each direction of the grid

  • interpolation – string (default: 'catrom'); the interpolation method to use: 'bilinear', 'bicubic', 'spline16', 'spline36', 'quadric', 'gaussian', 'sinc', 'bessel', 'mitchell', 'lanczos', 'catrom', 'hermite', 'hanning', 'hamming', 'kaiser'

Any additional parameters will be passed to show(), as long as they’re valid.

Note

Matplotlib colormaps can be chosen or customized to cater to different types of vision. The colormaps ‘cividis’ and ‘viridis’ in matplotlib are designed to be perceptually uniform to a broader audience. The colormap ‘turbo’ is similar to the default but with more even contrast. See [NAR2018] for more information about colormap choice for scientific visualization.

EXAMPLES:

Here we plot a couple of simple functions:

sage: complex_plot(sqrt(x), (-5, 5), (-5, 5))
Graphics object consisting of 1 graphics primitive
../../_images/complex_plot-1.svg
sage: complex_plot(sin(x), (-5, 5), (-5, 5))
Graphics object consisting of 1 graphics primitive
../../_images/complex_plot-2.svg
sage: complex_plot(log(x), (-10, 10), (-10, 10))
Graphics object consisting of 1 graphics primitive
../../_images/complex_plot-3.svg
sage: complex_plot(exp(x), (-10, 10), (-10, 10))
Graphics object consisting of 1 graphics primitive
../../_images/complex_plot-4.svg

A plot with a different choice of colormap:

sage: complex_plot(exp(x), (-10, 10), (-10, 10), cmap='viridis')
Graphics object consisting of 1 graphics primitive
../../_images/complex_plot-5.svg

A function with some nice zeros and a pole:

sage: f(z) = z^5 + z - 1 + 1/z
sage: complex_plot(f, (-3, 3), (-3, 3))
Graphics object consisting of 1 graphics primitive
../../_images/complex_plot-6.svg

The same function as above, but with contours. Contours render poorly with few plot points, so we use 300 here:

sage: f(z) = z^5 + z - 1 + 1/z
sage: complex_plot(f, (-3, 3), (-3, 3), plot_points=300, contoured=True)
Graphics object consisting of 1 graphics primitive
../../_images/complex_plot-7.svg

The same function as above, but tiled and with the plasma colormap:

sage: f(z) = z^5 + z - 1 + 1/z
sage: complex_plot(f, (-3, 3), (-3, 3),
....:              plot_points=300, tiled=True, cmap='plasma')
Graphics object consisting of 1 graphics primitive
../../_images/complex_plot-8.svg

When using tiled=True, the number of phase subdivisions can be controlled by adjusting nphases. We make the same plot with fewer tilings:

sage: f(z) = z^5 + z - 1 + 1/z
sage: complex_plot(f, (-3, 3), (-3, 3), plot_points=300,
....:              tiled=True, nphases=5, cmap='plasma')
Graphics object consisting of 1 graphics primitive
../../_images/complex_plot-9.svg

It is also possible to use linear contours. We plot the same function above on an inset, setting contours to appear \(1\) apart:

sage: f(z) = z^5 + z - 1 + 1/z
sage: complex_plot(f, (0, 1), (0, 1), plot_points=300,
....:              contoured=True, contour_type='linear', contour_base=1)
Graphics object consisting of 1 graphics primitive
../../_images/complex_plot-10.svg

Note that tightly spaced contours can lead to Moiré patterns and aliasing problems. For example:

sage: f(z) = z^5 + z - 1 + 1/z
sage: complex_plot(f, (-3, 3), (-3, 3), plot_points=300,
....:              contoured=True, contour_type='linear', contour_base=1)
Graphics object consisting of 1 graphics primitive
../../_images/complex_plot-11.svg

When choosing colormaps, cyclic colormaps such as twilight or hsv might be considered more appropriate for showing changes in phase without sharp color contrasts:

sage: f(z) = z^5 + z - 1 + 1/z
sage: complex_plot(f, (-3, 3), (-3, 3), plot_points=300, cmap='twilight')
Graphics object consisting of 1 graphics primitive
../../_images/complex_plot-12.svg

Passing matplotlib as the colormap gives a special colormap that is similar to the default:

sage: f(z) = z^5 + z - 1 + 1/z
sage: complex_plot(f, (-3, 3), (-3, 3),
....:              plot_points=300, contoured=True, cmap='matplotlib')
Graphics object consisting of 1 graphics primitive
../../_images/complex_plot-13.svg

