List Plots¶

sage.plot.plot3d.list_plot3d.
list_plot3d
(v, interpolation_type='default', texture='automatic', point_list=None, **kwds)¶ A 3dimensional plot of a surface defined by the list \(v\) of points in 3dimensional space.
INPUT:
v
 something that defines a set of points in 3 space, for example: a matrix
 a list of 3tuples
 a list of lists (all of the same length)  this is treated the same as a matrix.
texture
 (default: “automatic”, a solid light blue)
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 CloughTocher 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 Bspline.
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 valuespoint_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, wheninterpolation_type
is notdefault
. 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; list_plot3d(matrix(RDF, n, [(i+j)%n for i in [1..n] for j in [1..n]])) Graphics3d Object
We plot a matrix of values of sin.
sage: pi = float(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, texture='yellow', frame_aspect_ratio=[1, 1, 1/3]) Graphics3d Object
Though it doesn’t change the shape of the graph, increasing num_points can increase the clarity of the graph.
sage: list_plot3d(m, texture='yellow', frame_aspect_ratio=[1, 1, 1/3], num_points=40) Graphics3d Object
We can change the interpolation type.
sage: import warnings sage: warnings.simplefilter('ignore', UserWarning) sage: list_plot3d(m, texture='yellow', interpolation_type='clough', frame_aspect_ratio=[1, 1, 1/3]) Graphics3d Object
We can make this look better by increasing the number of samples.
sage: list_plot3d(m, texture='yellow', interpolation_type='clough', frame_aspect_ratio=[1, 1, 1/3], num_points=40) Graphics3d Object
Let’s try a spline.
sage: list_plot3d(m, texture='yellow', interpolation_type='spline', frame_aspect_ratio=[1, 1, 1/3]) Graphics3d Object
That spline doesn’t capture the oscillation very well; let’s try a higher degree spline.
sage: list_plot3d(m, texture='yellow', interpolation_type='spline', degree=5, frame_aspect_ratio=[1, 1, 1/3]) Graphics3d Object
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]]))
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, texture='yellow') Graphics3d Object
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', texture='yellow', num_points=100) sage: L Graphics3d Object
Check that no NaNs are produced (see trac ticket #13135):
sage: any(math.isnan(c) for v in L.vertices() for c in v) False
sage: list_plot3d([(2, 3, 4)]) Graphics3d Object
sage: list_plot3d([(0, 0, 1), (2, 3, 4)]) Graphics3d Object
However, if two points are given with the same x,y coordinates but different z coordinates, an exception will be raised:
sage: pts =[(4/5, 2/5, 2/5), (4/5, 2/5, 2/5), (4/5, 2/5, 2/5), (4/5, 2/5, 2/5), (2/5, 4/5, 2/5), (2/5, 4/5, 2/5), (2/5, 2/5, 4/5), (2/5, 2/5, 4/5), (2/5, 2/5, 4/5), (2/5, 2/5, 4/5), (2/5, 4/5, 2/5), (2/5, 4/5, 2/5), (2/5, 4/5, 2/5), (2/5, 4/5, 2/5), (2/5, 2/5, 4/5), (2/5, 2/5, 4/5), (2/5, 2/5, 4/5), (2/5, 2/5, 4/5), (2/5, 4/5, 2/5), (2/5, 4/5, 2/5), (4/5, 2/5, 2/5), (4/5, 2/5, 2/5), (4/5, 2/5, 2/5), (4/5, 2/5, 2/5)] sage: show(list_plot3d(pts, interpolation_type='clough')) Traceback (most recent call last): ... ValueError: Two points with same x,y coordinates and different z coordinates were given. Interpolation cannot handle this.
Additionally we need at least 3 points to do the interpolation:
sage: mat = matrix(RDF, 1, 2, [3.2, 1.550]) sage: show(list_plot3d(mat, interpolation_type='clough')) Traceback (most recent call last): ... ValueError: We need at least 3 points to perform the interpolation

sage.plot.plot3d.list_plot3d.
list_plot3d_array_of_arrays
(v, interpolation_type, texture, **kwds)¶ A 3dimensional plot of a surface defined by a list of lists
v
defining points in 3dimensional space. This is done by making the list of lists into a matrix and passing back tolist_plot3d()
. Seelist_plot3d()
for full details.INPUT:
v
 a list of lists, all the same lengthinterpolation_type
 (default: ‘linear’)texture
 (default: “automatic”, a solid light blue)
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
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]]))
With certain extra keywords (see
list_plot3d_matrix()
), this function will end up usinglist_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'))

sage.plot.plot3d.list_plot3d.
list_plot3d_matrix
(m, texture, **kwds)¶ A 3dimensional plot of a surface defined by a matrix
M
defining points in 3dimensional space. Seelist_plot3d()
for full details.INPUT:
M
 a matrixtexture
 (default: “automatic”, a solid light blue)
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; list_plot3d(matrix(RDF, n, [(i+j)%n for i in [1..n] for j in [1..n]])) # indirect doctest Graphics3d Object
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: pi = float(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, texture='yellow', frame_aspect_ratio=[1, 1, 1/3]) # indirect doctest Graphics3d Object sage: list_plot3d(m, texture='yellow', interpolation_type='linear') # indirect doctest Graphics3d Object
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*(xy)/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

sage.plot.plot3d.list_plot3d.
list_plot3d_tuples
(v, interpolation_type, texture, **kwds)¶ A 3dimensional plot of a surface defined by the list \(v\) of points in 3dimensional 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 seelist_plot3d_matrix()
.a list of 3tuples
a list of lists (all of the same length, under same conditions as a matrix)
texture
 (default: “automatic”, a solid light blue)
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 CloughTocher scheme. The interpolant is guaranteed to be continuously differentiable.
The option ‘spline’ interpolates using a bivariate Bspline.
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 valuespoint_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: pi = float(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, texture='yellow', interpolation_type='linear', num_points=5) # indirect doctest Graphics3d Object
sage: list_plot3d(m, texture='yellow', interpolation_type='spline', frame_aspect_ratio=[1, 1, 1/3]) Graphics3d Object
sage: show(list_plot3d([[1, 1, 1], [1, 2, 1], [0, 1, 3], [1, 0, 4]], point_list=True))
sage: list_plot3d([(1, 2, 3), (0, 1, 3), (2, 1, 4), (1, 0, 2)], texture='yellow', num_points=50) # long time Graphics3d Object