# matplotlib.tri#

Unstructured triangular grid functions.

An unstructured triangular grid consisting of npoints points and ntri triangles. The triangles can either be specified by the user or automatically generated using a Delaunay triangulation.

Parameters
x, y(npoints,) array-like

Coordinates of grid points.

triangles(ntri, 3) array-like of int, optional

For each triangle, the indices of the three points that make up the triangle, ordered in an anticlockwise manner. If not specified, the Delaunay triangulation is calculated.

Notes

For a Triangulation to be valid it must not have duplicate points, triangles formed from colinear points, or overlapping triangles.

Attributes
triangles(ntri, 3) array of int

For each triangle, the indices of the three points that make up the triangle, ordered in an anticlockwise manner. If you want to take the mask into account, use get_masked_triangles instead.

is_delaunaybool

Whether the Triangulation is a calculated Delaunay triangulation (where triangles was not specified) or not.

calculate_plane_coefficients(z)[source]#

Calculate plane equation coefficients for all unmasked triangles from the point (x, y) coordinates and specified z-array of shape (npoints). The returned array has shape (npoints, 3) and allows z-value at (x, y) position in triangle tri to be calculated using z = array[tri, 0] * x  + array[tri, 1] * y + array[tri, 2].

property edges#

Return integer array of shape (nedges, 2) containing all edges of non-masked triangles.

Each row defines an edge by it's start point index and end point index. Each edge appears only once, i.e. for an edge between points i and j, there will only be either (i, j) or (j, i).

get_cpp_triangulation()[source]#

Return the underlying C++ Triangulation object, creating it if necessary.

static get_from_args_and_kwargs(*args, **kwargs)[source]#

Return a Triangulation object from the args and kwargs, and the remaining args and kwargs with the consumed values removed.

There are two alternatives: either the first argument is a Triangulation object, in which case it is returned, or the args and kwargs are sufficient to create a new Triangulation to return. In the latter case, see Triangulation.__init__ for the possible args and kwargs.

Return an array of triangles that are not masked.

get_trifinder()[source]#

Return the default matplotlib.tri.TriFinder of this triangulation, creating it if necessary. This allows the same TriFinder object to be easily shared.

property neighbors#

Return integer array of shape (ntri, 3) containing neighbor triangles.

For each triangle, the indices of the three triangles that share the same edges, or -1 if there is no such neighboring triangle. neighbors[i, j] is the triangle that is the neighbor to the edge from point index triangles[i, j] to point index triangles[i, (j+1)%3].

Set or clear the mask array.

Parameters
maskNone or bool array of length ntri
class matplotlib.tri.TriContourSet(ax, *args, **kwargs)[source]#

Create and store a set of contour lines or filled regions for a triangular grid.

This class is typically not instantiated directly by the user but by tricontour and tricontourf.

Attributes
axAxes

The Axes object in which the contours are drawn.

collections

The Artists representing the contour. This is a list of PathCollections for both line and filled contours.

levelsarray

The values of the contour levels.

layersarray

Same as levels for line contours; half-way between levels for filled contours. See ContourSet._process_colors.

Draw triangular grid contour lines or filled regions, depending on whether keyword arg 'filled' is False (default) or True.

The first argument of the initializer must be an Axes object. The remaining arguments and keyword arguments are described in the docstring of tricontour.

class matplotlib.tri.TriFinder(triangulation)[source]#

Abstract base class for classes used to find the triangles of a Triangulation in which (x, y) points lie.

Rather than instantiate an object of a class derived from TriFinder, it is usually better to use the function Triangulation.get_trifinder.

Derived classes implement __call__(x, y) where x and y are array-like point coordinates of the same shape.

class matplotlib.tri.TrapezoidMapTriFinder(triangulation)[source]#

TriFinder class implemented using the trapezoid map algorithm from the book "Computational Geometry, Algorithms and Applications", second edition, by M. de Berg, M. van Kreveld, M. Overmars and O. Schwarzkopf.

The triangulation must be valid, i.e. it must not have duplicate points, triangles formed from colinear points, or overlapping triangles. The algorithm has some tolerance to triangles formed from colinear points, but this should not be relied upon.

class matplotlib.tri.TriInterpolator(triangulation, z, trifinder=None)[source]#

Abstract base class for classes used to interpolate on a triangular grid.

Derived classes implement the following methods:

• __call__(x, y), where x, y are array-like point coordinates of the same shape, and that returns a masked array of the same shape containing the interpolated z-values.

• gradient(x, y), where x, y are array-like point coordinates of the same shape, and that returns a list of 2 masked arrays of the same shape containing the 2 derivatives of the interpolator (derivatives of interpolated z values with respect to x and y).

class matplotlib.tri.LinearTriInterpolator(triangulation, z, trifinder=None)[source]#

Linear interpolator on a triangular grid.