Here is the identity, useful for seeing what values map to what colors:

sage: complex_plot(lambda z: z, (-3, 3), (-3, 3))
Graphics object consisting of 1 graphics primitive
../../_images/complex_plot-14.svg

The Riemann Zeta function:

sage: complex_plot(zeta, (-30,30), (-30,30))
Graphics object consisting of 1 graphics primitive
../../_images/complex_plot-15.svg

For advanced usage, it is possible to tweak many parameters. Increasing dark_rate will make regions become darker/lighter faster when there are no contours:

sage: complex_plot(zeta, (-30, 30), (-30, 30), dark_rate=1.0)
Graphics object consisting of 1 graphics primitive
../../_images/complex_plot-16.svg

Decreasing dark_rate has the opposite effect. When there are contours, adjust dark_rate affects how visible contours are. Compare:

sage: complex_plot(zeta, (-1, 9), (10, 20), plot_points=200,  # long time
....:              contoured=True, cmap='twilight', dark_rate=0.2)
Graphics object consisting of 1 graphics primitive
../../_images/complex_plot-17.svg

and:

sage: complex_plot(zeta, (-1, 9), (10, 20), plot_points=200,  # long time
....:              contoured=True, cmap='twilight', dark_rate=0.75)
Graphics object consisting of 1 graphics primitive
../../_images/complex_plot-18.svg

In practice, different values of dark_rate will work well with different colormaps.

Extra options will get passed on to show(), as long as they are valid:

sage: complex_plot(lambda z: z, (-3, 3), (-3, 3), figsize=[1,1])
Graphics object consisting of 1 graphics primitive
sage: complex_plot(lambda z: z, (-3, 3), (-3, 3)).show(figsize=[1,1]) # These are equivalent

REFERENCES:

Plotting complex functions with colormaps follows the strategy from [LD2021] and incorporates contour techniques described in [WegSem2010].

sage.plot.complex_plot.complex_to_cmap_rgb(z_values, cmap='turbo', contoured=False, tiled=False, contour_type='logarithmic', contour_base=None, dark_rate=0.5, nphases=10)

Convert a grid of complex numbers to a grid of rgb values using colors taken from given colormap.

INPUT:

  • z_values – A grid of complex numbers, as a list of lists

  • cmap – the string name of a matplotlib colormap, or an instance of a matplotlib Colormap (default: 'turbo').

  • contoured – boolean (default: False); causes magnitude to be indicated through contour-like adjustments to lightness.

  • tiled – boolean (default: False); causes magnitude and argument to be indicated through contour-like adjustments to lightness.

  • nphases – a positive integer (default: \(10\)); when tiled=True, this is the number of divisions the phase is divided into.

  • contour_type – either 'logarithmic', or 'linear' (default: 'logarithmic'); causes added contours to be of given type when contoured=True.

  • contour_base – a positive integer; when contour_type is 'logarithmic', this sets logarithmic contours at multiples of contour_base apart. When contour_type is 'linear', this sets contours at distances of contour_base apart. If None, then a default is chosen depending on contour_type.

  • dark_rate – a positive number (default: \(0.5\)); affects how quickly magnitudes affect how light/dark the image is. When there are contours, this affects how visible each contour is. Large values (near \(1.0\)) have very strong, immediate effects, while small values (near \(0.0\)) have gradual effects.

OUTPUT:

An \(N \times M \times 3\) floating point Numpy array X, where X[i,j] is an (r, g, b) tuple.

EXAMPLES:

We can call this on grids of complex numbers:

sage: from sage.plot.complex_plot import complex_to_cmap_rgb
sage: complex_to_cmap_rgb([[0, 1, 1000]])  # abs tol 1e-4
array([[[0.        , 0.        , 0.        ],
        [0.49669808, 0.76400071, 0.18024425],
        [0.87320419, 0.99643856, 0.72730967]]])
sage: complex_to_cmap_rgb([[0, 1, 1000]], cmap='viridis')  # abs tol 1e-4
array([[[0.        , 0.        , 0.        ],
        [0.0984475 , 0.4375291 , 0.42487821],
        [0.68959896, 0.84592555, 0.84009311]]])

