List plots#

sage.plot.plot3d.list_plot3d.list_plot3d(v, interpolation_type='default', point_list=None, **kwds)[source]#

A 3-dimensional plot of a surface defined by the list \(v\) of points in 3-dimensional space.

INPUT:

  • v – something that defines a set of points in 3 space:

    • a matrix

    • a list of 3-tuples

    • a list of lists (all of the same length) – this is treated the same as a matrix.

OPTIONAL KEYWORDS:

  • interpolation_type – ‘linear’, ‘clough’ (CloughTocher2D), ‘spline’

    ‘linear’ will perform linear interpolation

    The option ‘clough’ will interpolate by using a piecewise cubic interpolating Bezier polynomial on each triangle, using a Clough-Tocher scheme. The interpolant is guaranteed to be continuously differentiable. The gradients of the interpolant are chosen so that the curvature of the interpolating surface is approximatively minimized.

    The option ‘spline’ interpolates using a bivariate B-spline.

    When v is a matrix the default is to use linear interpolation, when v is a list of points the default is ‘clough’.

  • degree – an integer between 1 and 5, controls the degree of spline used for spline interpolation. For data that is highly oscillatory use higher values

  • point_list – If point_list=True is passed, then if the array is a list of lists of length three, it will be treated as an array of points rather than a 3xn array.

  • num_points – Number of points to sample interpolating function in each direction, when interpolation_type is not default. By default for an \(n\times n\) array this is \(n\).

  • **kwds – all other arguments are passed to the surface function

OUTPUT: a 3d plot

EXAMPLES:

We plot a matrix that illustrates summation modulo \(n\):

sage: n = 5
sage: list_plot3d(matrix(RDF, n, [(i+j)%n for i in [1..n] for j in [1..n]]))
Graphics3d Object

>>> from sage.all import *
>>> n = Integer(5)
>>> list_plot3d(matrix(RDF, n, [(i+j)%n for i in (ellipsis_range(Integer(1),Ellipsis,n)) for j in (ellipsis_range(Integer(1),Ellipsis,n))]))
Graphics3d Object
../../../_images/list_plot3d-1.svg

We plot a matrix of values of sin:

sage: from math import pi
sage: m = matrix(RDF, 6, [sin(i^2 + j^2)
....:                     for i in [0,pi/5,..,pi] for j in [0,pi/5,..,pi]])
sage: list_plot3d(m, color='yellow', frame_aspect_ratio=[1, 1, 1/3])
Graphics3d Object

>>> from sage.all import *
>>> from math import pi
>>> m = matrix(RDF, Integer(6), [sin(i**Integer(2) + j**Integer(2))
...                     for i in (ellipsis_range(Integer(0),pi/Integer(5),Ellipsis,pi)) for j in (ellipsis_range(Integer(0),pi/Integer(5),Ellipsis,pi))])
>>> list_plot3d(m, color='yellow', frame_aspect_ratio=[Integer(1), Integer(1), Integer(1)/Integer(3)])
Graphics3d Object
../../../_images/list_plot3d-2.svg

Though it does not change the shape of the graph, increasing num_points can increase the clarity of the graph:

sage: list_plot3d(m, color='yellow', num_points=40,
....:             frame_aspect_ratio=[1, 1, 1/3])
Graphics3d Object

>>> from sage.all import *
>>> list_plot3d(m, color='yellow', num_points=Integer(40),
...             frame_aspect_ratio=[Integer(1), Integer(1), Integer(1)/Integer(3)])
Graphics3d Object
../../../_images/list_plot3d-3.svg

We can change the interpolation type:

sage: import warnings
sage: warnings.simplefilter('ignore', UserWarning)
sage: list_plot3d(m, color='yellow', interpolation_type='clough',
....:             frame_aspect_ratio=[1, 1, 1/3])
Graphics3d Object

>>> from sage.all import *
>>> import warnings
>>> warnings.simplefilter('ignore', UserWarning)
>>> list_plot3d(m, color='yellow', interpolation_type='clough',
...             frame_aspect_ratio=[Integer(1), Integer(1), Integer(1)/Integer(3)])
Graphics3d Object
../../../_images/list_plot3d-4.svg

We can make this look better by increasing the number of samples:

sage: list_plot3d(m, color='yellow', interpolation_type='clough',
....:             frame_aspect_ratio=[1, 1, 1/3], num_points=40)
Graphics3d Object