Each triangle is represented by a plane so that an interpolated value at point (x, y) lies on the plane of the triangle containing (x, y). Interpolated values are therefore continuous across the triangulation, but their first derivatives are discontinuous at edges between triangles.

Parameters
triangulationTriangulation

The triangulation to interpolate over.

z(npoints,) array-like

Array of values, defined at grid points, to interpolate between.

trifinderTriFinder, optional

If this is not specified, the Triangulation's default TriFinder will be used by calling Triangulation.get_trifinder.

Methods

 __call__ (x, y) (Returns interpolated values at (x, y) points.) gradient (x, y) (Returns interpolated derivatives at (x, y) points.)

Returns a list of 2 masked arrays containing interpolated derivatives at the specified (x, y) points.

Parameters
x, yarray-like

x and y coordinates of the same shape and any number of dimensions.

Returns
dzdx, dzdynp.ma.array

2 masked arrays of the same shape as x and y; values corresponding to (x, y) points outside of the triangulation are masked out. The first returned array contains the values of $$\frac{\partial z}{\partial x}$$ and the second those of $$\frac{\partial z}{\partial y}$$.

class matplotlib.tri.CubicTriInterpolator(triangulation, z, kind='min_E', trifinder=None, dz=None)[source]#

Cubic interpolator on a triangular grid.

In one-dimension - on a segment - a cubic interpolating function is defined by the values of the function and its derivative at both ends. This is almost the same in 2D inside a triangle, except that the values of the function and its 2 derivatives have to be defined at each triangle node.

The CubicTriInterpolator takes the value of the function at each node - provided by the user - and internally computes the value of the derivatives, resulting in a smooth interpolation. (As a special feature, the user can also impose the value of the derivatives at each node, but this is not supposed to be the common usage.)

Parameters
triangulationTriangulation

The triangulation to interpolate over.

z(npoints,) array-like

Array of values, defined at grid points, to interpolate between.

kind{'min_E', 'geom', 'user'}, optional

Choice of the smoothing algorithm, in order to compute the interpolant derivatives (defaults to 'min_E'):

• if 'min_E': (default) The derivatives at each node is computed to minimize a bending energy.

• if 'geom': The derivatives at each node is computed as a weighted average of relevant triangle normals. To be used for speed optimization (large grids).

• if 'user': The user provides the argument dz, no computation is hence needed.

trifinderTriFinder, optional

If not specified, the Triangulation's default TriFinder will be used by calling Triangulation.get_trifinder.

dztuple of array-likes (dzdx, dzdy), optional

Used only if kind ='user'. In this case dz must be provided as (dzdx, dzdy) where dzdx, dzdy are arrays of the same shape as z and are the interpolant first derivatives at the triangulation points.

Notes

This note is a bit technical and details how the cubic interpolation is computed.

The interpolation is based on a Clough-Tocher subdivision scheme of the triangulation mesh (to make it clearer, each triangle of the grid will be divided in 3 child-triangles, and on each child triangle the interpolated function is a cubic polynomial of the 2 coordinates). This technique originates from FEM (Finite Element Method) analysis; the element used is a reduced Hsieh-Clough-Tocher (HCT) element. Its shape functions are described in [1]. The assembled function is guaranteed to be C1-smooth, i.e. it is continuous and its first derivatives are also continuous (this is easy to show inside the triangles but is also true when crossing the edges).

In the default case (kind ='min_E'), the interpolant minimizes a curvature energy on the functional space generated by the HCT element shape functions - with imposed values but arbitrary derivatives at each node. The minimized functional is the integral of the so-called total curvature (implementation based on an algorithm from [2] - PCG sparse solver):

$E(z) = \frac{1}{2} \int_{\Omega} \left( \left( \frac{\partial^2{z}}{\partial{x}^2} \right)^2 + \left( \frac{\partial^2{z}}{\partial{y}^2} \right)^2 + 2\left( \frac{\partial^2{z}}{\partial{y}\partial{x}} \right)^2 \right) dx\,dy$

If the case kind ='geom' is chosen by the user, a simple geometric approximation is used (weighted average of the triangle normal vectors), which could improve speed on very large grids.

References

1

Michel Bernadou, Kamal Hassan, "Basis functions for general Hsieh-Clough-Tocher triangles, complete or reduced.", International Journal for Numerical Methods in Engineering, 17(5):784 - 789. 2.01.

2

C.T. Kelley, "Iterative Methods for Optimization".

Methods

 __call__ (x, y) (Returns interpolated values at (x, y) points.) gradient (x, y) (Returns interpolated derivatives at (x, y) points.)

Returns a list of 2 masked arrays containing interpolated derivatives at the specified (x, y) points.

Parameters
x, yarray-like

x and y coordinates of the same shape and any number of dimensions.

Returns
dzdx, dzdynp.ma.array

2 masked arrays of the same shape as x and y; values corresponding to (x, y) points outside of the triangulation are masked out. The first returned array contains the values of $$\frac{\partial z}{\partial x}$$ and the second those of $$\frac{\partial z}{\partial y}$$.

class matplotlib.tri.TriRefiner(triangulation)[source]#

Abstract base class for classes implementing mesh refinement.

