A 3-dimensional plot of a surface defined by the list \(v\) of points in 3-dimensional space.
INPUT:
OPTIONAL KEYWORDS:
interpolation_type - ‘linear’, ‘nn’ (nearest neighbor), ‘spline’
‘linear’ will perform linear interpolation
The option ‘nn’ will interpolate by averaging the value of the nearest neighbors, this produces an interpolating function that is smoother than a linear interpolation, it has one derivative everywhere except at the sample points.
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 nearest neighbor.
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; list_plot3d(matrix(RDF,n,[(i+j)%n for i in [1..n] for j in [1..n]]))
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])
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)
We can change the interpolation type.
sage: list_plot3d(m, texture='yellow', interpolation_type='nn',frame_aspect_ratio=[1,1,1/3])
We can make this look better by increasing the number of samples.
sage: list_plot3d(m, texture='yellow', interpolation_type='nn',frame_aspect_ratio=[1,1,1/3],num_points=40)
Let’s try a spline.
sage: list_plot3d(m, texture='yellow', interpolation_type='spline',frame_aspect_ratio=[1,1,1/3])
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])
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')
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: list_plot3d(l,interpolation_type='nn',texture='yellow',num_points=100)
TESTS:
We plot 0, 1, and 2 points:
sage: list_plot3d([])
sage: list_plot3d([(2,3,4)])
sage: list_plot3d([(0,0,1), (2,3,4)])
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='nn'))
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='nn'))
Traceback (most recent call last):
...
ValueError: We need at least 3 points to perform the interpolation
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:
OPTIONAL KEYWORDS:
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 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'))
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:
OPTIONAL KEYWORDS:
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
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
sage: list_plot3d(m, texture='yellow', interpolation_type='linear') # indirect doctest
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)
texture - (default: “automatic”, a solid light blue)
OPTIONAL KEYWORDS:
interpolation_type - ‘linear’, ‘nn’ (nearest neighbor), ‘spline’
‘linear’ will perform linear interpolation
The option ‘nn’ will interpolate by averaging the value of the nearest neighbors, this produces an interpolating function that is smoother than a linear interpolation, it has one derivative everywhere except at the sample points.
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 nearest neighbor.
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: 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
sage: list_plot3d(m, texture='yellow', interpolation_type='spline',frame_aspect_ratio=[1,1,1/3])
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)