We can change contour types and the distances between contours:

sage: complex_to_cmap_rgb([[0, 1 + 1j, 3 + 4j]], contoured=True,  # abs tol 1e-4
....:                     contour_type="logarithmic", contour_base=3)
array([[[0.64362   , 0.98999   , 0.23356   ],
        [0.93239357, 0.81063338, 0.21955399],
        [0.95647342, 0.74861225, 0.14963982]]])
sage: complex_to_cmap_rgb([[0, 1 + 1j, 3 + 4j]], cmap='turbo',   # abs tol 1e-4
....:                     contoured=True, contour_type="linear", contour_base=3)
array([[[0.71246796, 0.9919238 , 0.3816262 ],
        [0.92617785, 0.79322304, 0.14779989],
        [0.95156284, 0.72025117, 0.05370383]]])

We see that changing dark_rate affects how visible contours are. In this example, we set contour_base=5 and note that the points \(0\) and \(1 + i\) are far away from contours, but \(2.9 + 4i\) is near (and just below) a contour. Raising dark_rate should have strong effects on the last coloration and weaker effects on the others:

sage: complex_to_cmap_rgb([[0, 1 + 1j, 2.9 + 4j]], cmap='turbo',  # abs tol 1e-4
....:                     contoured=True, dark_rate=0.05, contour_base=5)
array([[[0.64362   , 0.98999   , 0.23356   ],
        [0.93334746, 0.81330523, 0.23056563],
        [0.96357185, 0.75337736, 0.19440913]]])
sage: complex_to_cmap_rgb([[0, 1 + 1j, 2.9 + 4j]], cmap='turbo',  # abs tol 1e-4
....:                     contoured=True, dark_rate=0.85, contour_base=5)
array([[[0.64362   , 0.98999   , 0.23356   ],
        [0.93874682, 0.82842892, 0.29289564],
        [0.57778954, 0.42703289, 0.02612716]]])
sage.plot.complex_plot.complex_to_rgb(z_values, contoured=False, tiled=False, contour_type='logarithmic', contour_base=None, dark_rate=0.5, nphases=10)

Convert a grid of complex numbers to a grid of rgb values using a default choice of colors.

INPUT:

  • z_values – A grid of complex numbers, as a list of lists

  • contoured – boolean (default: False); causes magnitude to be indicated through contour-like adjustments to lightness.

  • tiled – boolean (default: False); causes magnitude and argument to be indicated through contour-like adjustments to lightness.

  • nphases – a positive integer (default: \(10\)); when tiled=True, this is the number of divisions the phase is divided into.

  • contour_type – either 'logarithmic', or 'linear' (default: 'logarithmic'); causes added contours to be of given type when contoured=True.

  • contour_base – a positive integer; when contour_type is 'logarithmic', this sets logarithmic contours at multiples of contour_base apart. When contour_type is 'linear', this sets contours at distances of contour_base apart. If None, then a default is chosen depending on contour_type.

  • dark_rate – a positive number (default: \(0.5\)); affects how quickly magnitudes affect how light/dark the image is. When there are contours, this affects how visible each contour is. Large values (near \(1.0\)) have very strong, immediate effects, while small values (near \(0.0\)) have gradual effects.

OUTPUT:

An \(N \times M \times 3\) floating point Numpy array X, where X[i,j] is an (r,g,b) tuple.

EXAMPLES:

We can call this on grids of complex numbers:

sage: from sage.plot.complex_plot import complex_to_rgb
sage: complex_to_rgb([[0, 1, 1000]])  # abs tol 1e-4
array([[[0.        , 0.        , 0.        ],
        [0.77172568, 0.        , 0.        ],
        [1.        , 0.64421177, 0.64421177]]])
sage: complex_to_rgb([[0, 1j, 1000j]])  # abs tol 1e-4
array([[[0.        , 0.        , 0.        ],
        [0.38586284, 0.77172568, 0.        ],
        [0.82210588, 1.        , 0.64421177]]])
sage: complex_to_rgb([[0, 1, 1000]], contoured=True)   # abs tol 1e-4
array([[[1.        , 0.        , 0.        ],
        [1.        , 0.15      , 0.15      ],
        [0.66710786, 0.        , 0.        ]]])
sage: complex_to_rgb([[0, 1, 1000]], tiled=True)   # abs tol 1e-4
array([[[1.        , 0.        , 0.        ],
        [1.        , 0.15      , 0.15      ],
        [0.90855393, 0.        , 0.        ]]])