>>> from sage.all import *
>>> list_plot3d(m, color='yellow', interpolation_type='clough',
...             frame_aspect_ratio=[Integer(1), Integer(1), Integer(1)/Integer(3)], num_points=Integer(40))
Graphics3d Object
../../../_images/list_plot3d-5.svg

Let us try a spline:

sage: list_plot3d(m, color='yellow', interpolation_type='spline',
....:             frame_aspect_ratio=[1, 1, 1/3])
Graphics3d Object

>>> from sage.all import *
>>> list_plot3d(m, color='yellow', interpolation_type='spline',
...             frame_aspect_ratio=[Integer(1), Integer(1), Integer(1)/Integer(3)])
Graphics3d Object
../../../_images/list_plot3d-6.svg

That spline does not capture the oscillation very well; let’s try a higher degree spline:

sage: list_plot3d(m, color='yellow', interpolation_type='spline', degree=5,
....:             frame_aspect_ratio=[1, 1, 1/3])
Graphics3d Object

>>> from sage.all import *
>>> list_plot3d(m, color='yellow', interpolation_type='spline', degree=Integer(5),
...             frame_aspect_ratio=[Integer(1), Integer(1), Integer(1)/Integer(3)])
Graphics3d Object
../../../_images/list_plot3d-7.svg

We plot a list of lists:

sage: show(list_plot3d([[1, 1, 1, 1], [1, 2, 1, 2], [1, 1, 3, 1], [1, 2, 1, 4]]))

>>> from sage.all import *
>>> show(list_plot3d([[Integer(1), Integer(1), Integer(1), Integer(1)], [Integer(1), Integer(2), Integer(1), Integer(2)], [Integer(1), Integer(1), Integer(3), Integer(1)], [Integer(1), Integer(2), Integer(1), Integer(4)]]))
../../../_images/list_plot3d-8.svg

We plot a list of points. As a first example we can extract the (x,y,z) coordinates from the above example and make a list plot out of it. By default we do linear interpolation:

sage: l = []
sage: for i in range(6):
....:     for j in range(6):
....:         l.append((float(i*pi/5), float(j*pi/5), m[i, j]))
sage: list_plot3d(l, color='red')
Graphics3d Object

>>> from sage.all import *
>>> l = []
>>> for i in range(Integer(6)):
...     for j in range(Integer(6)):
...         l.append((float(i*pi/Integer(5)), float(j*pi/Integer(5)), m[i, j]))
>>> list_plot3d(l, color='red')
Graphics3d Object
../../../_images/list_plot3d-9.svg

Note that the points do not have to be regularly sampled. For example:

sage: l = []
sage: for i in range(-5, 5):
....:     for j in range(-5, 5):
....:         l.append((normalvariate(0, 1),
....:                   normalvariate(0, 1),
....:                   normalvariate(0, 1)))
sage: L = list_plot3d(l, interpolation_type='clough',
....:                 color='orange', num_points=100)
sage: L
Graphics3d Object

>>> from sage.all import *
>>> l = []
>>> for i in range(-Integer(5), Integer(5)):
...     for j in range(-Integer(5), Integer(5)):
...         l.append((normalvariate(Integer(0), Integer(1)),
...                   normalvariate(Integer(0), Integer(1)),
...                   normalvariate(Integer(0), Integer(1))))
>>> L = list_plot3d(l, interpolation_type='clough',
...                 color='orange', num_points=Integer(100))
>>> L
Graphics3d Object
../../../_images/list_plot3d-10.svg

Check that no NaNs are produced (see Issue #13135):

sage: any(math.isnan(c) for v in L.vertices() for c in v)
False

>>> from sage.all import *
>>> any(math.isnan(c) for v in L.vertices() for c in v)
False
sage.plot.plot3d.list_plot3d.list_plot3d_array_of_arrays(v, interpolation_type, **kwds)[source]#

A 3-dimensional plot of a surface defined by a list of lists v defining points in 3-dimensional space.

This is done by making the list of lists into a matrix and passing back to list_plot3d(). See list_plot3d() for full details.