A TriRefiner encapsulates a Triangulation object and provides tools for mesh refinement and interpolation.

Derived classes must implement:

• refine_triangulation(return_tri_index=False, **kwargs) , where the optional keyword arguments kwargs are defined in each TriRefiner concrete implementation, and which returns:

• a refined triangulation,

• optionally (depending on return_tri_index), for each point of the refined triangulation: the index of the initial triangulation triangle to which it belongs.

• refine_field(z, triinterpolator=None, **kwargs), where:

• z array of field values (to refine) defined at the base triangulation nodes,

• triinterpolator is an optional TriInterpolator,

• the other optional keyword arguments kwargs are defined in each TriRefiner concrete implementation;

and which returns (as a tuple) a refined triangular mesh and the interpolated values of the field at the refined triangulation nodes.

class matplotlib.tri.UniformTriRefiner(triangulation)[source]#

Uniform mesh refinement by recursive subdivisions.

Parameters
triangulationTriangulation

The encapsulated triangulation (to be refined)

refine_field(z, triinterpolator=None, subdiv=3)[source]#

Refine a field defined on the encapsulated triangulation.

Parameters
z(npoints,) array-like

Values of the field to refine, defined at the nodes of the encapsulated triangulation. (n_points is the number of points in the initial triangulation)

triinterpolatorTriInterpolator, optional

Interpolator used for field interpolation. If not specified, a CubicTriInterpolator will be used.

subdivint, default: 3

Recursion level for the subdivision. Each triangle is divided into 4**subdiv child triangles.

Returns
refi_triTriangulation

The returned refined triangulation.

refi_z1D array of length: refi_tri node count.

The returned interpolated field (at refi_tri nodes).

refine_triangulation(return_tri_index=False, subdiv=3)[source]#

Compute an uniformly refined triangulation refi_triangulation of the encapsulated triangulation.

This function refines the encapsulated triangulation by splitting each father triangle into 4 child sub-triangles built on the edges midside nodes, recursing subdiv times. In the end, each triangle is hence divided into 4**subdiv child triangles.

Parameters
return_tri_indexbool, default: False

Whether an index table indicating the father triangle index of each point is returned.

subdivint, default: 3

Recursion level for the subdivision. Each triangle is divided into 4**subdiv child triangles; hence, the default results in 64 refined subtriangles for each triangle of the initial triangulation.

Returns
refi_triangulationTriangulation

The refined triangulation.

found_indexint array

Index of the initial triangulation containing triangle, for each point of refi_triangulation. Returned only if return_tri_index is set to True.

class matplotlib.tri.TriAnalyzer(triangulation)[source]#

Define basic tools for triangular mesh analysis and improvement.

A TriAnalyzer encapsulates a Triangulation object and provides basic tools for mesh analysis and mesh improvement.

Parameters
triangulationTriangulation

The encapsulated triangulation to analyze.

Attributes
scale_factors

Factors to rescale the triangulation into a unit square.

circle_ratios(rescale=True)[source]#

Return a measure of the triangulation triangles flatness.

The ratio of the incircle radius over the circumcircle radius is a widely used indicator of a triangle flatness. It is always <= 0.5 and == 0.5 only for equilateral triangles. Circle ratios below 0.01 denote very flat triangles.

To avoid unduly low values due to a difference of scale between the 2 axis, the triangular mesh can first be rescaled to fit inside a unit square with scale_factors (Only if rescale is True, which is its default value).

Parameters
rescalebool, default: True

If True, internally rescale (based on scale_factors), so that the (unmasked) triangles fit exactly inside a unit square mesh.

Returns

Ratio of the incircle radius over the circumcircle radius, for each 'rescaled' triangle of the encapsulated triangulation. Values corresponding to masked triangles are masked out.

Eliminate excessively flat border triangles from the triangulation.

Returns a mask new_mask which allows to clean the encapsulated triangulation from its border-located flat triangles (according to their circle_ratios()). This mask is meant to be subsequently applied to the triangulation using Triangulation.set_mask. new_mask is an extension of the initial triangulation mask in the sense that an initially masked triangle will remain masked.

The new_mask array is computed recursively; at each step flat triangles are removed only if they share a side with the current mesh border. Thus no new holes in the triangulated domain will be created.

Parameters
min_circle_ratiofloat, default: 0.01

Border triangles with incircle/circumcircle radii ratio r/R will be removed if r/R < min_circle_ratio.

rescalebool, default: True

If True, first, internally rescale (based on scale_factors) so that the (unmasked) triangles fit exactly inside a unit square mesh. This rescaling accounts for the difference of scale which might exist between the 2 axis.

Returns
array of bool

Scaling factors (kx, ky) so that the triangulation [triangulation.x * kx, triangulation.y * ky] fits exactly inside a unit square.