We can change contour types and the distances between contours:

sage: complex_to_rgb([[0, 1 + 1j, 3 + 4j]],  # abs tol 1e-4
....:                contoured=True, contour_type="logarithmic", contour_base=3)
array([[[1.        , 0.        , 0.        ],
        [0.99226756, 0.74420067, 0.        ],
        [0.91751324, 0.81245954, 0.        ]]])
sage: complex_to_rgb([[0, 1 + 1j, 3 + 4j]],  # abs tol 1e-4
....:                contoured=True, contour_type="linear", contour_base=3)
array([[[1.        , 0.15      , 0.15      ],
        [0.91429774, 0.6857233 , 0.        ],
        [0.81666667, 0.72315973, 0.        ]]])

Lowering dark_rate causes colors to go to black more slowly near \(0\):

sage: complex_to_rgb([[0, 0.5, 1]], dark_rate=0.4)  # abs tol 1e-4
array([[[0.        , 0.        , 0.        ],
        [0.65393731, 0.        , 0.        ],
        [0.77172568, 0.        , 0.        ]]])
sage: complex_to_rgb([[0, 0.5, 1]], dark_rate=0.2)  # abs tol 1e-4
array([[[0.        , 0.        , 0.        ],
        [0.71235886, 0.        , 0.        ],
        [0.77172568, 0.        , 0.        ]]])
sage.plot.complex_plot.hls_to_rgb(hls)

Convert array of hls values (each in the range \([0, 1]\)) to a numpy array of rgb values (each in the range \([0, 1]\))

INPUT:

  • hls – an \(N \times 3\) array of floats in the range \([0, 1]\); the hls values at each point. (Note that the input can actually be of any dimension, such as \(N \times M \times 3\), as long as the last dimension has length \(3\)).

OUTPUT:

An \(N \times 3\) Numpy array of floats in the range \([0, 1]\), with the same dimensions as the input array.

EXAMPLES:

We convert a row of floats and verify that we can convert back using rgb_to_hls:

sage: from sage.plot.complex_plot import rgb_to_hls, hls_to_rgb
sage: hls = [[0.2, 0.4, 0.5], [0.1, 0.3, 1.0]]
sage: rgb = hls_to_rgb(hls)
sage: rgb  # abs tol 1e-4
array([[0.52, 0.6 , 0.2 ],
       [0.6 , 0.36, 0.  ]])
sage: rgb_to_hls(rgb)  # abs tol 1e-4
array([[0.2, 0.4, 0.5],
       [0.1, 0.3, 1. ]])

Multidimensional inputs can be given as well:

sage: multidim_arr = [[[0, 0.2, 0.4], [0, 1, 0]], [[0, 0, 0], [0.5, 0.6, 0.9]]]
sage: hls_to_rgb(multidim_arr)  # abs tol 1e-4
array([[[0.28, 0.12, 0.12],
        [1.  , 1.  , 1.  ]],
       [[0.  , 0.  , 0.  ],
        [0.24, 0.96, 0.96]]])
sage.plot.complex_plot.rgb_to_hls(rgb)

Convert array of rgb values (each in the range \([0, 1]\)) to a numpy array of hls values (each in the range \([0, 1]\))

INPUT:

  • rgb – an \(N \times 3\) array of floats with values in the range \([0, 1]\); the rgb values at each point. (Note that the input can actually be of any dimension, such as \(N \times M \times 3\), as long as the last dimension has length \(3\)).

OUTPUT:

An \(N \times 3\) Numpy array of floats in the range \([0, 1]\), with the same dimensions as the input array.

EXAMPLES:

We convert a row of floats and verify that we can convert back using hls_to_rgb:

sage: from sage.plot.complex_plot import rgb_to_hls, hls_to_rgb
sage: rgb = [[0.2, 0.4, 0.5], [0.1, 0.3, 1.0]]
sage: hls = rgb_to_hls(rgb)
sage: hls   # abs tol 1e-4
array([[0.55555556, 0.35      , 0.42857143],
       [0.62962963, 0.55      , 1.        ]])
sage: hls_to_rgb(hls)  # abs tol 1e-4
array([[0.2, 0.4, 0.5],
       [0.1, 0.3, 1. ]])

Multidimensional inputs can be given as well:

sage: multidim_arr = [[[0, 0.2, 0.4], [1, 1, 1]], [[0, 0, 0], [0.5, 0.6, 0.9]]]
sage: rgb_to_hls(multidim_arr)   # abs tol 1e-4
array([[[0.58333333, 0.2       , 1.        ],
        [0.        , 1.        , 0.        ]],
       [[0.        , 0.        , 0.        ],
        [0.625     , 0.7       , 0.66666667]]])