INPUT:

  • v – a list of lists, all the same length

  • interpolation_type – (default: ‘linear’)

OPTIONAL KEYWORDS:

  • **kwds – all other arguments are passed to the surface function

OUTPUT: a 3d plot

EXAMPLES:

The resulting matrix does not have to be square:

sage: show(list_plot3d([[1, 1, 1, 1], [1, 2, 1, 2], [1, 1, 3, 1]])) # indirect doctest

>>> from sage.all import *
>>> show(list_plot3d([[Integer(1), Integer(1), Integer(1), Integer(1)], [Integer(1), Integer(2), Integer(1), Integer(2)], [Integer(1), Integer(1), Integer(3), Integer(1)]])) # indirect doctest
../../../_images/list_plot3d-11.svg

The normal route is for the list of lists to be turned into a matrix and use list_plot3d_matrix():

sage: show(list_plot3d([[1, 1, 1, 1], [1, 2, 1, 2], [1, 1, 3, 1], [1, 2, 1, 4]]))

>>> from sage.all import *
>>> show(list_plot3d([[Integer(1), Integer(1), Integer(1), Integer(1)], [Integer(1), Integer(2), Integer(1), Integer(2)], [Integer(1), Integer(1), Integer(3), Integer(1)], [Integer(1), Integer(2), Integer(1), Integer(4)]]))
../../../_images/list_plot3d-12.svg

With certain extra keywords (see list_plot3d_matrix()), this function will end up using list_plot3d_tuples():

sage: show(list_plot3d([[1, 1, 1, 1], [1, 2, 1, 2], [1, 1, 3, 1], [1, 2, 1, 4]],
....:                  interpolation_type='spline'))

>>> from sage.all import *
>>> show(list_plot3d([[Integer(1), Integer(1), Integer(1), Integer(1)], [Integer(1), Integer(2), Integer(1), Integer(2)], [Integer(1), Integer(1), Integer(3), Integer(1)], [Integer(1), Integer(2), Integer(1), Integer(4)]],
...                  interpolation_type='spline'))
../../../_images/list_plot3d-13.svg
sage.plot.plot3d.list_plot3d.list_plot3d_matrix(m, **kwds)[source]#

A 3-dimensional plot of a surface defined by a matrix M defining points in 3-dimensional space.

See list_plot3d() for full details.

INPUT:

  • M – a matrix

OPTIONAL KEYWORDS:

  • **kwds – all other arguments are passed to the surface function

OUTPUT: a 3d plot

EXAMPLES:

We plot a matrix that illustrates summation modulo \(n\):

sage: n = 5
sage: list_plot3d(matrix(RDF, n, [(i+j) % n         # indirect doctest
....:                             for i in [1..n] for j in [1..n]]))
Graphics3d Object

>>> from sage.all import *
>>> n = Integer(5)
>>> list_plot3d(matrix(RDF, n, [(i+j) % n         # indirect doctest
...                             for i in (ellipsis_range(Integer(1),Ellipsis,n)) for j in (ellipsis_range(Integer(1),Ellipsis,n))]))
Graphics3d Object
../../../_images/list_plot3d-14.svg

The interpolation type for matrices is ‘linear’; for other types use other list_plot3d() input types.

We plot a matrix of values of \(sin\):

sage: from math import pi
sage: m = matrix(RDF, 6, [sin(i^2 + j^2)
....:                     for i in [0,pi/5,..,pi] for j in [0,pi/5,..,pi]])
sage: list_plot3d(m, color='yellow', frame_aspect_ratio=[1, 1, 1/3])  # indirect doctest
Graphics3d Object

>>> from sage.all import *
>>> from math import pi
>>> m = matrix(RDF, Integer(6), [sin(i**Integer(2) + j**Integer(2))
...                     for i in (ellipsis_range(Integer(0),pi/Integer(5),Ellipsis,pi)) for j in (ellipsis_range(Integer(0),pi/Integer(5),Ellipsis,pi))])
>>> list_plot3d(m, color='yellow', frame_aspect_ratio=[Integer(1), Integer(1), Integer(1)/Integer(3)])  # indirect doctest
Graphics3d Object
../../../_images/list_plot3d-15.svg
::

sage: list_plot3d(m, color=’yellow’, interpolation_type=’linear’) # indirect doctest Graphics3d Object

../../../_images/list_plot3d-16.svg

Here is a colored example, using a colormap and a coloring function which must take values in (0, 1):

sage: cm = colormaps.rainbow
sage: n = 20
sage: cf = lambda x, y: ((2*(x-y)/n)**2) % 1
sage: list_plot3d(matrix(RDF, n, [cos(pi*(i+j)/n) for i in [1..n]
....:                             for j in [1..n]]), color=(cf,cm))
Graphics3d Object

>>> from sage.all import *
>>> cm = colormaps.rainbow
>>> n = Integer(20)
>>> cf = lambda x, y: ((Integer(2)*(x-y)/n)**Integer(2)) % Integer(1)
>>> list_plot3d(matrix(RDF, n, [cos(pi*(i+j)/n) for i in (ellipsis_range(Integer(1),Ellipsis,n))
...                             for j in (ellipsis_range(Integer(1),Ellipsis,n))]), color=(cf,cm))
Graphics3d Object
../../../_images/list_plot3d-17.svg
sage.plot.plot3d.list_plot3d.list_plot3d_tuples(v, interpolation_type, **kwds)[source]#

A 3-dimensional plot of a surface defined by the list \(v\) of points in 3-dimensional space.

INPUT:

  • v – something that defines a set of points in 3 space, for example:

    • a matrix

      This will be if using an interpolation_type other than 'linear', or if using num_points with 'linear'; otherwise see list_plot3d_matrix().

    • a list of 3-tuples

    • a list of lists (all of the same length, under same conditions as a matrix)

OPTIONAL KEYWORDS:

  • interpolation_type – one of 'linear', 'clough' (CloughTocher2D), 'spline'

    'linear' will perform linear interpolation

    The option ‘clough’ will interpolate by using a piecewise cubic interpolating Bezier polynomial on each triangle, using a Clough-Tocher scheme. The interpolant is guaranteed to be continuously differentiable.

    The option 'spline' interpolates using a bivariate B-spline.

    When v is a matrix the default is to use linear interpolation, when v is a list of points the default is 'clough'.

  • degree – an integer between 1 and 5, controls the degree of spline used for spline interpolation. For data that is highly oscillatory use higher values

  • point_list – If point_list=True is passed, then if the array is a list of lists of length three, it will be treated as an array of points rather than a \(3\times n\) array.

  • num_points – Number of points to sample interpolating function in each direction. By default for an \(n\times n\) array this is \(n\).

  • **kwds – all other arguments are passed to the surface function

OUTPUT: a 3d plot

EXAMPLES:

All of these use this function; see list_plot3d() for other list plots:

sage: from math import pi
sage: m = matrix(RDF, 6, [sin(i^2 + j^2)
....:                     for i in [0,pi/5,..,pi] for j in [0,pi/5,..,pi]])
sage: list_plot3d(m, color='yellow', interpolation_type='linear',  # indirect doctest
....:             num_points=5)
Graphics3d Object

>>> from sage.all import *
>>> from math import pi
>>> m = matrix(RDF, Integer(6), [sin(i**Integer(2) + j**Integer(2))
...                     for i in (ellipsis_range(Integer(0),pi/Integer(5),Ellipsis,pi)) for j in (ellipsis_range(Integer(0),pi/Integer(5),Ellipsis,pi))])
>>> list_plot3d(m, color='yellow', interpolation_type='linear',  # indirect doctest
...             num_points=Integer(5))
Graphics3d Object
../../../_images/list_plot3d-18.svg
sage: list_plot3d(m, color='yellow', interpolation_type='spline',
....:             frame_aspect_ratio=[1, 1, 1/3])
Graphics3d Object

>>> from sage.all import *
>>> list_plot3d(m, color='yellow', interpolation_type='spline',
...             frame_aspect_ratio=[Integer(1), Integer(1), Integer(1)/Integer(3)])
Graphics3d Object
../../../_images/list_plot3d-19.svg
sage: show(list_plot3d([[1, 1, 1], [1, 2, 1], [0, 1, 3], [1, 0, 4]],
....:                  point_list=True))

>>> from sage.all import *
>>> show(list_plot3d([[Integer(1), Integer(1), Integer(1)], [Integer(1), Integer(2), Integer(1)], [Integer(0), Integer(1), Integer(3)], [Integer(1), Integer(0), Integer(4)]],
...                  point_list=True))
../../../_images/list_plot3d-20.svg
sage: list_plot3d([(1, 2, 3), (0, 1, 3), (2, 1, 4), (1, 0, -2)],  # long time
....:             color='yellow', num_points=50)
Graphics3d Object

>>> from sage.all import *
>>> list_plot3d([(Integer(1), Integer(2), Integer(3)), (Integer(0), Integer(1), Integer(3)), (Integer(2), Integer(1), Integer(4)), (Integer(1), Integer(0), -Integer(2))],  # long time
...             color='yellow', num_points=Integer(50))
Graphics3d Object
../../../_images/list_plot3d-21.svg